Effective Equations in Mathematical Quantum
Mechanics
Von der Fakulta¨t Mathematik und Physik der Universita¨t
Stuttgart zur Erlangung der Wu¨rde eines
Doktors der Naturwissenschaften (Dr. rer. nat.)
genehmigte Abhandlung
Vorgelegt von
Steffen Gilg
aus Go¨ppingen
Hauptberichter: Prof. Dr. Guido Schneider
Mitberichter: Prof. Dr. Hannes Uecker
Tag der mu¨ndlichen Pru¨fung: 13. Juli 2017
Institut fu¨r Analysis, Dynamik und Modellierung der Universita¨t Stuttgart
2017

Inhaltsverzeichnis
Zusammenfassung v
Abstract vii
Danksagung ix
1. Introduction 1
2. Approximation of a nonlinear Schro¨dinger equation on periodic quantum graphs 5
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1. The periodic quantum graph . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2. The Floquet-Bloch spectrum . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3. The effective amplitude equation . . . . . . . . . . . . . . . . . . . . . . 11
2.2.4. The amplitude equations at the Dirac points . . . . . . . . . . . . . . . . 13
2.3. Local existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4. Bloch transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1. Bloch transform on the real line . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2. The system in Bloch space . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.3. Bloch transform for smooth functions . . . . . . . . . . . . . . . . . . . 18
2.5. Estimates for the residual terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5.1. Derivation of the effective amplitude equation . . . . . . . . . . . . . . . 19
2.5.2. The improved approximation . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.3. From Fourier space to Bloch space . . . . . . . . . . . . . . . . . . . . . 21
2.5.4. Estimates in Bloch space . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6. Estimates for the error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3. Approximation of a cubic Klein-Gordon equation on periodic quantum graphs 31
3.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1. The Floquet-Bloch spectrum . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2. The effective amplitude equation . . . . . . . . . . . . . . . . . . . . . . 34
3.3. Local existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4. Derivation of the NLS approximation . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.1. The system in Bloch space . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.2. Derivation of the effective amplitude equation . . . . . . . . . . . . . . . 38
3.5. The improved approximation and estimates for the residual terms . . . . . . . . 39
3.5.1. The improved approximation . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5.2. From Fourier space to Bloch space . . . . . . . . . . . . . . . . . . . . . 40
iii
3.5.3. Estimates in Bloch space . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6. Estimates for the error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4. Approximation of a two-dimensional Gross-Pitaevskii equation with a periodic po-
tential 45
4.1. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2. The spectral situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1. Wannier function decomposition in one dimension . . . . . . . . . . . . . 47
4.2.2. Properties of the harmonic oscillator . . . . . . . . . . . . . . . . . . . . 50
4.3. Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4. Computation of the residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4.1. Residual of the approximate solution . . . . . . . . . . . . . . . . . . . . 52
4.4.2. The improved approximation . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4.3. Estimates on the error term . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5. Local Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6. Control on the error bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A. Appendices to Chapter 3 67
A.1. Computation of the spectral bands ω(`) . . . . . . . . . . . . . . . . . . . . . . 67
A.2. Calculations for the derivation of the effective amplitude equation . . . . . . . . 69
B. Appendices to Chapter 4 71
B.1. The function space H1,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.2. Computation of the projection Πn,j . . . . . . . . . . . . . . . . . . . . . . . . . 76
Bibliography 77
iv
Zusammenfassung
Um die Dynamik von quantenmechanischen Systemen zu untersuchen, ist es oft sehr nu¨tzlich,
effektive Gleichungen als eine Na¨herung fu¨r das urspru¨ngliche System zu betrachten. Solche re-
duzierten Modelle lassen sich aus Vielteilchensystemen ebenso herleiten wie auch aus schon be-
kannten partiellen Differentialgleichungen. In dieser Arbeit studieren wir physikalische Probleme,
welche durch eine nichtlineare Differentialgleichung beschrieben werden und deren Dynamik mit
Hilfe einer einfacheren effektiven Gleichung approximiert werden soll.
Zuna¨chst betrachten wir eine nichtlineare Schro¨dingergleichung und eine kubische Klein-Gordon
Gleichung auf einem periodischen Quantengraph. Fu¨r die Amplitude eines sich auf dem Graph be-
wegenden Wellenpakets leiten wir in beiden Fa¨llen eine Na¨herungsgleichung her. Diese effektiven
Gleichungen haben ebenfalls die Form einer nichlinearen Schro¨dingergleichung, sind jedoch auf
einem homogenen Raum definiert. Wir rechtfertigen diese Na¨herungen durch den Beweis, dass
sich die Lo¨sungen der effektiven Gleichungen fu¨r lange Zeiten nahe der tatsa¨chlichen Lo¨sungen
der urspru¨nglichen Probleme befinden. Dafu¨r nutzen wir einen Blochwellenansatz und scha¨tzen
den Fehler zwischen beiden Lo¨sungen mit Hilfe eines Gronwall-Arguments ab. Im Falle der ku-
bischen Klein-Gordon Gleichung beno¨tigen wir noch eine zusa¨tzliche Energieabscha¨tzung fu¨r den
Fehlerterm.
Im zweiten Teil der Arbeit konzentrieren wir uns auf eine nichtlineare Schro¨dingergleichung mit
einem zusa¨tzlichen Potential, der sogenannten Gross-Pitaevskii Gleichung. Diese betrachten wir
auf dem zweidimensionalen homogenen Raum mit einem periodischen Potential in x-Richtung
und einem harmonischen Oszillatorpotential in y-Richtung. Die Periodizita¨t wird hier durch eine
unendliche Folge von endlich hohen Potentialwa¨nden eingefu¨hrt. Als Na¨herungsgleichung erhalten
wir eine diskrete nichtlineare Schro¨dingergleichung, deren Lo¨sungen lokalisierte Amplituden in den
einzelnen Potentialto¨pfen darstellen. Wir nutzen einen Ansatz aus Eigenfunktionen der entspre-
chenden linearen Anfangswertprobleme in beiden Raumrichtungen und beweisen einen Approxima-
tionssatz fu¨r das so hergeleitete effektive System. Erneut beno¨tigen wir eine Energieabscha¨tzung,
um den Fehlerterm mit Hilfe des Satzes von Gronwall zu beschra¨nken.
v

Abstract
In order to analyze the dynamics of quantum mechanical systems, it is often very useful to
consider effective equations as an approximation for the original system. Such reduced models
can be derived from many body systems as well as from partial differential equations already
known. In this thesis, we study physical problems described by a nonlinear differential equation
whose dynamics will be approximated by a simpler effective equation.
First we consider a nonlinear Schro¨dinger equation and a cubic Klein-Gordon equation on a
periodic quantum graph. In both cases, we derive an approximation equation for the amplitude of
a wave packet moving on the graph. These effective equations also have the form of a nonlinear
Schro¨dinger equation but on a homogeneous space. We justify these approximations by proving
that the solutions of the effective equations lie close to the true solutions of the original problem
on a long time scale. For that reason, we use a Bloch wave ansatz and estimate the error between
both solutions with the help of a Gronwall argument. In the case of the cubic Klein-Gordon
equation, we need an additional energy estimate for the error term.
In the second part of the thesis, we concentrate on the nonlinear Schro¨dinger equation with an
additional potential, the so-called Gross-Pitaevskii equation. We consider this equation on the two-
dimensional homogeneous space with a periodic potential in x-direction and a harmonic oscillator
potential in y-direction. The periodicity is introduced here by an infinite sequence of potential
walls of finite height. As an approximation equation, we obtain a discrete nonlinear Schro¨dinger
equation whose solutions represent localized amplitudes in the corresponding potential wells. We
use an ansatz built with eigenfunctions of the respective linear initial value problems in both space
directions and prove an approximation theorem for the derived effective system. Once more, we
need an energy estimate to bound the error term with Gronwall‘s theorem.
vii

Danksagung
Mein erster und gro¨ßter Dank gilt meinem Betreuer, Prof. Dr. Guido Schneider. Seine durchgehen-
de Unterstu¨tzung und die Gelegenheit, als Mitglied seiner Arbeitsgruppe zu promovieren, haben
diese Arbeit erst mo¨glich gemacht. Neben den unza¨hligen, sehr produktiven fachlichen Diskussio-
nen, Hinweisen und Anregungen weiß ich auch unsere Gespra¨che, welche u¨ber die mathematischen
Problemstellungen hinaus gingen, sehr zu scha¨tzen. Die daraus resultierenden Ratschla¨ge und Auf-
munterungen haben mir immer weitergeholfen.
Ich mo¨chte mich auch bei Prof. Dr. Hannes Uecker fu¨r die Gastfreundschaft wa¨hrend meines
Aufenthalts an der Universita¨t Oldenburg und die damit verbundenen Diskussionen bedanken, die
diese Arbeit ebenfalls vorangebracht haben.
Ebenso gilt mein Dank dem Graduiertenkolleg 1838
”
Spectral Theory and Dynamics of Quantum
Systems“ fu¨r die finanzielle Unterstu¨tzung und die Mo¨glichkeit, durch die angebotenen Veran-
staltungen und Kooperationen einen Blick u¨ber die Grenzen des eigenen Themas zu erhalten.
Bedanken mo¨chte ich mich auch bei allen meinen aktuellen und ehemaligen Kolleginnen und
Kollegen, welche durch ihre Hilfsbereitschaft und ihren freundschaftlichen Umgang zu einer an-
genehmen Arbeitsatmospha¨re beigetragen haben.
Fu¨r die gemeinsame Zeit innerhalb und außerhalb des universita¨ren Alltags mo¨chte ich mich noch
besonders bei den folgenden Personen bedanken: Roman Bauer, Markus Daub, Ulrich Linden,
Lenon Minorics, Bartosch Ruszkowski, Jochen Schmid, Sebastian Stegmu¨ller, Andreas Wu¨nsch
und Dominik Zimmermann. Auch den Mitgliedern des ISA gebu¨hrt mein ausdru¨cklicher Dank.
Ebenso mo¨chte ich mich noch bei Daniela Maier bedanken, die Teile der Arbeit Korrektur gelesen
hat und deren Aufmerksamkeit kein Fehler entgehen konnte.
Auch nicht vergessen mo¨chte ich Stefanie Siegert und Katja Engstler, die mich mehr als einmal
durch die Untiefen der Bu¨rokratie geleitet haben. Jo¨rg Ho¨rner danke ich fu¨r die Geduld, die er
immer wieder aufbringen musste um mich mit den Eigenheiten meines Computers vertraut zu
machen.
Schließlich mo¨chte ich an dieser Stelle auch meine Eltern Ute und Hartmut Gilg erwa¨hnen, fu¨r
deren vorbehaltlosen Ru¨ckhalt ich sehr dankbar bin. Ohne ihre Unterstu¨tzung, ihren Einsatz und
ihr Vorbild wa¨re das alles nicht mo¨glich gewesen. Auch meiner Schwester Andrea Gilg mo¨chte ich
an dieser Stelle fu¨r ihre Unterstu¨tzung danken.
Nicht zuletzt gilt mein ganz besonderer Dank meiner Frau Verena, die mich mit sehr viel Geduld
und Versta¨ndnis auf diesem Weg begleitet und mir damit vieles erleichtert hat. Nicht nur hat sie
es immer wieder geschafft in den richtigen Momenten die richtigen Worte zu finden, sie war mir in
allen Situationen auch immer eine verla¨ssliche Stu¨tze. Fu¨r all das kann ich mich gar nicht genug
bei ihr bedanken.
ix

1. Introduction
In mathematical physics, it is often necessary to approximate complex physical systems by effective
equations. Such simplified mathematical models often lead to a deeper understanding of the
physical problem and give us solutions we are unable to obtain in the original setting. For
example, in a mean-field approximation of the linear N-body Schro¨dinger equation the effective
dynamics of a Bose gas can be described by a Gross-Pitaevskii equation. We refer to [22] for
more details on this topic.
A different approach to obtain an effective equation is to consider a nonlinear partial differential
equation as original system, which then will be approximated by a simpler nonlinear evolution
problem. In such a situation, it is common to use a so-called multiple scaling expansion to derive
an effective equation. By proving that the solutions of the original system lie close to the used
multiscale ansatz, the validity of the approximation equation can be justified.
A simple application of this technique is used for the approximation of the cubic Klein-Gordon
equation
∂2t u = ∂
2
xu− u− u3, t ∈ R, x ∈ R,
where the ansatz
εΨnls(t, x) = εA(T,X)e
i`0xeiω0t + c.c.
leads to the nonlinear Schro¨dinger equation
2iω0∂TA = (1− c2g)∂2XA− 3|A|2A
as an effective equation describing the amplitude A(T,X) ∈ C of a spatially and temporarily
oscillating wave packet. Here, the small perturbation parameter 0 < ε  1 and the group
velocity cg of the wave packet define the slow time variable T = ε
2t and the rescaled space
variable X = ε(x− cgt). In order to obtain an approximation result of the form
sup
t∈[0,T0/ε2]
‖u(t, x)− εΨnls(t, x)‖B ≤ Cεβ, (1.1)
for β > 1, it is necessary to bound the error term εβR = u− εΨnls such that ‖R‖B = O(1) in a
suitable chosen Banach space B. In [19], a bound of the formal order O(ε3/2) in the space L2(R)
is proved by the use of Gronwall’s theorem.
This thesis is divided into two main parts, where we use the approach introduced above to obtain
effective amplitude equations in two different physical settings. In Chapter 2, we consider a
nonlinear Schro¨dinger equation
i∂tu = −∂2xu− |u|2u, t ∈ R, x ∈ Γ
as an original system acting on periodic quantum graphs Γ. An example for a such a periodic
quantum graph is shown in Figure 1.1.
1
Figure 1.1.: A periodic quantum graph Γ.
For this problem, a nonlinear Schro¨dinger equation
i∂TA = ν1∂
2
XA+ ν2|A|2A (1.2)
with T ∈ R, X ∈ R, ν1, ν2 ∈ R and A(T,X) ∈ C occurs as an universal amplitude equation
for slow modulations in time and space of an oscillating wave packet. Note that the effective
equation (1.2) is now defined on a homogeneous space. Using Bloch wave analysis and adapting
the approach mentioned above to periodic quantum graphs, we justify an approximation result of
the form (1.1). The content of this chapter is already published in [17].
In Chapter 3, we transfer these ideas to the problem of the cubic Klein-Gordon equation as the
original system on a periodic quantum graph Γ and justify a similar approximation theorem, where
the effective amplitude equation is also given by (1.2).
A more detailed view on the topic of quantum graphs is given in the introduction of Chapter 2.
The second part of this thesis is devoted to the Gross-Pitaevskii equation on a two-dimensional
homogeneous space,
i∂tu = −∆u+ V (r)u+ σ|u|2u, t ∈ R+, r ∈ R2, (1.3)
where u(t, r) : R+ × R2 → C and V (r) is given by a periodic sequence of potential wells with a
height of the formal order O(ε−2) in x-direction and a harmonic oscillator potential in y-direction.
An example for such a periodic well potential is shown in Figure 1.2. Thus, the Gross-Pitaevskii
equation can be seen as a nonlinear Schro¨dinger equation with a nonzero potential V (r).
Figure 1.2.: A one-dimensional periodic well potential.
The one-dimensional Gross-Pitaevskii equation with a periodic potential of the formal order
O(ε−2) is discussed in [32]. Here the authors justify that the original system can be approxi-
mated by a system of discrete nonlinear Schro¨dinger equations.
2
In Chapter 4, we transfer the analysis from [32] to our two-dimensional problem and also approx-
imate the original solutions of (1.3) for small values of ε by solutions of infinitely many coupled
discrete nonlinear Schro¨dinger equations
i∂Tam = α(am−1 + am+1) + σβ |am|2 am, (1.4)
where the amplitude functions am(T ) are located in the m-th potential well and evolve in the slow
time T = µt with µ = µ(ε) > 0. In contrast to the one-dimensional problem, higher regularity is
needed to control the nonlinearity of (1.3) in R2. For this reason, we introduce the anisotropic
Sobolev space H1,2(R2) to obtain a similar justification result of the formal order O(µ3/2) as in
[32].
For a more detailed introduction into the problem and a proper definition of the Sobolev space
H1,2(R2), we refer to Section 4.1.
3

2. Approximation of a nonlinear Schro¨dinger
equation on periodic quantum graphs 1
We consider a nonlinear Schro¨dinger (NLS) equation on a spatially extended periodic quantum
graph. With a multiple scaling expansion, an effective amplitude equation can be derived in
order to describe slow modulations in time and space of an oscillating wave packet. Using
Bloch wave analysis and Gronwall’s inequality, we estimate the distance between the macroscopic
approximation which is obtained via the amplitude equation and true solutions of the NLS equation
on the periodic quantum graph. Moreover, we prove an approximation result for the amplitude
equations which occur at the Dirac points of the system.
2.1. Introduction
A quantum graph is a network of bonds (or edges) connected at the vertices. Such systems appear
as models for the description of free electrons in organic molecules, in the study of waveguides,
photonic crystals, or Anderson localization, or as limit on shrinking thin wires [42]. Quantum
graphs are used in mesoscopic physics to obtain a theoretical understanding of nanotechnological
objects such as nanotubes or graphen, cf. [18, 20, 21]. A recent monograph [9] gives a good
introduction to the mathematics and physics of quantum graphs.
In the linear theory, partial differential equations (PDEs) are defined on the quantum graph
according to the following two ingredients. First, a differential operator acts on functions defined
on the bonds. Second, certain boundary conditions are applied to the functions at the vertices.
In particular, continuity of functions and conservation of flows through the vertices are expressed
by the so called Kirchhoff boundary conditions.
Here we are interested in nonlinear PDEs posed on an infinitely extended periodic chain of identical
quantum graphs. Nonlinear PDEs on quantum graphs have been only considered recently [26]
mostly in the context of unbounded graphs with finitely many vertices. Variational results on
existence of ground states on such unbounded graphs were obtained in a series of papers [2, 3, 4, 5].
It is the purpose of this chapter to derive and justify an effective amplitude equation for the
description of slow modulations in time and space of an oscillating wave packet. As a PDE toy
model on the periodic quantum graph, we consider a nonlinear Schro¨dinger (NLS) equation. The
effective amplitude equation also has the form of a NLS equation but on a homogeneous space. In
what follows, we refer to these two NLS equations as to the original system and to the amplitude
equation.
Hence, we consider the following NLS equation on the periodic quantum graph as the original
system,
i∂tu+ ∂
2
xu+ |u|2u = 0, t ∈ R, x ∈ Γ, (2.1)
1This chapter is a slightly modified version of the published article [17]. The contribution of the author to this
article is included in all main parts of the analysis.
5
where Γ is the quantum graph and u : R × Γ → C. The Kirchhoff boundary conditions at the
vertices are defined below in (2.2)-(2.3).
In order to explain our approach without too many technical details, we develop our subsequent
presentation to one special quantum graph shown in Figure 2.1. However, our approach can be
extended to other quantum graphs, as discussed in Section 2.7.
Figure 2.1.: The basic cell Γ0 (left) of the periodic quantum graph Γ (right).
The spectral problems associated with the linear Schro¨dinger operator on the periodic quantum
graph of Figure 2.1 and its modifications have been recently studied in the literature [20, 21, 25].
Our work is different in the sense that we are studying the time evolution (Cauchy) problem for the
nonlinear version of the Schro¨dinger equation associated with localized initial data. In the recent
work [33], the authors have studied the stationary NLS equation on the periodic quantum graph
Γ and constructed two families of localized bound states by reducing the differential equations to
the discrete maps.
The problem of localization in the periodic setting has been a fascinating topic of research with
several effective amplitude equations appearing in this context [29]. In particular, tight-binding
approximation [1, 32, 34] and coupled-mode approximation [39, 31, 13] were derived and justified
in the limit of large and small periodic potentials respectively. We are addressing here the envelope
approximation, which is the most universal approximation of modulated wave packets in nonlinear
dispersive PDEs [19]. The envelope approximation provides a homogenization of the NLS equation
(2.1) on the periodic quantum graph Γ with an effective homogeneous NLS equation derived for
a given wave packet.
Justification of the homogeneous NLS equation in the context of nonlinear Klein-Gordon equations
with smooth spatially periodic coefficients has been carried out in the work [10]. A modified
analytical approach with a similar result was developed in Section 2.3.1 in [29] in the context of
the Gross-Pitaevskii equation with a smooth periodic potential. Since the periodic quantum graph
introduces singularities in the effective potential (by means of the Kirchhoff boundary conditions),
it is an open question to be inspected here if the analytical techniques from [10, 29] can be made
applicable to the NLS equation (2.1) on the periodic quantum graph Γ. The answer to this
question turns out to be positive. With the same technique involving Bloch wave analysis and
Gronwall’s inequality, we prove estimates on the distance between the macroscopic approximation
via the amplitude equation and the true solutions of the original system. Moreover, we explain
that the same technique can also be used to prove an approximation result for the amplitude
equations which occur at the Dirac points associated with the periodic graph Γ. The amplitude
equations at the Dirac points take the form of the coupled-mode (Dirac) system.
The chapter is organized as follows. The main results are described in Section 2.2, after intro-
ducing the spectral problem associated with the periodic quantum graph on Figure 2.1 . Local
existence and uniqueness of solutions of the Cauchy problem for the NLS equation (2.1) is dis-
cussed in Section 2.3. The Bloch transform is introduced and studied in Section 2.4. In Section
2.5, we derive the effective amplitude equation, construct an improved approximation, and esti-
mate the residual for this improved approximation. The justification of the amplitude equation is
6
developed in Section 2.6. Discussion of other periodic quantum graphs is given in the concluding
Section 2.7.
Notation: We denote with Hs(R) the Sobolev space of s-times weakly differentiable functions on
the real line whose derivatives up to order s are in L2(R). The norm ‖u‖Hs for u in the Sobolev
space Hs(R) is equivalent to the norm ‖(I−∂2x)s/2u‖L2 in the Lebesgue space L2(R). Throughout
this chapter, many different constants are denoted by C if they can be chosen independently of
the small parameter 0 < ε 1.
2.2. Main result
2.2.1. The periodic quantum graph
The periodic quantum graph Γ shown on Figure 2.1 can be expressed as
Γ = ⊕n∈ZΓn, with Γn = Γn,0 ⊕ Γn,+ ⊕ Γn,−,
where Γn,0 represents the horizontal link of length pi between the circles and Γn,± represent the
upper and lower semicircles of the same length pi, for n ∈ Z. In what follows, Γn,0 is identified
isometrically with the interval In,0 = [2pin, 2pin + pi] and Γn,± are identified with the intervals
In,± = [2pin+ pi, 2pi(n+ 1)]. For a function u : Γ→ C, we denote the part on the interval In,0
associated to Γn,0 with un,0 and the parts on the intervals In,± associated to Γn,± with un,±.
The second-order differential operator ∂2x appearing on the right-hand side of the NLS equation
(2.1) is defined under certain boundary conditions at the vertex points {x = npi : n ∈ Z}. We
use so called Kirchhoff boundary conditions, which are given by the continuity of the functions at
the vertices {
un,0(t, 2pin+ pi) = un,+(t, 2pin+ pi) = un,−(t, 2pin+ pi),
un+1,0(t, 2pi(n+ 1)) = un,+(t, 2pi(n+ 1)) = un,−(t, 2pi(n+ 1)),
(2.2)
and the continuity of the fluxes at the vertices{
∂xun,0(t, 2pin+ pi) = ∂xun,+(t, 2pin+ pi) + ∂xun,−(t, 2pin+ pi),
∂xun+1,0(t, 2pi(n+ 1)) = ∂xun,+(t, 2pi(n+ 1)) + ∂xun,−(t, 2pi(n+ 1)).
(2.3)
Remark 2.2.1. The symmetry constraint un,+(t, x) = un,−(t, x) is an invariant reduction of the
NLS equation (2.1) provided the initial data of the corresponding Cauchy problem satisfies the
same reduction. In the case of symmetry reduction, the boundary conditions (2.2) and (2.3) can
be simplified as follows:{
un,0(t, 2pin+ pi) = un,+(t, 2pin+ pi),
un+1,0(t, 2pi(n+ 1)) = un,+(t, 2pi(n+ 1))
(2.4)
and {
∂xun,0(t, 2pin+ pi) = 2∂xun,+(t, 2pin+ pi),
∂xun+1,0(t, 2pi(n+ 1)) = 2∂xun,+(t, 2pi(n+ 1)).
(2.5)
In this way, the NLS equation (2.1) on the periodic graph Γ becomes equivalent to the NLS
equation with a singular periodic potential.
7
The scalar PDE problem on the periodic quantum graph Γ is transferred to a vector-valued PDE
problem on the real axis by introducing the functions
u0(x) =
{
un,0(x), x ∈ In,0,
0, x ∈ In,±, n ∈ Z, (2.6)
and
u±(x) =
{
un,±(x), x ∈ In,±,
0, x ∈ In,0, n ∈ Z. (2.7)
We introduce sets I0 and I± by
I0 =
⋃
n∈Z
In,0 = supp(u0) and I± =
⋃
n∈Z
In,± = supp(u±).
We collect the functions u0 and u± in the vector U = (u0, u+, u−) and rewrite the evolution
problem (2.1) as
i∂tU + ∂
2
xU + |U |2U = 0, t ∈ R, x ∈ R \ {kpi : k ∈ Z}, (2.8)
subject to the conditions (2.2)-(2.3) at the vertex points x ∈ {kpi : k ∈ Z}, where the cubic
nonlinear term stands for the vector |U |2U = (|u0|2u0, |u+|2u+, |u−|2u−).
2.2.2. The Floquet-Bloch spectrum
The spectral problem
ωW = −∂2xW, x ∈ R \ {kpi : k ∈ Z}, (2.9)
is obtained by inserting U(t, x) = W (x)e−iωt into the linearization associated to the NLS equation
(2.8). The components of W = (w0, w+, w−) satisfy the conditions (2.2)-(2.3) and have their
supports in (I0, I+, I−). The eigenfunctions W can be represented in the form of the so-called
Bloch waves
W (x) = ei`xf(`, x), `, x ∈ R, (2.10)
where f(`, ·) = (f0, f+, f−)(`, ·) is a 2pi-periodic function for every ` ∈ R. Since these functions
satisfy the continuation conditions
f(`, x) = f(`, x+ 2pi), f(`, x) = f(`+ 1, x)eix, `, x ∈ R, (2.11)
we can restrict the definition of f(`, x) to x ∈ T2pi = R/(2piZ) and ` ∈ T1 = R/Z. The torus
T2pi is isometrically parameterized with x ∈ [0, 2pi] and the torus T1 with ` ∈ [−1/2, 1/2], where
the endpoints of the intervals are identified to be the same for both tori.
Hence, f can be found as solution of the eigenvalue problem
−(∂x + i`)2f = ω(`)f, x ∈ T2pi, (2.12)
subject to the boundary conditions{
f0(`, pi) = f+(`, pi) = f−(`, pi),
f0(`, 0) = f+(`, 2pi) = f−(`, 2pi)
(2.13)
8
and {
(∂x + i`)f0(`, pi) = (∂x + i`)f+(`, pi) + (∂x + i`)f−(`, pi),
(∂x + i`)f0(`, 0) = (∂x + i`)f+(`, 2pi) + (∂x + i`)f−(`, 2pi).
(2.14)
The functions f0(`, ·) and f±(`, ·) have supports in I0,0 = [0, pi] ⊂ T2pi and I0,± = [pi, 2pi] ⊂ T2pi.
The boundary conditions (2.13)-(2.14) are derived from (2.2)-(2.3) by using the 2pi-periodicity
of the eigenfunction f(`, ·). Note that ei·xf(·, x) and ω(·) are 1-periodic functions on T1. The
extended variable U = (u0, u+, u−) is needed to give a meaning to ei`x which is defined for x ∈ R,
but not for x ∈ Γ.
The spectrum of the spectral problem (2.9) consists of two parts [20, 21, 33]. One part is
represented by the sequence of eigenvalues at {m2}m∈N of infinite multiplicity. For a fixed
m ∈ N, a bi-infinite sequence of eigenfunctions (Wm,k)k∈Z of the spectral problem (2.9) exists
and is supported compactly in each circle with the explicit representation:
wm,kn,0 (x) = 0, w
m,k
n,+(x) = −wm,kn,−(x) = δnk sin(m(x− 2pik)), n ∈ Z. (2.15)
The second part in the spectrum of the spectral problem (2.9) is represented by the union of a
countable set of spectral bands, which correspond to the real roots ρ1,2 of the transcendental
equation ρ2 − tr(M)(ω)ρ+ 1 = 0. Here
tr(M)(ω) :=
1
4
[
9 cos(2pi
√
ω)− 1]
is the trace of the monodromy matrix M associated with the linear difference equation obtained
after solving the differential equation (2.9) subject to the conditions (2.2)-(2.3), cf. [14, 33]. Real
roots are obtained when tr(M)(ω) ∈ [−2, 2].
The corresponding eigenfunctions of the spectral problem (2.9) are distributed over the entire
periodic graph Γ and satisfy the symmetry constraints wn,+(x) = wn,−(x), n ∈ Z and the
constrained boundary conditions (2.4)-(2.5).
The spectral bands of the periodic eigenvalue problem (2.12) are shown on Figure 2.2. The flat
bands at ω = m2, m ∈ N correspond to the eigenvalues of the spectral problem (2.9) of infinite
algebraic multiplicity. It is clear from the explicit representation (2.15) that the corresponding
eigenfunctions can also be written in the Bloch wave form (2.10) associated with the Bloch wave
number ` ∈ T1.
Let us confirm the spectral properties suggested by Figure 2.2. First, eigenvalues of infinite
multiplicity at ω = m2, m ∈ N, are at the end points of the spectral bands, because tr(M)(m2) =
2. Second, since
d
dω
tr(M)(ω)|ω=m2 = −
9pi
4
√
ω
sin(2pi
√
ω)|ω=m2 = 0,
the two adjacent spectral bands of σ(−∂2x) overlap at ω = m2 without a spectral gap. Coinciden-
tally, these so-called Dirac points of the dispersion relation happen to occur at the eigenvalues of
infinite multiplicities. Finally, the two adjacent spectral bands at tr(M)(ω) = −2 do not overlap
and the spectral band has a nonzero length because tr(M)(ω) has a minimum at ω = m
2
4 with
m ∈ Nodd and tr(M)
(
m2
4
)
= −52 < −2.
Let us now define the L2-based spaces, where the eigenfunctions of the periodic eigenvalue
problem (2.12) are properly defined. For fixed ` ∈ T1, we define
L2Γ := { U˜ = (u˜0, u˜+, u˜−) ∈ (L2(T2pi))3 : supp(u˜j) = I0,j , j ∈ {0,+,−}} (2.16)
9
Figure 2.2.: The spectral bands ω of the spectral problem (2.12) plotted versus the Bloch wave
number ` for the periodic quantum graph Γ.
and
H2Γ(`) := {U˜ ∈ L2Γ : u˜j ∈ H2(I0,j), j ∈ {0,+,−}, (2.13)− (2.14) are satisfied},
equipped with the norm
‖U˜‖H2Γ(`) =
(
‖u˜0‖2H2(I0,0) + ‖u˜+‖2H2(I0,+) + ‖u˜−‖2H2(I0,−)
)1/2
.
The parameter ` is defined in H2Γ(`) by means of the boundary conditions (2.13)-(2.14). We
obtain the following elementary result.
Lemma 2.2.2. For fixed ` ∈ T1, the operator L˜(`) := −(∂x + i`)2 is a self-adjoint, positive
semi-definite operator in L2Γ.
Proof. Using the conditions (2.13)-(2.14), we find for every f(`, ·), g(`, ·) ∈ H2Γ(`) and every
` ∈ T1:
〈L˜(`)f, g〉L2Γ =
∫ 2pi
0
(∂x + i`)f(`, x) · (∂x + i`)g(`, x)dx
− [∂xf0(`, pi) + i`f0(`, pi)] g0(`, pi) + [∂xf0(`, 0) + i`f0(`, 0)] g0(`, 0)
− [∂xf+(`, 2pi) + i`f+(`, 2pi)] g+(`, 2pi) + [∂xf+(`, pi) + i`f+(`, pi)] g+(`, pi)
− [∂xf−(`, 2pi) + i`f−(`, 2pi)] g−(`, 2pi) + [∂xf−(`, pi) + i`f−(`, pi)] g−(`, pi)
=
∫ 2pi
0
(∂x + i`)f(`, x) · (∂x + i`)g(`, x)dx.
Using another integration by parts with the conditions (2.13)-(2.14), we confirm that
〈L˜(`)f, g〉L2Γ = 〈f, L˜(`)g〉L2Γ .
Hence, L˜(`) is self-adjoint for every ` ∈ T1. Since
〈L˜(`)f, f〉L2Γ =
∫ 2pi
0
(∂x + i`)f · (∂x + i`)fdx ≥ 0,
10
the operator L˜(`) is positive semi-definite.
By Lemma 2.2.2 and the spectral theorem for self-adjoint operators with compact resolvent,
cf. [36], for each ` ∈ T1 there exists a Schauder base {f (m)(`, ·)}m∈N of L2Γ consisting of
eigenfunctions of L˜(`) with positive eigenvalues {ω(m)(`)}m∈N ordered as ω(m)(`) ≤ ω(m+1)(`).
By construction, the Bloch wave functions satisfy the continuation properties (2.11). They also
satisfy the orthogonality and normalization relations:
〈f (m)(`, ·), f (m′)(`, ·)〉L2Γ = δm,m′ , ` ∈ T1.
Note that we use superscripts for the count of the spectral bands, because the subscripts in
f
(m)
j (`, x), j ∈ {0,+,−} are reserved to indicate the component of f (m)(`, x) for x ∈ I0,j .
2.2.3. The effective amplitude equation
Slow modulations in time and space of a small-amplitude modulated Bloch mode are described
by the formal asymptotic expansion
U(t, x) = εΨnls(t, x) + higher-order terms, (2.17)
with
εΨnls(t, x) = εA(T,X)f
(m0)(`0, x)e
i`0xe−iω
(m0)(`0)t, (2.18)
where 0 < ε 1 is a small perturbation parameter, T = ε2t, X = ε(x− cgt), and A(T,X) ∈ C
is the wave amplitude. The parameter cg := ∂`ω
(m0)(`0) is referred to as the group velocity
associated with the Bloch wave and it corresponds to the velocity of the wave packet propagation.
The group velocity is different from the phase velocity cp := ω
(m0)(`0)/`0, which characterizes
movement of the carrier wave inside the wave packet. Figure 2.3 shows the characteristic scales
of the wave packet given by the asymptotic expansion (2.17) with (2.18).
O(1/ε)
O(ε)
-
cp
-
cg
Figure 2.3.: A schematic representation of the asymptotic solution (2.17)-(2.18) to the NLS equa-
tion (2.1) on the periodic quantum graph Γ. The envelope advances with the group
velocity cg and the underlying carrier wave advances with the phase velocity cp.
11
Formal asymptotic expansions show that at the lowest order in ε, the wave amplitude A satisfies
the following cubic NLS equation on the homogeneous space:
i∂TA− 1
2
∂2`ω
(m0)(`0)∂
2
XA+ ν|A|2A = 0, (2.19)
where the cubic coefficient is given by
ν =
‖f (m0)(`0, ·)‖4L4Γ
‖f (m0)(`0, ·)‖2L2Γ
.
Mathematical justification of the effective amplitude equation (2.19) by means of the error esti-
mates for the original system (2.8) is the main purpose of this work. The approximation result is
given by the following theorem.
Theorem 2.2.3. Pick m0 ∈ N and `0 ∈ T1 such that the following non-resonance condition is
satisfied:
ω(m)(`0) 6= ω(m0)(`0), for every m 6= m0. (2.20)
Then, for every C0 > 0 and T0 > 0, there exist ε0 > 0 and C > 0 such that for all solutions
A ∈ C(R, H3(R)) of the effective amplitude equation (2.19) with
sup
T∈[0,T0]
‖A(T, ·)‖H3 ≤ C0
and for all ε ∈ (0, ε0), there are solutions U ∈ C([0, T0/ε2], L∞(R)) of the original system (2.8)
satisfying the bound
sup
t∈[0,T0/ε2]
sup
x∈R
|U(t, x)− εΨnls(t, x)| ≤ Cε3/2, (2.21)
where εΨnls is given by (2.18).
Remark 2.2.4. Thanks to the global well-posedness and integrability of the cubic NLS equation
(2.19) in one space dimension [11, 40], a global solution A ∈ C(R, Hs(R)) for every integer s ≥ 0
exists and satisfies the bound
sup
T∈[0,T0]
‖A(T, ·)‖Hs ≤ C
for every T0 > 0, where C is T0-independent.
Remark 2.2.5. As it follows from the spectral bands shown on Figure 2.2, it is clear that the
non-resonance assumption (2.20) is satisfied for every m0 ∈ N and `0 6= 0 and it fails for every
m0 ∈ N and `0 = 0 with the exception of the lowest spectral band.
Remark 2.2.6. The approximation result of Theorem 2.2.3 should not be taken for granted.
There exists a number of counterexamples [37, 38], where a formally correctly derived amplitude
equation makes wrong predictions about the dynamics of the original system.
Remark 2.2.7. The new difficulty in the proof of Theorem 2.2.3 on the periodic quantum graph
Γ comes from the vertex conditions (2.2)-(2.3), which have to be incorporated into the functional
analytic set-up from [10, 29] used for the derivation of the amplitude equation (2.19). Since the
NLS equation (2.1) only contains cubic nonlinearities, the proof of Theorem 2.2.3 does not require
near-identity transformations and is based on a simple application of Gronwall’s inequality.
12
2.2.4. The amplitude equations at the Dirac points
Near Dirac points, which correspond to m0 ∈ N and `0 = 0 on Figure 2.2 with the exception of
the lowest spectral band, see Remark 2.2.5, the cubic NLS equation (2.19) cannot be justified.
However, we can find a coupled-mode (Dirac) system, as it is done for smooth periodic potentials
(see Section 2.2.1 in [29]). Eigenvalues of infinite multiplicities appearing as the flat bands in
Figure 2.2 represent an obstacle in the standard justification analysis.
To overcome the obstacle, we can consider solutions of the original system (2.8) which satisfy the
symmetry constraint un,+(t, x) = un,−(t, x), see Remark 2.2.1. In this way, all flat bands shown
on Figure 2.2 disappear as they violate the symmetry constraint.
Figure 2.4 shows the spectral bands of the spectral problem (2.12) under the symmetry constraint
un,+ = un,−. The flat bands are removed due to the symmetry constraints. Near the Dirac points,
we can now justify the coupled-mode (Dirac) system by using the analysis developed in the proof
of Theorem 2.2.3.
Figure 2.4.: The spectral bands ω of the spectral problem (2.12) plotted versus the Bloch wave
number ` for the periodic quantum graph Γ under the symmetry constraint un,+ =
un,−. The intersection points of the spectral curves at ` = 0 are called Dirac points.
To be specific, we consider an intersection point of the two spectral bands at ` = 0, as per Figure
2.4, such that ω(2m0)(0) = ω(2m0+1)(0) for some fixed m0 ∈ N. We relabel these two bands, and
introduce
ω+(`) =
{
ω(2m0)(`), ` ≤ 0,
ω(2m0+1)(`), ` > 0,
(2.22)
and
ω−(`) =
{
ω(2m0+1)(`), ` ≤ 0,
ω(2m0)(`), ` > 0.
(2.23)
We denote the associated eigenfunctions with f+(`, x) and f−(`, x). In order to derive the Dirac
system we make the ansatz
εΨdirac(t, x) = εA+(T,X)f
+(0, x)e−iω
+(0)t + εA−(T,X)f−(0, x)e−iω
−(0)t, (2.24)
13
where T = ε2t, X = ε2x, and A±(T,X) ∈ C. Formal asymptotic expansions show that at the
lowest order in ε, the wave amplitudes A± satisfy the cubic Dirac system on the homogeneous
space:
i∂TA+ + i∂`ω
+(0)∂XA+ +
∑
j1,j2,j3∈{+,−}
ν+j1j2j3Aj1Aj2Aj3 = 0, (2.25)
i∂TA− + i∂`ω−(0)∂XA− +
∑
j1,j2,j3∈{+,−}
ν−j1j2j3Aj1Aj2Aj3 = 0, (2.26)
where the coefficients ν±j1j2j3 ∈ C are given by
νjj1,j2,j3 =
〈f j1(0, ·)f j2(0, ·)f j3(0, ·), f j(0, ·)〉L2Γ
‖f j(0, ·)‖2
L2Γ
, j, j1, j2, j3 ∈ {+,−}.
The system (2.25)-(2.26) is invariant under the transformation (X,A+, A−) 7→ (−X,A−, A+).
The Cauchy problem is locally well-posed in Sobolev spaces. Depending on the nonlinear terms,
it is also globally well-posed in Sobolev spaces [30]. Assuming existence of a global solution to
the cubic Dirac system (2.25)-(2.26), the approximation result is given by the following theorem.
Theorem 2.2.8. For every C0 > 0 and T0 > 0, there exist ε0 > 0 and C > 0 such that for all
solutions A± ∈ C(R, H2(R)) of the Dirac-system (2.25)-(2.26) with
sup
T∈[0,T0]
‖A±(T, ·)‖H2 ≤ C0
and for all ε ∈ (0, ε0), there are solutions U ∈ C([0, T0/ε2], L∞(R)) of the original system (2.8)
satisfying the bound
sup
t∈[0,T0/ε2]
sup
x∈R
|U(t, x)− εΨdirac(t, x)| ≤ Cε3/2.
where εΨdirac is given by (2.24).
The proof of Theorem 2.2.8 is a straightforward modification of the proof of Theorem 2.2.3, cf.
Remark 2.6.1.
2.3. Local existence and uniqueness
Here we prove the local existence and uniqueness of solutions to the original system (2.8). We
consider the operator L = −∂2x in the space
L2 = {U = (u0, u+, u−) ∈ (L2(R))3 : supp(un,j) = In,j , n ∈ Z, j ∈ {0,+,−}}
with the domain of definition
H2 := {U ∈ L2 : un,j ∈ H2(In,j), n ∈ Z, j ∈ {0,+,−}, (2.2)− (2.3) are satisfied},
equipped with the norm
‖U‖H2 :=
(∑
n∈Z
‖un,0‖2H2(In,0) + ‖un,+‖2H2(In,+) + ‖un,−‖2H2(In,−)
)1/2
.
For the local existence and uniqueness of solutions to system (2.8), we need the following results.
14
Lemma 2.3.1. The space H2 is closed under pointwise multiplication.
Proof. For each open interval In,j for n ∈ Z, j ∈ {0,+,−}, the Sobolev space H2(In,j) is closed
under pointwise multiplication. Therefore, there is a positive constant C such that for every
u, v ∈ H2, we have
‖un,jvn,j‖H2(In,j) ≤ C‖un,j‖H2(In,j)‖vn,j‖H2(In,j).
If U and V are continuous at the vertices, then UV is also continuous at the vertices. If U and
V satisfy the flux continuity conditions (2.3), then by the product rule for continuous functions
U and V , the product UV also satisfies the flux continuity conditions (2.3). The support for U ,
V , and UV is identical. Finally, by the Cauchy-Schwarz inequality, we have
‖UV ‖2H2 =
∑
n∈Z,j∈{0,+,−}
‖un,jvn,j‖2H2(In,j)
≤ C2
∑
n∈Z,j∈{0,+,−}
‖un,j‖2H2(In,j)‖vn,j‖2H2(In,j)
≤ C2‖U‖2H2‖V ‖2H2 .
The statement of the lemma is proved.
Lemma 2.3.2. The operator L with the domain H2 is self-adjoint and positive semi-definite in
L2.
Proof. Using the Kirchhoff boundary conditions (2.2)-(2.3), it is an easy exercise to show that
〈U,LV 〉L2 = 〈LU, V 〉L2
is true for every U, V ∈ H2. Then, the operator L with the domain H2 is self-adjoint (similar to
Theorem 1.4.4 in [9]). Positivity and semi-definiteness of L follows from the integration by parts
〈U,LU〉L2 =
∑
n∈Z,j∈{0,±}
‖∂xun,j‖2L2(In,j) ≥ 0,
where the Kirchhoff boundary conditions (2.2)-(2.3) have been used again.
As a consequence of classical semigroup theory, cf. [28], we have
Corollary 2.3.3. The skew symmetric operator −iL with the domain H2 defines a unitary group
(e−iLt)t∈R in L2 such that ‖e−iLtU‖L2 = ‖U‖L2 for every t ∈ R.
By Corollary 2.3.3, we obtain another ingredient of the existence and uniqueness theory.
Lemma 2.3.4. There exists a positive constant CL such that
‖e−iLtU‖H2 ≤ CL‖U‖H2 (2.27)
for every U ∈ H2 and every t ∈ R.
15
Proof. We obtain the following chain of inequalities:
‖e−iLtU‖H2 ≤ C‖(1 + L)e−iLtU‖L2
≤ C‖e−iLt(1 + L)U‖L2
≤ C‖(1 + L)U‖L2
≤ C‖U‖H2 ,
where we have used the equivalence between ‖U‖H2 and ‖(1 +L)U‖L2 , the commutativity of L
and e−iLt, and the existence of the unitary group in Corollary 2.3.3.
We are now ready to prove the local existence and uniqueness of solutions of the Cauchy problem
associated with the original system (2.8) in H2.
Theorem 2.3.5. For every U0 ∈ H2, there exists a T0 = T0(‖U0‖H2) > 0 and a unique solution
U ∈ C([−T0, T0],H2) of the original system (2.8) with the initial data U |t=0 = U0.
Proof. The estimates from Lemma 2.3.1 and Lemma 2.3.4 allow us to proceed with the general
theory for semilinear dynamical systems [28]. Namely, by Duhamel’s principle, we rewrite the
Cauchy problem associated with the original system (2.8) as the integral equation
U(t, ·) = e−iLtU(0, ·) + i
∫ t
0
e−iL(t−τ)|U(τ, ·)|2U(τ, ·)dτ, (2.28)
where the solution is considered in the space
M := {U ∈ C([−T0, T0],H2) : sup
t∈[−T0,T0]
‖ U(t, ·)‖H2 ≤ 2CL‖U(0, ·)‖H2},
and the constant CL is defined by the bound (2.27) in Lemma 2.3.4. For every U0 ∈ H2,
there is a sufficiently small T0 = T0(‖U0‖H2) > 0 such that the right-hand side of the integral
equation (2.28) is a contraction in the space M. Therefore, the existence of a unique solution
U ∈ C([−T0, T0],H2) follows from Banach’s fixed-point theorem.
2.4. Bloch transform
The justification of the NLS approximation in the context of nonlinear Klein-Gordon equations
with smooth spatially periodic coefficients in [10] or in the context of the Gross-Pitaevskii equation
with a smooth periodic potential in [29] heavily relies on the use of the Bloch transform. In order
to transfer the evolution problem (2.8) to Bloch space, we first recall the fundamental properties
of Bloch transform on the real line. Next, we generalize Bloch transform to periodic quantum
graphs, first in L2 and then for smooth functions. In Section 2.7, we explain how to generalize
our approach developed for the periodic graph sketched in Figure 2.1 to other periodic graphs.
General Floquet-Bloch theory for spectral problems posed on periodic quantum graphs is reviewed
in [9, Chapter 4]. However, as far as we can see, the approach of [9, Chapter 4] does not allow us
to transfer the proof of [10] and [29] to the periodic quantum graphs. In what follows, we explain
the necessary modifications of the Bloch transform for the periodic quantum graphs.
16
2.4.1. Bloch transform on the real line
Bloch transform T generalizes Fourier transform F from spatially homogeneous problems to
spatially periodic problems. It was introduced by Gelfand [16] and it appears for instance in the
handling of the Schro¨dinger operator with a spatially periodic potential [36]. Bloch transform is
(formally) defined by
u˜(`, x) = (T u)(`, x) =
∑
n∈Z
u(x+ 2pin)e−i`x−2piin`. (2.29)
The inverse of Bloch transform is given by
u(x) = (T −1u˜)(x) =
∫ 1/2
−1/2
ei`xu˜(`, x)d`.
By construction, u˜(`, x) is extended from (`, x) ∈ T1 × T2pi to (`, x) ∈ R × R according to the
continuation conditions:
u˜(`, x) = u˜(`, x+ 2pi) and u˜(`, x) = u˜(`+ 1, x)eix. (2.30)
The following lemma specifies the well-known property of Bloch transform acting on Sobolev
function spaces, cf. [15, 29].
Lemma 2.4.1. Bloch transform T is an isomorphism between
Hs(R) and L2(T1, Hs(T2pi)),
where L2(T1, Hs(T2pi)) is equipped with the norm
‖u˜‖L2(T1,Hs(T2pi)) =
(∫ 1/2
−1/2
‖u˜(`, ·)‖2Hs(T2pi)d`
)1/2
.
Bloch transform T defined by (2.29) is related to the Fourier transform F by the following formula,
cf. [15, 29],
u˜(`, x) =
∑
j∈Z
eijxû(`+ j), (2.31)
where û(ξ) = (Fu) (ξ), ξ ∈ R, is the Fourier transform of u on the real axis.
Multiplication of two functions u(x) and v(x) in x-space corresponds to the convolution integral
in Bloch space:
(u˜ ? v˜)(`, x) =
1/2∫
−1/2
u˜(`−m,x)v˜(m,x)dm, (2.32)
where the continuation conditions (2.30) have to be used for |`−m| > 1/2.
If χ : R→ R is 2pi-periodic, then
T (χu)(`, x) = χ(x)(T u)(`, x). (2.33)
The relations (2.32) and (2.33) are well-known [10, 15] and can be proved from (2.29) and (2.31).
17
2.4.2. The system in Bloch space
Thanks to the definitions (2.6), (2.7), and (2.8), it is obvious how to transfer the evolution problem
(2.8) into Bloch space. We apply the Bloch transform T to all components of U = (u0, u+, u−)
and obtain
i∂tU˜(t, `, x) = L˜(`)U˜(t, `, x)− (U˜ ? U˜ ? U˜)(t, `, x), (2.34)
where the operator L˜(`) := −(∂x + i`)2 appears in the periodic spectral problem (2.12), the
function U˜(t, `, x) = (u˜0, u˜+, u˜−)(t, `, x) satisfies the continuation conditions
U˜(t, `, x) = U˜(t, `, x+ 2pi) and U˜(t, `, x) = U˜(t, `+ 1, x)eix,
and the convolution integrals are applied componentwise as in
U˜ ? U˜ ? U˜ =
(
u˜0 ? u˜0 ? u˜0, u˜+ ? u˜+ ? u˜+, u˜− ? u˜− ? u˜−
)
.
In order to guarantee that u˜j(t, `, ·) has support in I0,j for j ∈ {0,+,−}, we define periodic
cut-off functions
χj(x) =
{
1, x ∈ Ij ,
0, elsewhere,
j ∈ {0,+,−}. (2.35)
With the help of property (2.33), we obtain
T (uj)(`, x) = T (χjuj)(`, x) = χj(x)(T uj)(`, x), j ∈ {0,+,−}.
Therefore, the support of T (uj)(`, x) with respect to x is contained in Ij for any j ∈ {0,+,−}.
2.4.3. Bloch transform for smooth functions
Since we proved the local existence and uniqueness of solutions in H2, the domain of definition of
the operator L := −∂2x in L2, we have to work in Bloch space in its counterpart H˜2, the domain
of definition of the operator L˜(`) := −(∂x + i`)2 in the space L2(T1, L2Γ), where L2Γ is defined
by (2.16). We define
H˜2 = {U˜ ∈ L2(T1, L2Γ) : u˜j ∈ L2(T1, H2(I0,j)), j ∈ {0,+,−}, (2.13)− (2.14) are satisfied},
equipped with the norm
‖U˜‖H˜2 =
(∫ 1/2
−1/2
(
‖u˜0(`, ·)‖2H2(I0,0) + ‖u˜+(`, ·)‖2H2(I0,+) + ‖u˜−(`, ·)‖2H2(I0,−)
)
d`
)1/2
.
The following lemma presents an important result for the justification analysis in Bloch space.
Lemma 2.4.2. The Bloch transform T is an isomorphism between the spaces H2 and H˜2.
Proof. We start with the function u0 defined in (2.6). The L
2-function u0 which is in H
2 on the
intervals [2npi, 2npi+pi] for n ∈ Z is extended smoothly to a global H2 function u0,ext. According
to Lemma 2.4.1, we have T (u0,ext) ∈ L2(T1, H2(T2pi)). With the cut-off function χ0 defined in
(2.35), we find by using (2.33) that
u˜0 = T (u0) = T (χ0u0,ext) = χ0T (u0,ext).
Therefore, for fixed ` ∈ T1, we have supp(u˜0) = I0,0. From the properties of T (u0,ext), we
conclude that u˜0 ∈ L2(T1, H2(I0,0)). The components u± are handled with the same technique.
The boundary conditions (2.2)-(2.3) transfer in Bloch space into the boundary conditions (2.13)-
(2.14).
18
2.5. Estimates for the residual terms
Here we decompose the evolution problem (2.34) into two parts. The first part reduces to the
effective amplitude equation of the type (2.19) but written in Fourier space. The other part
satisfies the evolution problem where the residual terms can be estimated in the space H˜2. Since
the residual term after a standard decomposition similar to (2.17) and (2.18) is still large for
estimates, we will also introduce an improved approximation by singling out some terms in the
second part of the decomposition. Although the estimates are performed in Fourier and Bloch
space, they can be easily transferred back to physical space.
In order to recover the ansatz (2.17) and (2.18) used for the derivation of the effective amplitude
equation (2.19) in Bloch space, we split the solution to the evolution problem (2.34) into two
parts. We write
U˜(t, `, x) = V˜ (t, `)f (m0)(`, x) + U˜⊥(t, `, x), (2.36)
where the orthogonality condition 〈U˜⊥(t, `, ·), f (m0)(`, ·)〉L2Γ = 0 is used for uniqueness of the
decomposition. We find two parts of the evolution problem:
i∂tV˜ (t, `) = ω
(m0)(`)V˜ (t, `)−NV (V˜ , U˜⊥)(t, `) (2.37)
and
i∂tU˜
⊥(t, `, x) = L˜(`)U˜⊥(t, `, x)−N⊥(V˜ , U˜⊥)(t, `, x), (2.38)
where
NV (V˜ , U˜
⊥)(t, `) = 〈(U˜ ? U˜ ? U˜)(t, `, ·), f (m0)(`, ·)〉L2Γ
and
N⊥(V˜ , U˜⊥)(t, `, x) = (U˜ ? U˜ ? U˜)(t, `, x)−NV (V˜ , U˜⊥)(t, `)f (m0)(`, x).
Next, we estimate each part of the evolution problem.
2.5.1. Derivation of the effective amplitude equation
The effective amplitude equation (2.19) can be derived from equation (2.37) by evaluating it at
U˜⊥ = 0. To be precise, we write
NV (V˜ , U˜
⊥)(t, `) =
∫
T1
∫
T1
β(`, `1, `2, `1 + `2 − `)
× V˜ (t, `1)V˜ (t, `2)V˜ (t, `1 + `2 − `)d`1d`2 +NV,rest(V˜ , U˜⊥)(t, `)
where we used V˜ (t, `) = V˜ (t,−`), and introduced the kernel β by
β(`, `1, `2, `1 + `2 − `) :=
〈
f (m0)(`1, ·)f (m0)(`2, ·)f (m0)(`1 + `2 − `, ·), f (m0)(`, ·)
〉
L2Γ
.
We note that NV,rest(V˜ , 0) = 0. Let us now make the ansatz
V˜app(t, `) = A˜
(
ε2t,
`− `0
ε
)
E(t, `), (2.39)
19
with
E(t, `) := e−iω
(m0)(`0)te−i∂`ω
(m0)(`0)(`−`0)t,
insert (2.39) into the evolution problem (2.37), and set the coefficients of ε2E to zero. As a
result, we obtain the leading-order equation in the form
i∂T A˜(T, ξ) =
1
2
∂2`ω
(m0)(`0)ξ
2A˜(T, ξ)
− ν
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜(T, ξ1)A˜(T, ξ2)A˜(T, ξ1 + ξ2 − ξ)dξ1dξ2,
(2.40)
where ` = `0 + εξ, T = ε
2t, and ν = β(`0, `0, `0, `0) coincides with the definition of ν in the
amplitude equation (2.19).
By letting ε → 0, in particular ∫ 12ε− 1
2ε
dξ → ∫∞−∞ dξ, and A˜(T, ξ) → Â(T, ξ) as ε → 0, equation
(2.40) yields formally the NLS equation in Fourier space, namely
i∂T Â(T, ξ)− 1
2
∂2`ω
(m0)(`0)ξ
2Â(T, ξ) + ν(Â ∗ Â ∗ Â)(T, ξ) = 0. (2.41)
Equation (2.41) corresponds to the amplitude equation (2.19) in physical space. The formal
calculations will be made rigorous in Section 2.5.3.
Remark 2.5.1. If A(·) is defined on R and if it is scaled with the small parameter ε, then the
Fourier transform of A(ε·) is ε−1Â(ε−1·). Therefore, a small term of the formal order O(εr) in
physical space corresponds to a small term of the formal order O(εr−1) in Fourier space. Since
Bloch space is very similar to Fourier space, we have implemented the corresponding orders in
the representation (2.39) compared to the standard approximation (2.17).
2.5.2. The improved approximation
The simple approximation (2.39) produces a number of terms in the second equation (2.38) which
are of the formal order O(ε2) in Bloch space and which do not cancel out each other. These
terms are collected together in the so called residual. However, in order to bound the error with
a simple application of Gronwall’s inequality, as we do in Section 2.6, we need the residual to be
of the formal order O(ε3) in Bloch space.
As in [19], the O(ε2) terms can be canceled out by adding higher order terms to the approximation
(2.39) in (2.36). Therefore, we set
U˜⊥app(t, `, x) = ε
2B˜
(
ε2t,
`− `0
ε
, x
)
E(t, `). (2.42)
Inserting (2.42) into the evolution problem (2.38) and equating the coefficients of ε2E to zero
gives the following equation in the lowest order in ε:
ω(m0)(`0)B˜
(
ε2t, ξ, x
)
= L˜(`0)B˜
(
ε2t, ξ, x
)
− ε−2E−1(t, `)N⊥(V˜app, 0)(t, `, x),
(2.43)
where ` = `0+εξ. Note that all E-factors cancel each other out in the nonlinear terms. Moreover,
the pre-factor ε−2 cancels with the factor ε2 coming from the two times convolution of the scaled
20
ansatz functions. The equation (2.43) can be solved with respect to B˜ if L˜(`0) − ω(m0)(`0)I is
invertible. The invertibility condition
inf
m∈N\{m0}
∣∣∣ω(m)(`0)− ω(m0)(`0)∣∣∣ > 0
is satisfied for the spectral problem (2.9) under the condition (2.20) of Theorem 2.2.3. Substituting
A˜ and B˜ obtained from (2.40) and (2.43) into (2.39) and (2.42), and inserting the approximation
(V˜app, U˜
⊥
app) into the evolution problem (2.37) and (2.38) cancel out all terms of the formal order
O(ε2). According to Remark 2.5.1, this corresponds to the cancelation of all terms of the formal
order O(ε3) in physical space. Hence the residual is formally of the order O(ε3) in Bloch space
and of the order O(ε4) in physical space.
2.5.3. From Fourier space to Bloch space
As in Theorem 2.2.3, let A ∈ C(R, H3(R)) be a solution of the effective amplitude equation
(2.19). Here we show that the residual of the evolution problem (2.34) given by
R˜es(U˜)(t, `, x) = −i∂tU˜(t, `, x) + L˜(`)U˜(t, `, x)− (U˜ ? U˜ ? U˜)(t, `, x),
can be estimated in H˜2 to be of order O(ε7/2) if the improved approximation is constructed by
using the decomposition (2.36) with (V˜app, U˜
⊥
app) given by (2.39) and (2.42).
Before we start, we introduce some weights with respect to the `-variable, namely
ρ`0,ε,s(`) =
[
1 +
(
`− `0
ε
)2]s/2
.
Remark 2.5.2. Regularity of functions in physical space corresponds to decay rates of their Fourier
transforms at infinity. Due to Parseval’s identity, Fourier transform is an isomorphism between
Hs and L2 equipped with a weight ρ0,1,s. Furthermore, weights ρ∗,1,∗ appear with functions
which are not scaled with respect to ε, whereas weights ρ∗,ε,∗ appear with functions which are
scaled with respect to ε. The scaled weights ρ∗,ε,∗ are necessary to transfer the smallness property
∂xA(εx) = ε∂XA(X) = O(ε) from physical space into Fourier space, cf. Lemma 2.5.4.
As a consequence of the assumptions on A ∈ C(R, H3(R)), the Fourier transform  is a solution
of the NLS equation in Fourier space (2.41) and satisfies Âρ0,1,3 ∈ L2(R). By the Cauchy-Schwarz
inequality, we have
‖Âρ0,1,2‖L1 ≤ ‖Âρ0,1,3‖L2‖ρ0,1,−1‖L2 ≤ C‖Âρ0,1,3‖L2 , (2.44)
hence, Âρ0,1,2 ∈ L1(R). For such a function  in Fourier space, we define a function A˜ in Bloch
space by
A˜(T, ε−1(`− `0)) = χ˜`0(`)Â(T, ε−1(`− `0)),
where χ˜`0 is defined as the cutoff function
χ˜`0(`) =
{
1, `− `0 ∈ [−δ, δ] ,
0, otherwise,
21
with δ > 0 being sufficiently small but independent of the small parameter ε. Using the periodicity
condition
A˜
(
T, ε−1(`+ 1− `0)
)
= A˜
(
T, ε−1(`− `0)
)
, ` ∈ R,
we extend A˜(T, ε−1(`− `0)) periodically in ` over R. By construction, the leading-order approx-
imation
V˜appf
(m0)ρ`0,ε,3 ∈ H˜2
is of the order O(ε1/2) due to the scaling properties of the L2-norm. Therefore, we are losing
ε1/2 when we perform estimates in H˜2. In order to avoid losing ε1/2, let us consider estimates in
the following L1-based space
C˜2 = {U˜ ∈ L1(T1, L2Γ) : u˜j ∈ L1(T1, H2(I0,j)), j ∈ {0,+,−}, (2.13)− (2.14) is satisfied},
equipped with the norm
‖U˜‖C˜2 =
∫ 1/2
−1/2
(
‖u˜0(`, ·)‖H2(I0,0) + ‖u˜+(`, ·)‖H2(I0,+) + ‖u˜−(`, ·)‖H2(I0,−)
)
d`.
Compared to the estimates in H˜2, the leading-order approximation
V˜appf
(m0)ρ`0,ε,2 ∈ C˜2
is of the order O(ε). Due to Young’s inequality and (2.44) we have
‖V˜ ? W˜‖H˜2 ≤ ‖V˜ ‖C˜2‖W˜‖H˜2 ,
respectively with weights
‖(V˜ ? W˜ )ρ`0,ε,2‖H˜2 ≤ C‖V˜ ρ`0,ε,2‖C˜2‖W˜ρ`0,ε,2‖H˜2 ,
with a constant C independent of the small parameter ε. Using these estimates shows that
E−1(t, ·)N⊥(V˜app, 0)(t, ·, ·)ρ`0,ε,2(·) ∈ H˜2
is of the order O(ε5/2) in H˜2 and of the order O(ε3) in C˜2. Moreover, we have
supp
(
E−1(t, ·)N⊥(V˜app, 0)(t, ·, ·)
)
⊂ [`0 − 3δ, `0 + 3δ].
Hence, we drop (2.43) and define
B˜
(
ε2t, ξ, x
)
= (L˜(`)− ω(m0)(`0)I)−1ε−2E−1(t, `)N⊥(V˜app, 0)(t, `, x), (2.45)
where again ` = `0 + εξ. The inverse (L˜(`) − ω(m0)(`0)I)−1 exists due to the non-resonance
condition (2.20) for δ > 0 sufficiently small, but independent of the small parameter ε > 0. The
change from L˜(`0) in equation (2.43) to L˜(`) here allows us to avoid an expansion of L˜(`) at
` = `0, which would correspond to a loss of regularity.
By construction in (2.42), we have that U˜⊥appρ`0,ε,2 ∈ H˜2 is of the order O(ε5/2) and U˜⊥appρ`0,ε,1 ∈
C˜2 is of the order O(ε3). Thus, we set
εΨ˜(t, `, x) = V˜app(t, `)f
(m0)(`, x) + U˜⊥app(t, `, x), (2.46)
with V˜app and U˜
⊥
app defined in (2.39) and (2.42).
Remark 2.5.3. In contrast to the approximation εΨnls the approximation εΨ = T −1(εΨ˜) satisfies
the Kirchhoff boundary conditions (2.2)-(2.3).
22
2.5.4. Estimates in Bloch space
By construction of εΨ˜, the lower order terms are canceled out so that R˜es(εΨ˜) is formally of
the order O(ε4) in physical space and of the order of O(ε3) in Bloch space. In order to put this
formal count on a rigorous footing, we use the following elementary result.
Lemma 2.5.4. Let m, s ≥ 0 and let g : T1 → R satisfy
|g(`)| ≤ C|`− `0|s, ` ∈ T1,
for some C > 0. Then, we have
‖ρ0,1,m(·)g(·)A˜(ε−1(· − `0))‖L2(T1) ≤ Cεs+1/2‖ρ0,1,m+sÂ‖L2(R).
Proof. We estimate the left-hand side as follows:
‖ρ0,1,m(·)g(·)A˜(ε−1(· − `0))‖2L2(T1) =
∫
T1
|g(`)|2(1 + `2)m
∣∣∣∣A˜(`− `0ε
)∣∣∣∣2 d`
≤ sup
`∈T1
|g(`)|2(1 + ε−2|`− `0|2)−s−m(1 + `2)m
∫
T1
(1 + ε−2(`− `0)2)m+s
∣∣∣∣A˜(`− `0ε
)∣∣∣∣2 d`
≤ C2ε2sε‖ρ0,1,m+sÂ‖2L2(R)
where the last inequality follows from the scaling transformation for the squared L2-norm, cf. also
the subsequent Remark 2.5.6.
By using Lemma 2.5.4, we obtain the estimate on R˜es(εΨ˜) given by (2.46).
Lemma 2.5.5. Let A ∈ C([0, T0], H3) be a solution of the amplitude equation (2.19) for some
T0 > 0. Then, there is a positive ε-independent constant CRes that only depends on the norm of
the solution A such that
sup
t∈[0,T0/ε2]
‖R˜es(εΨ˜)‖H˜2 ≤ CResε7/2, (2.47)
or equivalently,
sup
t∈[0,T0/ε2]
‖Res(εΨ)‖H2 ≤ CResε7/2. (2.48)
Proof. We define
R˜esV (V˜ , U˜
⊥)(t, `) = −i∂tV˜ (t, `) + ω(m0)(`)V˜ (t, `)−NV (V˜ , U˜⊥)(t, `),
R˜es
⊥
(V˜ , U˜⊥)(t, `, x) = −i∂tU˜⊥(t, `, x) + L˜(`)U˜⊥(t, `, x)−N⊥(V˜ , U˜⊥)(t, `, x).
By construction we have
R˜es
⊥
(V˜app, U˜
⊥
app)(t, `, x) = s1 + s2,
where
s1 = (−i∂t + ω(m0)(`0))U˜⊥app(t, `, x)
= (−(ε2∂`ω(m0)(`0)(`− `0) + ε4∂T )B˜ (T, ξ, x))E(t, `)
= (−(ε3∂`ω(m0)(`0)ξ + ε4∂T )B˜ (T, ξ, x))E(t, `)
23
and
s2 = N
⊥(V˜ , 0)(t, `, x)−N⊥(V˜ , U˜⊥)(t, `, x),
again with ` = `0 + εξ. Via (2.45) the term ∂T B˜ in s1 can be expressed in terms of ∂T V˜app,
respectively in terms of ∂TA, where ∂TA can be expressed by the right-hand side of the amplitude
equation (2.19). Similarly, the term ξB˜ (T, ξ, x) can be estimated in terms of ξÂ(T, ξ). Since
U˜⊥app obviously is in H˜2, we eventually have the estimate
‖s1‖H˜2 ≤ Cε7/2‖Â‖2L1‖Âρ0,1,1‖L2
+ Cε9/2‖Â‖2L1(‖Âρ0,1,2‖L2 + ‖Â‖2L1‖Â‖L2).
In s2 by pure counting of powers of ε we find the formal order O(ε3) in Bloch space and due to
the scaling properties of the L2-norm, we have
‖s2‖H˜2 ≤ CAε7/2,
where the constant CA depend on ‖Âρ0,1,3‖L2 .
Next we have
R˜esV (V˜app, U˜
⊥
app)(t, `) = r1 + r2,
where
r1 = −i∂tV˜app(t, `) + ω(m0)(`)V˜app(t, `)−NV (V˜app, 0)(t, `)
+Eχ˜`0(`)(i∂T Â(T, ξ)−
1
2
∂2`ω
(m0)(`0)ξ
2Â(T, ξ) + ν(Â ∗ Â ∗ Â)(T, ξ))
and
r2 = NV (V˜app, 0)(t, `)−NV (V˜app, U˜⊥app)(t, `).
The term r2 is of the formal order O(ε3) in Bloch space and due to the scaling properties of the
L2-norm, it is of the order O(ε7/2) in L2. The second line in r1 vanishes identically since it is a
multiple of the effective amplitude equation (2.19). The prefactor E is necessary to compare the
second line in r1 with the first line in r1. The cut-off function χ˜`0 is needed to bring (2.19) from
Fourier space to Bloch space.
The comparison of the terms of the first and of the second line in r1 condense in estimates for
the difference between ω(m0)(`) and its second Taylor polynomial at `0,
T2(`; `0) = ω
(m0)(`0) + ∂`ω
(m0)(`0)(`− `0) + 1
2
∂2`ω
(m0)(`0)(`− `0)2,
the difference between the nonlinear coefficient β = β(`, `1, `2, `1 + `2 − `) defined in (2.39) and
the coefficient ν = β(`0, `0, `0, `0), and the difference between  and A˜.
In detail, we use the estimate∣∣∣ω(m0)(`)− T2(`; `0)∣∣∣ ≤ C|`− `0|3
and apply Lemma 2.5.4 with m = 0 and s = 3 to find
‖(ω(m0)(·)− T2(·; `0))A˜(ε−1(· − `0))‖L2(T1) ≤ Cε7/2‖ρ0,1,3Â‖L2(R).
24
For the difference between the nonlinear coefficients, we use the estimate
|β(`, `1, `2, `1 + `2 − `)− β(`0, `0, `0, `0)|
≤ C(|`− `0|+ |`1 − `0|+ |`2 − `0|+ |`1 + `2 − `− `0|)
and apply an obvious generalization of Lemma 2.5.4 to multilinear terms. It remains to estimate
the difference between  and A˜. Since |χ˜`0(`) − 1| ≤ C|` − `0|m for every m ≥ 0, we have for
m = 3,
‖A˜(ε−1(· − `0))− Â(ε−1(· − `0))‖L2 = ‖(1− χ˜`0)Â(ε−1(· − `0))‖L2
≤ ε1/2 sup
`∈R
|(1− χ˜0(ε`))(1 + |`|)−3|‖Âρ0,1,3‖L2
≤ Cε7/2‖Âρ0,1,3‖L2 .
By using these expansions, we derive the bound (2.47). Bound (2.48) holds thanks to the
isomorphism of Bloch transform T between H2 and H˜2.
Remark 2.5.6. Compared to Remark 2.5.1 on the formal order in physical and Bloch space, we
note that bounds (2.47) and (2.48) are identical in physical and Bloch space. This is because
we gain ε1/2 in the H˜2-norm due to the concentration and lose ε−1/2 in the H2-norm due to the
long wave scaling.
Let us now recall that the approximation εΨnls given by (2.18) that leads to the effective amplitude
equation (2.19) is different from the improved approximation εΨ, which is given by (2.46) in
Bloch space. The next result compares the two approximations. It is obtained by an elementary
application of the Lemmas 2.3.1, 2.4.2 and 2.5.4.
Lemma 2.5.7. Let A ∈ C([0, T0], H3) be a solution of the amplitude equation (2.19) for some
T0 > 0. Then, there exist positive ε-independent constants C and Cψ that only depend on the
norm of the solution A such that
sup
t∈[0,T0/ε2]
‖εΨ˜‖C˜2 ≤ CΨε (2.49)
and
sup
t∈[0,T0/ε2]
‖εΨ− εΨnls‖L∞ ≤ Cε3/2. (2.50)
Proof. The first estimate (2.49) immediately follows by the previous estimates on each component
of Ψ˜. The second estimate (2.50) follows by applying a slight generalization of Lemma 2.5.4 to
the difference f (m0)(`, ·) − f (m0)(`0, ·) and using the triangle inequality, since the term U˜⊥app is
very small compared to the term V˜appf
(m0) in (2.46). Since the boundary conditions for the
derivatives of the eigenfunctions depend on ` they can only be compared in H1(T2pi). We have
‖f (m0)(`, ·)− f (m0)(`0, ·)‖H1(T2pi) ≤ C|`− `0|.
With the obvious generalization of Lemma 2.5.4 we obtain
‖(f (m0)(`, ·)− f (m0)(`0, ·))A˜(ε−1(· − `0))‖L2(T1,H1(T2pi)) ≤ Cε3/2‖ρ0,1,1Â‖L2(R).
Lemma 2.4.1 and Sobolev’s embedding theorem yield estimate (2.50).
25
2.6. Estimates for the error term
Here we complete the proof of Theorem 2.2.3. The proof of the approximation result is based on
a simple application of Gronwall’s inequality.
First we note that, by the standard energy estimates, the local solution U to the evolution problem
(2.8) constructed in Theorem 2.3.5 can be continued to the global solution U inH2 with a possible
growth of the H2-norm as t → ∞. We do not worry about the possible growth of the global
solution U because the approximation result of Theorem 2.2.3 is obtained on finite but long time
intervals with a precise control of the error terms, cf. bound (2.21).
We write the solution U to the evolution problem (2.8) as a sum of the approximation term εΨ
controlled by Lemma 2.5.7 and the error term ε3/2R, i.e.,
U = εΨ + ε3/2R. (2.51)
Inserting this decomposition into the evolution problem (2.8) gives
∂tR = −iLR+ iG(Ψ, R) (2.52)
where the linear operator L = −∂2x is studied in Lemma 2.3.2 and the nonlinear terms are expanded
as
G(Ψ, R) = ε−3/2Res(εΨ) + ε2Ψ2R+ 2ε2ΨRΨ + 2ε5/2ΨRR+ ε5/2R2Ψ + ε3R2R.
The product terms in the definition ofG(Ψ, R) are understood componentwise withR = (r0, r+, r−)
and Ψ = (ψ0, ψ+, ψ−). Using the bounds
‖ΨR‖H2 ≤ C‖Ψ˜R˜‖H˜2 ≤ C‖Ψ˜‖C˜2‖R˜‖H˜2 ≤ CCΨ‖R˜‖H˜2 ≤ C2CΨ‖R‖H2 ,
where CΨ appears in (2.49) of Lemma 2.5.7, we estimate each term of G with the help of Lemmas
2.3.1 and 2.5.5:
‖ε−3/2Res(εΨ)‖H2 ≤ CResε2,
‖2ε2ΨRΨ‖H2 ≤ 2C1ε2‖R‖H2 ,
‖ε2Ψ2R‖H2 ≤ C1ε2‖R‖H2 ,
‖ε5/2R2Ψ‖H2 ≤ C1ε5/2‖R‖2H2 , ,
‖2ε5/2ΨRR‖H2 ≤ 2C1ε5/2‖R‖2H2 ,
‖ε3R2R‖H2 ≤ C1ε3‖R‖3H2 ,
where C1 is a constant independent of ‖R‖H2 and the small parameter ε > 0. Therefore, we find
‖G(Ψ, R)‖H2 ≤ CResε2 + 3C1ε2‖R‖H2 + 3C1ε5/2‖R‖2H2 + C1ε3‖R‖3H2 .
For simplicity, we assume R(0) = 0. Then, the variation of constant formula for the evolution
system (2.52) yields the integral formula
R(t) =
t∫
0
e−iL(t−τ)iG(Ψ, R)(τ)dτ.
26
By Lemma 2.3.4, the operator e−iLt forms a group in H2 which is uniformly bounded with respect
to t. Using Gronwall’s inequality finally allows us to estimate the error term on the time scale
T = ε2t for T ∈ [0, T0] by
sup
t∈[0,T0/ε2]
‖R(t)‖H2 ≤ CResT0e4C1T0 =: M
for all ε ∈ (0, ε0), if ε0 > 0 is chosen so small that 3ε1/20 M + ε0M2 ≤ 1. Sobolev’s embedding
theorem, bound (2.50), and the decomposition (2.51) complete the proof of the approximation
result (2.21) of Theorem 2.2.3.
Remark 2.6.1. We explain how the proof of Theorem 2.2.3 has to be modified in order to prove
Theorem 2.2.8. We only need H2 for the Dirac case instead of H3 in the NLS case due to the
fact that the functions ω± given by (2.22) and (2.23) have to be expanded in ` up to quadratic
order for estimating the residual terms. The decomposition formula (2.36) is replaced by
U˜(t, `, x) = V˜+(t, `)f
+(`, x) + V˜−(t, `)f−(`, x) + U˜⊥(t, `, x),
subject to the orthogonality constraints 〈U˜⊥(t, `, ·), f+(`, ·)〉L2Γ = 〈U˜
⊥(t, `, ·), f−(`, ·)〉L2Γ = 0.
For the derivation of the coupled-mode system (2.25)-(2.26) we then make the ansatz
V˜app,±(t, `) = ε−1A˜±
(
ε2t, ε−2`
)
e−iω±(0)t.
Straightforward modifications of this kind can be performed at each step in the proof of Theorem
2.2.3. This procedure yields the proof of Theorem 2.2.8.
2.7. Discussion
Here we discuss why the previously presented theory applies to other periodic quantum graphs.
The general strategy is as follows. Rescale the length of the bonds in such a way that the
basic cell of the periodic graph has a length of 2pi. The differential operators and the Kirchhoff
boundary conditions at the vertices have to be rescaled, too. We refrain from greatest generality
and explain this approach for two periodic quantum graphs, cf. Figure 2.5, which are slightly
more complicated than the periodic graph plotted in Figure 2.1 .
a) b)
Figure 2.5.: a) Generalization of the periodic quantum graph sketched in Figure 2.1. The central
segment Γn,0 has length L0 and the circular segments Γn,± have lengths L+ and L−.
b) A periodic quantum graph with a vertical pendant and a horizontal bond, each of
length pi, with Dirichlet boundary conditions at the dead end.
In order to bring the quantum graph plotted in Figure 2.5(a) into a form for which our previous
theory applies, we first identify Γ0,0 with [0, L0], Γ0,+ with [0, L+], and Γ0,− with [0, L−]. The
coordinates in these bonds are denoted with y. Then on Γ0,0 we introduce piy = L0x and on
27
Γ0,± we introduce piy = L±(x− pi). Hence we are back on our original quantum graph, but with
different equations and different vertex conditions, namely:
i∂tU +
L20
pi2
∂2xU + |U |2U = 0, for x ∈ (2pin, 2pin+ pi)
and
i∂tU +
L2±
pi2
∂2xU + |U |2U = 0, for x ∈ (2pin+ pi, 2pi(n+ 1)),
subject to {
un,0(t, 2pin+ pi) = un,+(t, 2pin+ pi) = un,−(t, 2pin+ pi),
un+1,0(t, 2pi(n+ 1)) = un,+(t, 2pi(n+ 1)) = un,−(t, 2pi(n+ 1)),
and{
L0∂xun,0(t, 2pin+ pi) = L+∂xun,+(t, 2pin+ pi) + L−∂xun,−(t, 2pin+ pi),
L0∂xun+1,0(t, 2pi(n+ 1)) = L+∂xun,+(t, 2pi(n+ 1)) + L−∂xun,−(t, 2pi(n+ 1)).
The spectral bands of the linear operator for the periodic graph on Figure 2.5(a) depend on
parameter L0, L+, and L−.
In the case L0 6= L+ = L− (left panel on Fig. 2.6), the Dirac points disappear and all spectral
bands but the flat bands are disjoint. The flat bands still intersect with the interior points of the
spectral bands of L. As a result, the justification of the amplitude equation (2.19) can still be
developed for the NLS equation on the periodic quantum graph but the non-resonance condition
(2.20) is satisfied for every m0 ∈ N and `0 ∈ T1, for which ω(m0)(`0) is different from the
eigenvalue corresponding to the flat spectral bands.
In the case L0 = L+ 6= L− (right panel on Fig. 2.6), the degeneracy of all flat bands is broken
and all spectral bands have nonzero curvature and are disjoint from each other. As a result,
the non-resonance condition (2.20) is now satisfied for every m0 ∈ N and `0 ∈ T1 without any
reservations.
Figure 2.6.: The Floquet-Bloch spectrum of the linear operator L = −∂2x for the periodic quantum
graph plotted on Figure 2.5(a) with L0 = pi + 0.3 and L+ = L− = pi (left) and
L0 = pi, L+ = pi, and L− = pi + 0.3 (right).
28
In a similar way, the quantum graph plotted in Figure 2.5(b) can be handled. We refrain here from
details and only show the spectral picture in Figure 2.7. Dirac points appear now at ` = ±12 and
the flat bands are now disjoint from the other bands. Correspondingly, both the NLS amplitude
equation and the coupled-mode (Dirac) equations can be justified for the periodic quantum graph
at the corresponding points in the spectral bands.
Figure 2.7.: The Floquet-Bloch spectrum of the linear operator L = −∂2x for the periodic quantum
graph plotted in Figure 2.5(b).
Finally, we can think of transferring the ideas of the justification analysis to other nonlinear evo-
lution equations, which would include the nonlinear wave equations and systems with quadratic
nonlinearities. Since the eigenfunctions are not smooth at the graph vertices due to the Kirchhoff
boundary conditions, we may face difficulties with analysis of convolution terms and near-identity
transformations, in comparison with a similar analysis for smooth periodic potentials [10]. Addi-
tionally, more complicated non-resonance conditions may appear in the analysis of the nonlinear
wave equation without the gauge covariance compared to the case of the cubic NLS equation
(2.1). Thus, it will be a purpose of subsequent works to extend the justification analysis to other
nonlinear evolution equations.
29

3. Approximation of a cubic Klein-Gordon
equation on periodic quantum graphs
After proving an approximation result for the NLS equation on a spatially extended periodic
quantum graph, we now consider a cubic Klein-Gordon (cKG) equation as original system in
the same setting. In Chapter 2, a multiple scaling expansion yields a NLS equation as effective
amplitude equation describing slow modulations in time and space of an oscillating wave packet.
It is the goal of this chapter to transfer the analysis from [17] to this situation. A similar result
for the spatially homogeneous case is given in [10].
Here we have to deal with two major differences compared to the problem described in Chapter 2.
First of all, the cKG equation is a second order differential equation with respect to time in contrast
to the NLS equation, and is also not invariant under the transformation u(t, x) 7→ u(t, x)eiφ0 for
every φ0 ∈ R. Both points force us to modify the arguments for the construction of the solutions
and the higher order approximations which are necessary to get rid of some regularity problems
associated with the Kirchhoff boundary conditions at the vertices.
Our strategy to obtain an approximation theorem for the cKG equation is the same as in [17]
and Chapter 2, respectively. We describe the model in Section 3.1 and the main result is stated
in Section 3.2 after introducing the spectral situation on the periodic quantum graph. The local
existence and uniqueness of solutions of the original system is discussed in Section 3.3. Then, in
Section 3.4, we transfer the system into Bloch space and derive an effective amplitude equation.
We also construct an improved approximation and estimate the corresponding residual. Finally,
the amplitude equation is justified by estimating the error terms in Section 3.6.
Notation: According to Chapter 2, we use the standard notations for the Sobolev space Hs(R)
and the Lebesgue space Lp(R) for s ≥ 0 and p ≥ 1.
3.1. The model
We consider the cKG equation
∂2t u− ∂2xu+ u+ u3 = 0, t ∈ R, x ∈ Γ, (3.1)
as original system on the periodic quantum graph
Γ = ⊕n∈ZΓn, with Γn = Γn,0 ⊕ Γn,+ ⊕ Γn,−,
which was already introduced in Section 2.2. At the vertex points x ∈ {kpi : k ∈ Z} of the
quantum graph Γ, we again use the Kirchoff boundary conditions{
un,0(t, 2pin+ pi) = un,+(t, 2pin+ pi) = un,−(t, 2pin+ pi),
un+1,0(t, 2pi(n+ 1)) = un,+(t, 2pi(n+ 1)) = un,−(t, 2pi(n+ 1)),
(3.2)
31
and {
∂xun,0(t, 2pin+ pi) = ∂xun,+(t, 2pin+ pi) + ∂xun,−(t, 2pin+ pi),
∂xun+1,0(t, 2pi(n+ 1)) = ∂xun,+(t, 2pi(n+ 1)) + ∂xun,−(t, 2pi(n+ 1)).
(3.3)
Remark 3.1.1. Similar to the discussion in Section 2.7, we can extend the subsequent approach
to other one-dimensional quantum graphs by rescaling the length of the bonds, the differential
operators and the boundary conditions.
Studying the NLS equation on the periodic quantum graph Γ, it turned out to be advantageous
to transfer the scalar partial differential equation on Γ to a vector-valued PDE on the real axis by
introducing the functions
u0(x) =
{
un,0(x), x ∈ In,0,
0, x ∈ In,±, n ∈ Z,
and
u±(x) =
{
un,±(x), x ∈ In,±,
0, x ∈ In,0, n ∈ Z.
Hence, we get I0 = supp(u0) and I± = supp(u±). We collect the functions u0 and u± in the
vector U = (u0, u+, u−) and rewrite the evolution problem (3.1) as
∂2t U − ∂2xU + U + U3 = 0, t ∈ R, x ∈ R \ {kpi : k ∈ Z}, (3.4)
subject to the conditions (2.2)-(2.3) at the vertex points x ∈ {kpi : k ∈ Z}. Here, the cubic
nonlinear term U3 stands for the vector (u30, u
3
+, u
3−).
3.2. Main result
The first part of this section is devoted to the analysis of the spectral situation of the linearized
problem associated to the cKG equation (3.4), namely
∂2t U = ∂
2
xU − U. (3.5)
In the second part, we introduce the multiscale ansatz εΨnls and state the resulting amplitude
equation on the homogenous space. We finish the section with the formulation of the approxi-
mation theorem.
3.2.1. The Floquet-Bloch spectrum
The linear partial differential equation (3.5) is solved by the Bloch modes
U(t, x) = eiωtei`xf(`, x), `, x ∈ R,
where f(`, ·) = (f0, f+, f−)(`, ·) is a 2pi-periodic function for every ` ∈ R and satisfies the
continuation conditions
f(`, x) = f(`, x+ 2pi), f(`, x) = f(`+ 1, x)eix, `, x ∈ R.
32
Therefore, we can restrict the definition of f(`, x) to x ∈ T2pi and ` ∈ T1 as in Section 2.2.2 and
the Bloch function f(`, x) is a solution of the eigenvalue problem
−(∂x + i`)2f(`, x) + f(`, x) = ω2(`)f(`, x), x ∈ T2pi, (3.6)
subject to the boundary conditions{
f0(`, pi) = f+(`, pi) = f−(`, pi),
f0(`, 0) = f+(`, 2pi) = f−(`, 2pi)
(3.7)
and {
(∂x + i`)f0(`, pi) = (∂x + i`)f+(`, pi) + (∂x + i`)f−(`, pi),
(∂x + i`)f0(`, 0) = (∂x + i`)f+(`, 2pi) + (∂x + i`)f−(`, 2pi).
(3.8)
The functions f0(`, ·) and f±(`, ·) have their supports in I0,0 = [0, pi] ⊂ T2pi and I0,± = [pi, 2pi] ⊂
T2pi, respectively.
As in Section 2.2.2, we define for the periodic eigenvalue problem (3.6) the L2-based spaces
L2Γ := { U˜ = (u˜0, u˜+, u˜−) ∈ (L2(T2pi))3 : supp(u˜j) = I0,j , j ∈ {0,+,−}}
and
H2Γ(`) := {U˜ ∈ L2Γ : u˜j ∈ H2(I0,j), j ∈ {0,+,−}, (3.7)− (3.8) are satisfied},
equipped with the norm
‖U˜‖H2Γ(`) =
(
‖u˜0‖2H2(I0,0) + ‖u˜+‖2H2(I0,+) + ‖u˜−‖2H2(I0,−)
)1/2
,
where ` ∈ T1 is fixed and defined in H2Γ(`) by means of the boundary conditions (3.7)-(3.8).
Next, we present an elementary result for the linear operator L˜(`) := −(∂x + i`)2 + 1 similar to
the one of Lemma 2.2.2.
Lemma 3.2.1. For fixed ` ∈ T1, the operator L˜(`) is a self-adjoint, positive semi-definite operator
in L2Γ.
Proof. We find for every f(`, ·), g(`, ·) ∈ H2Γ(`) and every ` ∈ T1 the following representation of
the operator L˜(`):
〈L˜(`)f, g〉L2Γ = 〈
(−(∂x + i`)2 + 1) f, g〉L2Γ
= 〈−(∂x + i`)2f, g〉L2Γ + 〈f, g〉L2Γ .
Using Lemma 2.2.2, we conclude that
〈−(∂x + i`)2f, g〉L2Γ = 〈f,−(∂x + i`)
2g〉L2Γ
and with
〈f,−(∂x + i`)2g〉L2Γ + 〈f, g〉L2Γ = 〈f, L˜(`)g〉L2Γ .
the operator L˜(`) is self-adjoint for every ` ∈ T1 due to the conditions (3.7)-(3.8), see [9, Theorem
1.4.4]. Since
〈L˜(`)f, f〉L2Γ =
∫ 2pi
0
(∂x + i`)f · (∂x + i`)fdx+
∫ 2pi
0
ffdx ≥ 0,
the operator L˜(`) is positive semi-definite.
33
As in the argumentation for the linear problem of the NLS equation on the periodic quantum
graph Γ, we conclude that by Lemma 3.2.1 and the spectral theorem for self-adjoint operators with
compact resolvent, cf. [36], for each ` ∈ T1 there exists a Schauder base {f (m)(`, ·)}m∈N of L2Γ
consisting of eigenfunctions of L˜(`) with positive eigenvalues {λ(m)(`)}m∈N ordered as λ(m)(`) ≤
λ(m+1)(`). By construction, the Bloch wave functions satisfy the continuation properties (3.6)
and the orthogonality and normalization relations
〈f (m)(`, ·), f (m′)(`, ·)〉L2Γ = δm,m′ , ` ∈ T1.
Here we use again superscripts to count the spectral curves. The subscripts in f
(m)
j (`, x), j ∈
{0,+,−} are once more reserved to indicate the component of f (m)(`, x) for x ∈ I0,j . Via the
λ(m) we find ω = ω(±m) with ω(m) =
√
λ(m) and ω(−m) = −ω(m).
The spectral bands of the periodic eigenvalue problem (3.6) are shown on Figure 3.1. The flat
spectral curves correspond to the eigenvalues of infinite algebraic multiplicity {±√m2 + 1}m∈N.
For these eigenvalues, we again obtain eigenfunctions localized in the circles of the graph. A
detailed calculation of the spectral curves can be found in Appendix A.1.
Figure 3.1.: The spectral curves ω of the spectral problem (3.6) plotted versus the Bloch wave
number ` for the periodic quantum graph Γ.
3.2.2. The effective amplitude equation
We represent slow modulations in time and space of a small-amplitude modulated Bloch mode
by the formal asymptotic expansion
U(t, x) = εΨnls(t, x) + higher-order terms, (3.9)
with
εΨnls(t, x) = εA(T,X)f
(m0)(`0, x)e
i`0xeiω
(m0)(`0)t + c.c., (3.10)
where 0 < ε  1 is a small perturbation parameter, T = ε2t is the slow time variable, X =
ε(x − cgt) is the large space variable and A(T,X) ∈ C describes the amplitude function. The
parameter cg := ∂`ω
(m0)(`0) is defined as the group velocity associated with the Bloch wave
number `0.
34
We again take use of formal asymptotic expansions to show that, at the lowest order in ε, the
amplitude function A satisfies the NLS equation
i∂TA+
1
2
∂2`ω
(m0)(`0)∂
2
XA+ ν|A|2A = 0, (3.11)
where the cubic coefficient is given by
ν =
3〈f (m0)(`0, ·)f (m0)(`0, ·)f (m0)(−`0, ·), f (m0)(`0, ·)〉L2Γ
2ω(m0)(`0)
.
Our main goal is the mathematical justification of the effective amplitude equation (3.11) by
means of error estimates. The approximation result is similar to Theorem 2.2.3. The main
difference lies in the given non-resonance conditions.
Theorem 3.2.2. Pick m0 ∈ Z and `0 ∈ T1 such that the following non-resonance conditions are
satisfied:
ω(m)(`0) 6= ω(m0)(`0) for every m 6= m0 (3.12)
and
ω(m)(3`0) 6= 3ω(m0)(`0) for every m. (3.13)
Then, for every C0 > 0 and T0 > 0, there exist ε0 > 0 and C > 0 such that for all solutions
A ∈ C(R, H3(R)) of the effective amplitude equation (3.11) with
sup
T∈[0,T0]
‖A(T, ·)‖H3 ≤ C0
and for all ε ∈ (0, ε0), there are solutions U ∈ C([0, T0/ε2], L∞(R)) of the original system (3.4)
satisfying the bound
sup
t∈[0,T0/ε2]
sup
x∈R
|U(t, x)− εΨnls(t, x)| ≤ Cε3/2, (3.14)
where εΨnls is given by (3.10).
Remark 3.2.3. The justification of the NLS approximation for the spatially homogeneous cKG
equation is rather trivial and follows by a simple application of Gronwall’s inequality, cf. [19]. In
the context with smooth spatially periodic coefficients, the justification of the NLS approximation
was carried out in [10]. This analysis has to be adjusted to the present non-smooth situation by
using the methods introduced in Chapter 2.
3.3. Local existence and uniqueness
In order to prove the local existence and uniqueness of solutions of the original system (3.4), we
first show that L = −∂2x + 1 with the domain H2 is self-adjoint and positive semi-definite in L2
by adapting the proof of Lemma 2.3.2 to the linear differential operator L used in this problem.
Therefore, we define the function spaces
L2 = {U = (u0, u+, u−) ∈ (L2(R))3 : supp(un,j) = In,j , n ∈ Z, j ∈ {0,+,−}}
35
and
H2 = {U ∈ L2 : un,j ∈ H2(In,j), n ∈ Z, j ∈ {0,+,−}, (3.2)− (3.3) is satisfied},
identically to the ones in Section 2.3. Hence, there exists a self-adjoint and positive semi-definite
root Ω in L2 with L = Ω2 and we can rewrite (3.4) as
∂tW = ΛW +N(W ) (3.15)
with
W =
(
U
V
)
, Λ =
(
0 −Ω
Ω 0
)
and N(W ) =
(
0
Ω−1U3
)
.
Now we show that the initial-value problem for the first-order system (3.15) is locally well-posed
in the space Y2 := H2 ×H2.
Theorem 3.3.1. For every W0 ∈ Y2, there exists a T0 = T0(‖W0‖Y2) > 0 and a unique solution
W ∈ C([−T0, T0],Y2) of the first-order system (3.15) with the initial data W |t=0 = W0.
Proof. The skew symmetric operator Λ with domain D(Λ) defines a unitary group (eΛt)t∈R in
X 2 := L2×L2 such that ‖eΛtW‖X 2 = ‖W‖X 2 for every t ∈ R. This is a consequence of classical
semigroup theory, see Stone’s theorem in [28]. Moreover, there exists a positive constant CL such
that
‖eΛtW‖Y2 ≤ CL‖W‖Y2 (3.16)
for every W ∈ Y2 and every t ∈ R because the following chain of inequalities holds:
‖eΛtW‖Y2 ≤ C‖Λ2eΛtW‖X 2
≤ C‖eΛtΛ2W‖X 2
≤ C‖Λ2W‖X 2
≤ C‖W‖Y2 .
(3.17)
Alongside the existence of the unitary group (eΛt)t∈R, we used the commutativity of Λ and eΛt
and the equivalence between ‖W‖Y2 and ‖Λ2W‖X 2 to obtain (3.17).
Using the fact from Lemma 2.3.1 that the function space H2 is closed under pointwise multipli-
cation, we have U3 ∈ H2 for every U ∈ H2 and therefore we estimate
‖Ω−1U3‖H2 = ‖ΩU3‖L2 ≤ C‖Ω2U3‖L2 ≤ C‖U3‖H2 (3.18)
such that the nonlinearity is locally Lipschitz continuous from H2 to H2.
By Duhamel‘s principle, we now rewrite the Cauchy problem associated with the first-order system
(3.15) as the integral equation
W (t, ·) = eΛtW (0, ·) + i
∫ t
0
eΛ(t−τ)N(W )(τ)dτ, (3.19)
where the solution is considered in the space
M := {W ∈ C ([−T0, T0],Y2) : sup
t∈[−T0,T0]
‖ W (t, ·)− eΛtW (0, ·)‖Y2 ≤ CW },
and the constant CW > 0 is fixed. For every W0 ∈ Y2, there is a sufficiently small T0 =
T0(‖W0‖Y2) > 0 such that the right-hand side of the integral equation (3.19) is a contraction
in the space M, where we used (3.17) and (3.18). The existence of a unique solution W ∈
C([−T0, T0],Y2) then follows from Banach’s fixed-point theorem.
36
Remark 3.3.2. Using Theorem 3.3.1, it is easy to see that there exists a unique solution U ∈
C([−T0, T0],H2) of the original system (3.4) with the initial conditions W0 = (U0,−Ω−1∂tU0).
3.4. Derivation of the NLS approximation
The Bloch transform is the main tool to justify the NLS approximation on a periodic quantum
graph. Here, we will transfer the approach from Chapter 2 to the cKG equation (3.4), write
the system in Bloch space and use this representation to derive the amplitude equation for this
problem.
For an overview on the basic properties of the Bloch transform on periodic quantum graphs, we
refer to Section 2.4.
3.4.1. The system in Bloch space
First, we apply the Bloch transform T to (3.4) and get
∂2t U˜(t, `, x) = −L˜(`)U˜(t, `, x)− (U˜ ? U˜ ? U˜)(t, `, x), (3.20)
with the linear operator L˜(`) = −(∂x + i`)2 + 1 defined in Section 3.2.1 and the function
U˜(t, `, x) = (u˜0, u˜+, u˜−)(t, `, x) satisfying the continuation conditions
U˜(t, `, x) = U˜(t, `, x+ 2pi) and U˜(t, `, x) = U˜(t, `+ 1, x)eix.
The convolution integrals in (3.20) are defined componentwise:
U˜ ? U˜ ? U˜ = (u˜0 ? u˜0 ? u˜0, u˜+ ? u˜+ ? u˜+, u˜− ? u˜− ? u˜−) .
As in Section 2.4.2, every function u˜j(t, `, ·) with j ∈ {0,+,−} has support in I0,j due to the
fact that the equality
T (uj)(`, x) = T (χjuj)(`, x) = χj(x)(T uj)(`, x), j ∈ {0,+,−}
holds for the periodic cut-off functions χj stated in (2.35).
Next, we consider the underlying function space for (3.20) and recall the definition of the Bloch
space
H˜2 = {U˜ ∈ L2(T1, L2Γ) : u˜j ∈ L2(T1, H2(I0,j)), j ∈ {0,+,−}, (3.7)− (3.8) is satisfied},
which is equipped with the norm
‖U˜‖H˜2 =
(∫ 1/2
−1/2
(
‖u˜0(`, ·)‖2H2(I0,0) + ‖u˜+(`, ·)‖2H2(I0,+) + ‖u˜−(`, ·)‖2H2(I0,−)
)
d`
)1/2
.
With the same arguments as in the previous chapter, the space H˜2 is the domain of definition of
the operator L˜(`) in L2(T1, L2Γ) and Lemma 2.4.2 guarantees that the Bloch transform T is an
isomorphism between the spaces H˜2 and H2, in which the initial value problem for the system
(3.4) is locally well-posed.
37
3.4.2. Derivation of the effective amplitude equation
We recover the ansatz (3.9) and (3.10) and derive an effective amplitude equation of the type
(3.11) in Bloch space. Therefore, we first decompose the solution of the problem (3.20) into two
parts and write
U˜(t, `, x) = V˜ (t, `)f (m0)(`, x) + U˜⊥(t, `, x), (3.21)
where the orthogonality condition 〈f (m0)(`, ·), U˜⊥(t, `, ·)〉L2Γ = 0 is again used for uniqueness of
the decomposition. We find
∂2t V˜ (t, `) = −(ω(m0)(`))2V˜ (t, `)−NV (V˜ , U˜⊥)(t, `), (3.22)
∂2t U˜
⊥(t, `, x) = −L˜(`)U˜⊥(t, `, x)−N⊥(V˜ , U˜⊥)(t, `, x), (3.23)
where
NV (V˜ , U˜
⊥)(t, `) = 〈(U˜ ? U˜ ? U˜)(t, `, ·), f (m0)(`, ·)〉L2Γ ,
and
N⊥(V˜ , U˜⊥)(t, `, x) = (U˜ ? U˜ ? U˜)(t, `, x)−NV (V˜ , U˜⊥)(t, `)f (m0)(`, x).
Since the original system has no quadratic terms in the nonlinearity, it is sufficent to evaluate
equation (3.22) at U˜⊥ = 0 as in Section 2.5.1. In the case of quadratic nonlinearities, the terms
of this order have to be first eliminated by near identity transformations, cf. [10]. For a cubic
nonlinearity, we directly get
NV (V˜ , U˜
⊥)(t, `) =
∫
T1
∫
T1
β(`, `1, `2, `− `1 − `2)
× V˜ (t, `1)V˜ (t, `2)V˜ (t, `− `1 − `2)d`1d`2 +NV,rest(V˜ , U˜⊥)(t, `)
(3.24)
where NV,rest(V˜ , 0) = 0 and the kernel β is given by
β(`, `1, `2, `1 + `2 − `) :=
〈
f (m0)(`1, ·)f (m0)(`2, ·)f (m0)(`− `1 − `2, ·), f (m0)(`, ·)
〉
L2Γ
. (3.25)
In order to derive a NLS equation as effective amplitude equation, we now insert the ansatz
V˜app(t, `) = A˜1
(
ε2t,
`− `0
ε
)
E1(t, `) + A˜−1
(
ε2t,
`+ `0
ε
)
E−1(t, `), (3.26)
with
E±1(t, `) := e±iω
(m0)(`0)te±i∂`ω
(m0)(`0)(`∓`0)t
into (3.22) and obtain in the leading order ε2E1 the equation
i∂T A˜1(T, ξ) = −1
2
∂2`ω
(m0)(`0)ξ
2A˜1(T, ξ)
− ν
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜1(T, ξ1)A˜1(T, ξ2)A˜−1(T, ξ − ξ1 − ξ2)dξ1dξ2,
(3.27)
38
where ` = `0 +εξ, T = ε
2t and ν = 3β(`0, `0, `0,−`0)/2ω(m0)(`0). We also find additional terms
of the formal order O(ε2) which do not cancel each other. A detailed calculation of these terms
is given in Appendix A.2.
By taking the limit ε→ 0, the amplitude equation (3.27) yields to the equation
i∂T Â1(T, ξ) +
1
2
∂2`ω
(m0)(`0)ξ
2Â1(T, ξ) + ν(Â1 ∗ Â1 ∗ Â−1)(T, ξ) = 0 (3.28)
in Fourier space which corresponds to the amplitude equation (3.11) in physical space. This will
be made rigorous in Section 3.5.2.
3.5. The improved approximation and estimates for the residual
terms
As we mentioned above, the ansatz (3.26) yields to remaining terms in the equation (3.22) which
are of the formal order O(ε2) in Bloch space and furthermore produces terms of this order in the
second equation (3.23), too. Since we want to bound the error with a Gronwall argument as in
Chapter 2, we again need the residual to be of the formal order O(ε3) in Bloch space. Therefore,
we add higher order terms to the approximation to cancel out the terms of the formal order O(ε2)
in (3.22)-(3.23) and then estimate the residual of this improved ansatz.
3.5.1. The improved approximation
In order to improve the ansatz (3.26), we define the terms of the decomposition (3.21) in the
following way:
V˜app(t, `) = V˜app,1(t, `) + V˜app,−1(t, `) + V˜app,3(t, `) + V˜app,−3(t, `)
= A˜1
(
ε2t,
`− `0
ε
)
E1(t, `) + A˜−1
(
ε2t,
`+ `0
ε
)
E−1(t, `)
+ ε2A˜3
(
ε2t,
`− 3`0
ε
)
E3(t, `) + ε2A˜−3
(
ε2t,
`+ 3`0
ε
)
E−3(t, `)
(3.29)
and
U˜⊥app(t, `, x) = U˜
⊥
app,1(t, `, x) + U˜
⊥
app,−1(t, `, x) + U˜
⊥
app,3(t, `, x) + U˜
⊥
app,−3(t, `, x)
= ε2B˜1
(
ε2t,
`− `0
ε
, x
)
E1(t, `) + ε2B˜−1
(
ε2t,
`+ `0
ε
, x
)
E−1(t, `)
+ ε2B˜3
(
ε2t,
`− 3`0
ε
, x
)
E3(t, `) + ε2B˜−3
(
ε2t,
`+ 3`0
ε
, x
)
E−3(t, `).
(3.30)
Remark 3.5.1. In contrast to the analysis in Section 2.5.2, we need to improve the ansatz
V˜app(t, `) as well. This is necessary because we have to eliminate the terms concentrated in
neighborhoods of the Bloch numbers −`0 and ±3`0 which also appear in (3.22) for the simple
approximation (3.26), cf. Appendix A.2.
Inserting (3.29) and (3.30) into the system (3.22)-(3.23), we set the coefficients in the leading
order ε2Ej with
Ej(t, `) := ejiω
(m0)(`0)teji∂`ω
(m0)(`0)(`−j`0)t
39
for j = ±1,±3 to zero and thus obtain the following equalities:
−9(ω(m0)(`0))2A˜3
(
ε2t, ξ3
)
= −(ω(m0)(3`0))2A˜3
(
ε2t, ξ3
)
− β(`0, `0, `0, `0)(A˜1 ? A˜1 ? A˜1)(ε2t, ξ3),
−(ω(m0)(`0))2B˜1
(
ε2t, ξ1, x
)
= −L˜(`0)B˜1
(
ε2t, ξ1, x
)
− ε−2 (E1(t, `))−1N⊥1 (V˜app, 0)(t, `, x),
−9(ω(m0)(`0))2B˜3
(
ε2t, ξ3, x
)
= −L˜(3`0)B˜3
(
ε2t, ξ3, x
)
− ε−2 (E3(t, `))−1N⊥3 (V˜app, 0)(t, `, x),
(3.31)
where ξ3 = ε
−1(`− 3`0), ξ1 = ε−1(`− `0) and the nonlinear terms are given by
N⊥1 (V˜app, 0)(t, `, x) = 3(V˜app,1 ? V˜app,1 ? V˜app,−1)(t, `, x)
− 3ε2β(`0, `0, `0,−`0)(A˜1 ? A˜1 ? A˜−1)(ε2t, ξ1)f (m0)(`, x)E1(t, `)
and
N⊥3 (V˜app, 0)(t, `, x) = (V˜app,1 ? V˜app,1 ? V˜app,1)(t, `, x)
− ε2β(`0, `0, `0, `0)(A˜1 ? A˜1 ? A˜1)(ε2t, ξ3)f (m0)(`, x)E3(t, `).
For the functions A˜−3, B˜−1 and B˜−3 we find the associated complex conjugated equations to
the ones in (3.31).
Note that the prefactor ε−2 of the terms N⊥1 and N⊥3 cancels with the factor ε2 coming from
the two-times convolution and therefore all terms in the respective equations (3.31) are of the
same formal order. These equations can be solved by using the implicit function theorem, if the
invertibility conditions
9(ω(m0)(`0))
2 6= (ω(m0)(3`0))2,
−(ω(m0)(`0))2 6∈ σ(−L˜(`0))|{f (m0)(`0,·)}⊥ ,
−9(ω(m0)(`0))2 6∈ σ(−L˜(3`0))|{f (m0)(3`0,·)}⊥ .
hold. These conditions are satisfied due to the non-resonance conditions (3.12) and (3.13) of
Theorem 3.2.2. With the same arguments we solve the complex conjugate equations for A˜−3,
B˜−1 and B˜−3.
Hence, by inserting the approximation (V˜app, U˜
⊥
app) defined by the improved ansatz (3.29) and
(3.30) into the system (3.22)-(3.23), we get rid of all the remaining terms of the formal order
O(ε2) in the residual.
3.5.2. From Fourier space to Bloch space
Using the arguments from Section 2.5.3 to link Fourier analysis and Bloch wave analysis, we
define for every j = ±1,±3 the amplitude function in Bloch space by
A˜j(T, ε
−1(`− j`0)) = χ˜j`0(`)Âj(T, ε−1(`− j`0)), (3.32)
where we write the cutoff functions χ˜j`0 as
χ˜j`0(`) =
{
1, `− j`0 ∈ [−δ, δ] ,
0, otherwise,
40
with δ > 0 being sufficiently small but independent of the small parameter ε. The condition
A˜j
(
T, ε−1(`+ 1− j`0)
)
= A˜j
(
T, ε−1(`− j`0)
)
, ` ∈ R,
then extends A˜j(T, ε
−1(`− j`0)) periodically in ` over R.
As mentioned already in the last chapter, the assumption Aj ∈ C(R, H3(R)) yields to the fact that
the corresponding Fourier transform Âj satisfies Âjρ0,1,3 ∈ L2(R) and therefore Âjρ0,1,2 ∈ L1(R).
By the construction of the Bloch functions in (3.32) and using the scaled weights ρ∗,ε,∗, we thus
get for the leading order terms in V˜app that
V˜app,1f
(m0)ρ`0,ε,3 ∈ H˜2 and V˜app,−1f (m0)ρ−`0,ε,3 ∈ H˜2
and that these weighted terms are of the formal order O(ε1/2) in H˜2 due to the scaling properties
of the L2-norm. In order to avoid losing the factor ε1/2, we recall the L1-based space
C˜2 = {U˜ ∈ L1(T1, L2Γ) : u˜j ∈ L1(T1, H2(I0,j)), j ∈ {0,+,−}, (3.7)− (3.8) is satisfied},
equipped with the norm
‖U˜‖C˜2 =
∫ 1/2
−1/2
(
‖u˜0(`, ·)‖H2(I0,0) + ‖u˜+(`, ·)‖H2(I0,+) + ‖u˜−(`, ·)‖H2(I0,−)
)
d`
from Section 2.5.3 and obtain that the weighted functions
V˜app,1f
(m0)ρ`0,ε,2 ∈ C˜2 and V˜app,−1f (m0)ρ−`0,ε,2 ∈ C˜2
are of the formal order O(ε) in C˜2. With the same idea we get that the higher-order terms
V˜app,±3f (m0)ρ±3`0,ε,3 ∈ H˜2 are of the formal order O(ε5/2) in H˜2 and V˜app,±3f (m0)ρ±3`0,ε,2 ∈ C˜2
are of the formal order O(ε3) in C˜2.
Next, we study the second part U˜⊥app of the decomposition (3.21) and see that the terms(
Ej(t, ·))−1N⊥j (V˜app, 0)(t, ·, ·)ρj`0,ε,2(·) ∈ H˜2
are of the formal order O(ε5/2) in H˜2 and of the formal order O(ε3) in C˜2 due to Young‘s inequality
‖(V˜ ? W˜ )ρj`0,ε,2‖H˜2 ≤ C‖V˜ ρj`0,ε,2‖C˜2‖W˜ρj`0,ε,2‖H˜2
for weighted functions. The supports of the nonlinear terms in (3.31) and in the respective
complex conjugate system are given by
supp
((
Ej(t, ·))−1N⊥j (V˜app, 0)(t, ·, ·)) ⊂ [j`0 − 3δ, j`0 + 3δ].
and similar to (2.45), we define
B˜j
(
ε2t, ξj , x
)
= (−L˜(`) + (jω(m0)(`0))2I)−1ε−2
(
Ej(t, `)
)−1
N⊥j (V˜app, 0)(t, `, x), (3.33)
where ξj = ε
−1(` − j`0) for every j = ±1,±3. The inverse operators on the right hand side of
(3.33) exist due to the non-resonance conditions (3.12) and (3.13) for δ > 0 sufficiently small.
Therefore, we obtain that the weighted functions U˜⊥app,jρj`0,ε,2 ∈ H˜2 are of the formal order
O(ε5/2) in H˜2 and U˜⊥app,jρj`0,ε,1 ∈ C˜2 are of the formal order O(ε3) in C˜2.
41
The estimates for the terms V˜app,j and U˜
⊥
app,j now yield formally to the improved ansatz
εΨ˜(t, `, x) = V˜app(t, `)f
(m0)(`, x) + U˜⊥app(t, `, x), (3.34)
with V˜app and U˜
⊥
app defined in (3.29) and (3.30), where the amplitude function Â1 fulfills the
NLS equation (3.28) in Fourier space and Â−1 the complex conjugate equation respectively.
Moreover, the improved approximation εΨ = T −1(εΨ˜) in physical space satisfies by construction
the Kirchhoff boundary conditions (3.2) - (3.3).
3.5.3. Estimates in Bloch space
Here we show that residual
R˜es(U˜)(t, `, x) = −∂2t U˜(t, `, x)− L˜(`)U˜(t, `, x)− (U˜ ? U˜ ? U˜)(t, `, x)
of the evolution problem (3.20) can be estimated in H˜2 to be of the formal order O(ε7/2) if we
approximate U˜(t, `, x) with the ansatz (3.34). We also get the same result in physical space.
Lemma 3.5.2. Let A ∈ C([0, T0], H3) be a solution of the amplitude equation (3.11) for some
T0 > 0. Then, there is a positive ε-independent constant CRes that only depends on the norm of
the solution A such that
sup
t∈[0,T0/ε2]
‖R˜es(εΨ˜)‖H˜2 ≤ CResε7/2. (3.35)
or equivalently,
sup
t∈[0,T0/ε2]
‖Res(εΨ)‖H2 ≤ CResε7/2. (3.36)
Proof. The proof is straightforward and follows almost line for line the one for Lemma 2.5.5 in
Chapter 2 using the decompositions
V˜app(t, `) = V˜app,1(t, `) + V˜app,−1(t, `) + V˜app,3(t, `) + V˜app,−3(t, `)
and
U˜⊥app(t, `, x) = U˜
⊥
app,1(t, `, x) + U˜
⊥
app,−1(t, `, x) + U˜
⊥
app,3(t, `, x) + U˜
⊥
app,−3(t, `, x).
Moreover, we obtain a similar result as Lemma 2.5.7 for the difference between the two approxi-
mations εΨnls, given by (3.10), and εΨ given by (3.34) in Bloch space.
Lemma 3.5.3. Let A ∈ C([0, T0], H3) be a solution of the amplitude equation (3.11) for some
T0 > 0. Then, there exist positive ε-independent constants C and Cψ that only depend on the
norm of the solution A such that
sup
t∈[0,T0/ε2]
‖εΨ˜‖C˜2 ≤ CΨε (3.37)
and
sup
t∈[0,T0/ε2]
‖εΨ− εΨnls‖L∞ ≤ Cε3/2. (3.38)
Proof. Here we refer to the proof of Lemma 2.5.7 in Chapter 2.
42
3.6. Estimates for the error term
In this section we want to complete the proof of Theorem 3.2.2 which is based on an energy
estimate and a simple approximation of Gronwall‘s inequality.
Therefore, we write again the solution U of the equation as a sum of the approximation term εΨ
and the error term ε3/2R, i.e.,
U = εΨ + ε3/2R.
Inserting this decomposition into (3.4) we obtain the equation
∂2tR = −LR+G(Ψ, R) (3.39)
with the linear operator L = −∂2x + 1 and the nonlinear terms
G(Ψ, R) = ε−3/2Res(εΨ) + 3ε2Ψ2R+ 3ε5/2ΨR2 + ε3R3. (3.40)
As in Section 2.7, the product terms in the definition of G(Ψ, R) are understood componentwise
with R = (r0, r+, r−) and Ψ = (ψ0, ψ+, ψ−). We recall Young‘s inequality
‖V˜ ? W˜‖H˜2 ≤ ‖V˜ ‖C˜2‖W˜‖H˜2
as it is defined in Section 2.5.3 and thus get the bound
‖ΨR‖H2 ≤ C‖Ψ˜R˜‖H˜2 ≤ C‖Ψ˜‖C˜2‖R˜‖H˜2 ≤ CCΨ‖R˜‖H˜2 ≤ C2CΨ‖R‖H2 ,
where the constant CΨ is defined in Lemma 3.5.3. Hence, for each term of (3.40) we find
‖ε−3/2Res(εΨ)‖H2 ≤ CResε2,
‖3ε2Ψ2R‖H2 ≤ 3C1ε2‖R‖H2 ,
‖3ε5/2ΨR2‖H2 ≤ 3C1ε5/2‖R‖2H2 ,
‖ε3R3‖H2 ≤ C1ε3‖R‖3H2 ,
where we used (3.36) and Lemma 2.3.1. Here C1 is a constant independent of ‖R‖H2 and the
small parameter ε > 0. Therefore, we get
‖G(Ψ, R)‖H2 ≤ CResε2 + 3C1ε2‖R‖H2 + 3C1ε5/2‖R‖2H2 + C1ε3‖R‖3H2 .
In order to obtain an energy estimate, we multiply (3.39) with ∂tLR, take the scalar product in
L2 and obtain
〈∂tLR, ∂2tR〉L2 = −〈∂tLR,LR〉L2 + 〈∂tLR,G(Ψ, R)〉L2 (3.41)
We rewrite the left-hand side of (3.41) to
〈∂tLR, ∂2tR〉L2 = ∂t〈∂tLR, ∂tR〉L2 − 〈∂2t LR, ∂tR〉L2
and with
−〈∂2t LR, ∂tR〉L2 = −〈L∂2tR, ∂tR〉L2
= −〈L(−LR+G(Ψ, R)), ∂tR〉L2
= 〈LR, ∂tLR〉L2 − 〈G(Ψ, R), ∂tLR〉L2 ,
43
the equation (3.41) changes to
∂t〈∂tLR, ∂tR〉L2 + ∂t〈LR,LR〉L2 = 〈G(Ψ, R), ∂tLR〉L2 + 〈∂tLR,G(Ψ, R)〉L2 .
By introducing the representation L = Ω2, we have
∂t〈∂tΩR, ∂tΩR〉L2 + ∂t〈LR,LR〉L2 = 2 |〈∂tΩR,ΩG(Ψ, R)〉L2 | . (3.42)
Since 〈LR,LR〉L2 = ‖LR‖L2 = ‖R‖H2 and with the estimate
|〈∂tΩR,ΩG(Ψ, R)〉L2 | ≤ ‖∂tΩR‖L2‖G(Ψ, R)‖H2 ,
equation (3.42) is given by
∂t
(‖∂tΩR‖2L2 + ‖R‖2H2) ≤ 2‖∂tΩR‖L2‖G(Ψ, R)‖H2 . (3.43)
Now we define the energy ER = ‖∂tΩR‖2L2 + ‖R‖2H2 , insert (3.41) into (3.43) and thus find
∂tER ≤ 2‖∂tΩR‖L2
(
CResε
2 + 3C1ε
2‖R‖H2 + 3C1ε5/2‖R‖2H2 + C1ε3‖R‖3H2
)
≤ 2E1/2R
(
CResε
2 + 3C1ε
2E
1/2
R + 3C1ε
5/2ER + C1ε
3E
3/2
R
)
.
Using the estimate E
1/2
R ≤ 1 + ER, we finally get
∂tER ≤ 2CResε2 + 2(3C1 + CRes)ε2ER + 6C1ε5/2E3/2R + 2C1ε3E2R. (3.44)
For simplicity, we assume R(0) = ∂tR(0) = 0 and therefore ER(0) = 0. Using Gronwall’s
inequality for (3.44) allows us to estimate the energy ER on the time scale T = ε
2t for T ∈ [0, T0]
by
sup
t∈[0,T0/ε2]
ER ≤ 2CRese(6C1+2CRes+1)T0 =: M
for all ε ∈ (0, ε0), if ε0 > 0 is chosen so small that 6C1ε1/2E1/2R + 2C1εER ≤ 1. This energy
estimate yields to the bound
sup
t∈[0,T0/ε2]
‖R‖H2 ≤M1/2,
and by using Sobolev‘s embedding theorem, bound (3.38) and the decomposition (3.39), we
complete the proof of Theorem 3.2.2.
44
4. Approximation of a two-dimensional
Gross-Pitaevskii equation with a periodic
potential
In this chapter, we change the topic from quantum graphs to a different kind of approximation
problem for a NLS equation. Here we consider the two-dimensional Gross-Pitaevskii (GP) equation
with a periodic potential of the formal order O(ε−2) in one space direction on the homogenous
space and justify the validity of the discrete nonlinear Schro¨dinger (dNLS) equation as an effective
amplitude equation in the asymptotic limit ε→ 0.
We start with a short introduction into the model and define the function space we need to prove
our approximation result in Section 4.1. In Section 4.2, we give a short recap on the spectral
theory of the one-dimensional Schro¨dinger operator with a periodic potential and the theory of
Wannier function decomposition. This part is based on the more detailed introductions to this
topic in Section 2.1. of [29] and [34]. We also shortly discuss the spectral properties of the har-
monic oscillator problem in one space dimension. The main theorem is formulated in Section 4.3.
This result can be interpreted as an extension of [32] to the two-dimensional case. In Section 4.4,
we use a multiple scaling expansion to derive the effective amplitude equation. With an improved
approximation we compute the residual and estimate the nonlinear terms. Local existence and
uniqueness results for the error equation and the derived dNLS equation are stated in Section 4.5.
An energy estimate using Gronwall’s inequality, similar to the one in the updated version of [32],
completes the proof in Section 4.6.
Notation: According to Chapters 2 and 3, we use the standard notations for the Sobolev space
Hs(R2) and the Lebesgue space Lp(R2) for s ≥ 0 and p ≥ 1. We also denote with l1(Z) and
l2(Z) the complex-valued sequence spaces equipped with the norms ‖~a‖l1 =
∑
m∈Z |am| and
‖~a‖l2 =
(∑
m∈Z |am|2
)1/2
.
4.1. The model
The GP equation is well-known in physics literature [35] as a mean-field model of the Bose-Einstein
condensation. Looking at a condensate in an optical gap, a periodic structure is very common
in experimental and in theoretical research, see for example [23] and [24]. From a mathematical
point of view, we have to deal with a nonlinear Schro¨dinger equation with an external periodic
potential. The book [29] gives a good overview on the analysis of such systems.
Here, we consider the GP equation in two space dimensions,
i∂tu = −∆u+ V (r)u+ σ|u|2u, t ∈ R+, r ∈ R2, (4.1)
where u(t, r) : R+ × R2 → C decays to zero sufficiently fast as |r| → ∞ and the potential
V (r) = Vx(x)+Vy(y) is given by a piecewise-constant periodic potential Vx(x) in the x-direction
45
and a harmonic oscillator potential of the form
Vy(y) = y
2 (4.2)
in the y-direction. In particular, the potential Vx is bounded, real-valued, 2pi-periodic and defined
by
Vx(x) =
{
ε−2 x ∈ (0, a) mod(2pi),
0 x ∈ (a, 2pi) mod(2pi), (4.3)
where a ∈ (0, 2pi) is fixed and 0 < ε 1. Both potentials are shown in Figure 4.1. The parameter
σ = ±1 is normalized by convenience.
Figure 4.1.: The potential function Vx(x) with a = pi and ε = 0.2 (left) and the potential function
Vy(y) = y
2 (right).
Our goal in this chapter is to derive an effective equation in the tight-binding limit for the
description of the dynamics of the wave function u, in detail we reduce the continuous equation
(4.1) to a nonlinear lattice problem, cf. [7, 41]. In the one-dimensional case, this has recently
been done by Pelinovsky and Schneider in [32], where a GP equation with the periodic potential
(4.3) is approximated by a dNLS equation in the asymptotic limit ε → 0. More results on the
reduction of a one-dimensional GP equation to a dNLS equation can be found in [6, 8, 34].
In our two-dimensional problem, the periodicity of the potential is restricted to one space dimen-
sion and with the unbounded harmonic potential (4.2) in y-direction, we obtain in the slow time
scale T = µt with µ = εe−
a
ε an effective equation of the form
i∂Tam = α(am−1 + am+1) + σβ |am|2 am, (4.4)
where the sequence {am(T )}m∈Z = ~a(T ) represents a small-amplitude solution of (4.1) with
amplitude functions am(T ) located in the m-th potential well. The numerical coefficients α and
β are ε-independent and will be explained later on.
In order to justify the effective equation (4.4) with the methods from [32], we need a suitable
Banach space for our error estimates. As in the one-dimensional case, we have ‖Vx‖C0b → ∞ in
the limit ε → 0 and since Vx ≥ 0 for all x ∈ R, it is convenient to work in the function space
H1,2(R2) equipped with the norm
‖u‖2H1,2 = 〈(Lx + I)u, u〉L2 + 〈(Ly + I)2u, u〉L2 + 〈u, u〉L2 , (4.5)
where the occuring scalar products are defined by the linear Schro¨dinger operators Lx = −∂2x+Vx
and Ly = −∂2y + Vy. This space also helps us to control the nonlinearity of the original problem
46
(4.1) in two space dimensions. The norm representation (4.5) makes it reasonable to use an
approximation ansatz which decomposes the original solutions u(t, x) of the GP equation (4.1)
by the eigenfunctions of the linear problems corresponding to the one-dimensional operators Lx
and Ly. Therefore, we first study the spectrum of these operators.
Remark 4.1.1. Note that it is not possible to use the spaceH1(R2) with ‖u‖2H1 = 〈(L+I)u, u〉L2
in our analysis due to Sobolev’s embedding theorem in two space dimensions among other reasons.
Also the function space H2(R2) with ‖u‖2H2 = 〈(L+ I)2u, u〉L2 is inappropriate for this problem
because we cannot prove that this space is closed under pointwise multiplication with an ε-
independent constant for a potential V (r) of the formal order O(ε−2). Here the linear operator
L is defined by L = Lx + Ly = −∆ + V (r).
4.2. The spectral situation
Let us consider the spectral problem
Lu = λu (4.6)
of the two-dimensional linear Schro¨dinger operator with the potential V = Vx+Vy defined in (4.3)
and (4.2). By using the product ansatz u(r) = ϕ(x)ψ(y), the equation (4.6) can be seperated
as follows:
Lxϕ = Eϕ (4.7)
and
Lyψ = ωψ. (4.8)
Now we look at the spectral properties of the linear operators Lx = −∂2x+Vx and Ly = −∂2y +Vy
seperately.
4.2.1. Wannier function decomposition in one dimension
First, we focus on the one-dimensional eigenvalue problem (4.7). According to the Bloch wave
ansatz (2.10) in Section 2.2.2, we set
ϕ(x) = ei`xφ(`, x), `, x ∈ R,
where φ(`, ·) is a 2pi-periodic function for every ` ∈ R. These functions also satisfy the contin-
uation conditions (2.11) and therefore we can restrict the definition of φ(`, x) to x ∈ T2pi and
` ∈ T1. With this ansatz, φ(`, ·) is a solution to the eigenvalue problem
−(∂x + i`)2φ(`, x) + V (x)φ(`, x) = E(`)φ(`, x)
with the linear operator L˜x(`, x) := −(∂x + i`)2 + V (x).
Now we obtain an elementary result on the Schro¨dinger operator L˜x similar to Lemma 2.2.2.
Lemma 4.2.1. For fixed ` ∈ T1, the operator L˜x(`, x) is a self-adjoint, positive semi-definite
operator in L2(T2pi).
47
Proof. For every φ(`, ·), ψ(`, ·) ∈ H2(T2pi) and every ` ∈ T1, the relation
〈−(∂x + i`)2φ, ψ〉L2(T2pi) =
∫ 2pi
0
(∂x + i`)φ(`, x) · (∂x + i`)ψ(`, x)dx
− [∂xφ(`, 2pi) + i`φ(`, 2pi)]ψ(`, 2pi)
+ [∂xφ(`, 0) + i`φ(`, 0)]ψ(`, 0)
=
∫ 2pi
0
(∂x + i`)φ(`, x) · (∂x + i`)ψ(`, x)dx
holds by using the continuation conditions φ(`, 0) = φ(`, 2pi) and ψ(`, 0) = ψ(`, 2pi). As it is
shown in Lemma 2.2.2, we apply the integration by parts a second time and this leads to
〈−(∂x + i`)2φ, ψ〉L2(T2pi) = 〈φ,−(∂x + i`)2ψ〉L2(T2pi).
Because the potential function V (x) is real-valued, we directly obtain
〈L˜x(`, x)φ, ψ〉L2(T2pi) = 〈φ, L˜x(`, x)ψ〉L2(T2pi),
and thus the operator L˜x(`, x) is self-adjoint.
With the same conclusion as in the proof of Lemma 2.2.2, the operator L˜x(`, x) is positive
semi-definite.
Referring to the argumentation in Section 2.2.2 and Section 3.2.1, by Lemma 4.2.1 and the spectral
theorem for self-adjoint operators with compact resolvent, cf. [36], for each ` ∈ T1 there exists
a Schauder base {φn(`, ·)}n∈N of L2(T2pi) consisting of eigenfunctions of L˜x(`, x) with positive
eigenvalues {En(`)}n∈N ordered as En(`) ≤ En+1(`). Hence, we define the corresponding Bloch
wave function in L2(R) to the eigenvalue En(`) by φ˜n(`, x) := ei`xφn(`, x) and each of these
pairs solves the spectral problem
Lxφ˜n(`, x) = En(`)φ˜n(`, x) (4.9)
for every n ∈ N. By (2.11), the condition φ˜n(`, x + 2pi) = φ˜n(`, x)ei2pi` holds for all x ∈ R,
and also all Bloch wave functions φ˜n(`, ·) satisfy the following orthogonality and normalization
relation:
〈φ˜n(`, ·), φ˜n′(`′, ·)〉L2(R) = δn,n′δ(l − l′), n, n′ ∈ N, `, `′ ∈ T1. (4.10)
In order to normalize the phase factors of the Bloch wave functions as in [34], we set φ˜n(`, x) =
φ˜n(−`, x) as eigenfunctions for the eigenvalues En(`) = En(`) = En(−`). Note that in contrast
to Section 2.2.2 and Section 3.2.1, we use subscripts for the count of the spectral bands En. In
our problem, it is more useful to use the Wannier function decomposition to approximate the
continuous PDE (4.1) with the lattice equation (4.4). We follow very closely the well-written
introduction to this topic in [32] and for a more detailed approach we again refer to [29] and [34].
The band function En(`) and the Bloch wave function φn(`, x) introduced above are 1-periodic
with respect to ` ∈ T1 for any n ∈ N. Therefore, we can represent them by the Fourier series
En(`) =
∑
m∈Z
Ên,me
i2pim`, ` ∈ T1, (4.11)
48
and
φ˜n(`, x) =
∑
m∈Z
φ̂n,m(x)e
i2pim`, ` ∈ T1, x ∈ R. (4.12)
The Fourier coefficients in (4.11) and (4.12) are defined by the integrals
Ên,m =
∫
T1
En(`)e
−i2pim`dl, m ∈ Z,
and
φ̂n,m(x) =
∫
T1
φ˜n(`, x)e
−i2pim`dl, m ∈ Z, x ∈ R,
and satisfy the constraints
Ên,m = Ên,−m = Ên,−m, φ̂n,m(x) = φ̂n,m(x), ∀m ∈ Z, ∀n ∈ N, ∀x ∈ R.
Because of the quasi-periodicity φ˜n(`, x + 2pi) = φ˜n(`, x)e
i2pi` for any x ∈ R and ` ∈ T1, we
obtain the property
φ̂n,m(x) = φ̂n,m−1(x− 2pi) = φ̂n,0(x− 2pim), ∀m ∈ Z, ∀n ∈ N, ∀x ∈ R.
and the so-defined real-valued functions φ̂n,m are called Wannier functions.
Although the Wannier functions are no eigenfunctions of the Schro¨dinger operator Lx, we can
substitute the Fourier series representations (4.11) and (4.12) into the one-dimensional linear
problem (4.9) for a fixed n ∈ N. Thereby, in every spectral band En the set of functions {φ̂n,m}m∈Z
satisfies the system
Lxφ̂n,m =
∑
m′∈Z
Ên,m−m′ φ̂n,m′ , ∀m ∈ Z, (4.13)
with the corresponding coefficients {Ên,m}m∈Z. Orthogonality and normalization of Wannier
functions, given by
〈φ̂n,m(·), φ̂n′,m′(·)〉L2(R) = δn,n′δm,m′ , n, n′ ∈ N, m,m′ ∈ Z, (4.14)
directly follows from the relation (4.10) for Bloch functions.
We complete this part with the following two lemmas from [34], which state important properties
of the band functions Ên(`) and the Wannier functions φ̂n,m(x) of the linear one-dimensional
spectral problem (4.7).
Lemma 4.2.2. Let Vx be given by (4.3) and µ = εe−a/ε. For any fixed n0 ∈ N, there exist
ε0, Cs, E0, C
±
1 , C2, C0, Cm > 0, such that, for any ε ∈ [0, ε0), the band functions and the Wannier
functions of the operator Lx = −∂2x + Vx satisfy the properties.
(i) (band separation)
min
n∈N\{n0}
inf
`∈T1
|En(`)− Ên0,0| ≥ Cs (4.15)
49
(ii) (band boundedness)
|Eˆn0,0| ≤ E0 (4.16)
(iii) (tight-binding approximation)
C−1 µ ≤ |Eˆn0,1| ≤ C+1 µ,
|Eˆn0,m| ≤ C2µ2 for m ≥ 2
(4.17)
(iv) (compact support)
|φ̂n0,0(x)− φ̂0| ≤ C0ε, ∀x ∈ [0, 2pi], (4.18)
where
φ̂0(x) =
{
0, ∀x ∈ [0, a],√
2√
2pi−a sin
(
pin0(2pi−x)
2pi−a
)
, ∀x ∈ [a, 2pi].
(v) (exponential decay)
|φ̂n0,0(x)| ≤ Cmµm, ∀x ∈ [−2pim,−2pi(m− 1)] ∪ [2pim, 2pi(m+ 1)],m ∈ N (4.19)
Proof. The proof of (i)-(iii) is given in Appendix B and (iv)-(v) are shown in Appendix C of
[34].
Lemma 4.2.3. Let En be the invariant closed subspace of L2(R) associated with the n-th spectral
band and assume that En∩En′ for a fixed n ∈ N and all n′ 6= n. Then, 〈φ̂n,m, φ̂n,m′〉L2(R) = δm,m′
for any m,m′ ∈ Z and there exists constants ηn > 0 and Cn > 0, such that
|φ̂n,m(x)| ≤ Cne−ηn|x−2pim| (4.20)
Proof. The orthogonality relation for the Wannier functions is already mentioned in (4.14). The
exponential decay bound (4.20) follows from complex integration. For a detailed calculation we
refer to the proof of Proposition 2 in [34].
Remark 4.2.4. The exponential decay (4.20) is proven in [34] for the spectrum of an operator
L = −∂2x + V (x) consisting of the union of disjoint spectral bands. However, for the proof of
Lemma 4.2.3 we do not need the assumption that all spectral bands are disjoint. It is sufficient that
the particular n-th spectral band is disjoint from the other spectral bands of the linear operator.
According to property (4.15), this condition is satisfied for the spectrum of the operator Lx in
the asymptotic limit ε→ 0, cf. [32].
4.2.2. Properties of the harmonic oscillator
Here we give a short recap on the well-known spectral problem (4.8). The eigenvalue problem
−∂2yψ + Vyψ = ωψ
has an infinite set of isolated simple eigenvalues ωj = 1 + 2j with j ∈ N0 and the L2-normalized
orthogonal eigenfunctions of this system are given by
ψj(y) =
1
(pi)1/4(2jj!)1/2
Hj (y) e
−y2/2,
where the functions Hj represent the Hermite polynomials. For a more detailed derivation and
calculation of the spectrum of the one-dimensional harmonic oscillator, we refer to classical text-
books in quantum mechanics like [12] and [27].
50
4.3. Main result
We represent an approximate solution of the given GP equation (4.1) by the formal asymptotic
expansion
u(t, r) = µ1/2Ψ0(t, r) + higher-order terms,
with
µ1/2Ψ0(t, r) = µ
1/2
∑
m∈Z
am(T )φ̂n,m(x)ψj(y)e
−i(Eˆn,0+ωj)t
= µ1/2ϕ0(T, r)e
−i(Eˆn,0+ωj)t
= µ1/2ϕ0(T, r)E(t),
(4.21)
where T = µt with µ = εe−a/ε  1 is the slow time variable and r = (x, y) ∈ R2. The set
of Wannier functions {φ̂n,m}m∈Z belongs to the subspace of L2(R) associated with the n-th
spectral band of the one-dimensional Schro¨dinger operator Lx. The function ψj represents the
eigenfunction of the operator Ly with the corresponding energy ωj .
Substituting the ansatz (4.21) into the original equation (4.1) shows that the amplitudes {am(T )}m∈Z
satisfy the dNLS equation (4.4) where the parameters are given by
α =
Ên,1
µ
and β = ‖φ̂n,0ψj‖4L4 . (4.22)
The values of both constants α and β are uniformly bounded and nonzero as µ→ 0. For a bound
on the constant α we refer to (4.17). Using the embedding result from Remark B.1.4,
‖φ̂n,0ψj‖L4 ≤ C‖φ̂n,0ψj‖H1 ≤ C‖φ̂n,0ψj‖H1,2 ≤ C‖φ̂n,0ψj‖H1,2 ,
and with (B.15), we immediately get that β = ‖φ̂n,0ψj‖4L4 < ∞. Properties (4.17) and (4.18)
guarantee that α, β 6= 0 for 0 < µ 1.
The mathematical justification of the effective amplitude equation (4.4) by means of error esti-
mates is the main purpose of this chapter. The approximation result is given by the following
theorem.
Theorem 4.3.1. Pick n ∈ N, j ∈ N0 such that the following non-resonance condition is satisfied
for all ` ∈ T1:(
En′(`)− Ên,0
)
+ 2
(
j′ − j) 6= 0, for every n′ ∈ N\{n} and j′ ∈ N0\{j}. (4.23)
Let ~a(T ) ∈ C1 ([0, T0], l1(Z)) be a solution of the dNLS equation (4.4) with initial data ~a(0) = ~a0
satisfying the bound∣∣∣∣∣
∣∣∣∣∣u0 − µ1/2 ∑
m∈Z
am(0)φ̂n,m(x)ψj(y)
∣∣∣∣∣
∣∣∣∣∣
H1,2
≤ C0µ3/2
for some C0 > 0. Then, for any µ ∈ (0, µ0) with sufficiently small µ0 > 0, there exists
a µ-independent constant C > 0 such that the GP equation (4.1) admits a solution u(t) ∈
C1
(
[0, T0/µ] ,H1,2
(
R2
))
satisfying the bound∣∣∣∣∣
∣∣∣∣∣u(·, t)− µ1/2 ∑
m∈Z
am(µt)φ̂n,m(x)ψj(y)e
−i(Eˆn,0+ωj)t
∣∣∣∣∣
∣∣∣∣∣
H1,2
≤ Cµ3/2 (4.24)
for all t ∈ [0, T0/µ].
51
Remark 4.3.2. Since µ = εe−a/ε, the finite time intervall [0, T0/µ] is exponentially large with
respect to ε, c.f. [8, 32].
Remark 4.3.3. Such an approximation result was shown for the GP equation in one space di-
mension in [29] and [32]. Similar to the approach in the updated version of [32], we compute
the residual of the approximation ansatz and use a simple application of Gronwall’s inequality to
estimate the error bound. The main difficulty in the proof of Theorem 4.3.1 lies in the defini-
tion of the associated function space H1,2 in two space dimensions, which has to satisfy certain
conditions that are summarized in Appendix B.1.
Remark 4.3.4. The appearance of the non-resonance condition (4.23) in Theorem 4.3.1 is another
main difference to the one-dimensional result in [29] and [32]. This additional condition is needed
due to the orthogonal projection on the spectral bands which is introduced in Lemma 4.4.2.
4.4. Computation of the residual
Here we use the multi-scale expansion ansatz (4.21) and compute the corresponding residual.
In order to control the remaining terms of Res(µ1/2Ψ0), we will also introduce an improved
approximation by adding higher-order terms to our ansatz (4.21). With a projection on the
subspace En,j and on its complement E⊥n,j , we can decompose and simplify the problematic
terms. By estimating the difference between the solution of the evolution problem (4.1) and
the approximation Ψ0, we get a time evolution problem for this error and obtain bounds for the
respective terms.
4.4.1. Residual of the approximate solution
Our goal is to compute the remaining terms of (4.25), which do not cancel after inserting the
approximate solution µ1/2Ψ0 into the GP equation (4.1). These terms are collected in the residual
Res(µ1/2Ψ0) = −i∂tµ1/2Ψ0 −∆µ1/2Ψ0 + V (r)Ψ0 + σ|µ1/2Ψ0|2µ1/2Ψ0. (4.25)
Substitution of the ansatz (4.21) into (4.25) leads to the relation
Res(µ1/2Ψ0) = −iµ3/2
∑
m∈Z
∂Tam(T )φ̂n,m(x)ψj(y)E(t)
− µ1/2
∑
m∈Z
am(T )
(
−∂2x + Vx(x)− Ên,0
)
φ̂n,m(x)ψj(y)E(t)
− µ1/2
∑
m∈Z
am(T )φ̂n,m(x)
(−∂2y + Vy(y)− ωj)ψj(y)E(t)
+ σµ3/2 |ϕ0(T, r)|2 ϕ0(T, r)E(t),
(4.26)
where the time derivation is given by ∂tam(T ) = µ∂Tam(T ). Now we rewrite the Wannier
condition (4.13) as follows:(
−∂2x + Vx(x)− Ên,0
)
φ̂n,m(x) =
∑
m′∈Z
m 6=m′
Ên,m−m′ φ̂n,m′(x).
(4.27)
52
Using (4.27) and the one-dimensional harmonic oscillator equation
(−∂2y + Vy(y)− ωj)ψj(y) = 0,
we simplify the residual (4.26) to
Res(µ1/2Ψ0) = −iµ3/2
∑
m∈Z
∂Tam(T )φ̂n,m(x)ψj(y)E(t)
− µ1/2
∑
m∈Z
am(T )
∑
m′∈Z
m 6=m′
Ên,m−m′ φ̂n,m′(x)ψj(y)E(t)
+ σµ3/2 |ϕ0(T, r)|2 ϕ0(T, r)E(t).
Since the amplitude functions {am(T )}m∈Z satisfy the dNLS equation (4.4) with the coefficents
α and β stated in (4.22) and with an additional change of variables in the O(µ1/2)-term, we
obtain
Res(µ1/2Ψ0) = µ
1/2
∑
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )φ̂n,m(x)ψj(y)E(t)
+ σµ3/2
(
|ϕ0(T, r)|2 ϕ0(T, r)−
∑
m∈Z
β |am(T )|2 am(T )φ̂n,m(x)ψj(y)
)
E(t).
(4.28)
In order to get the residual sufficently small, we extend the ansatz (4.21) by adding higher-order
terms to the approximation.
4.4.2. The improved approximation
The residual of the simple approximation (4.21) contains remaining terms, which are of the
formal order O(µ1/2). However, we need the residual (4.28) to be of the formal order O(µ3/2)
and therefore we want to control the nonlinear O(µ3/2)-term by adding higher-order terms µ3/2Ψµ
to the ansatz (4.21). We set
µ3/2Ψµ(t, r) = µ
3/2ϕµ(T, r)E(t)
and thus define the improved approximation as follows:
µ1/2Ψ(t, r) = µ1/2Ψ0(t, r) + µ
3/2Ψµ(t, r)
= µ1/2ϕ0(T, r)E(t) + µ
3/2ϕµ(T, r)E(t).
(4.29)
53
Computing the residual of the improved ansatz, we insert (4.29) into the GP equation (4.1) and
get
Res(µ1/2Ψ) = Res(µ1/2Ψ0 + µ
3/2Ψµ)
= −i∂t
(
µ1/2Ψ0 + µ
3/2Ψµ
)
−∆
(
µ1/2Ψ0 + µ
3/2Ψµ
)
+ V (r)
(
µ1/2Ψ0 + µ
3/2Ψµ
)
+ σ
∣∣∣µ1/2Ψ0 + µ3/2Ψµ∣∣∣2 (µ1/2Ψ0 + µ3/2Ψµ)
= −i∂tµ1/2Ψ0 − i∂tµ3/2Ψµ −∆µ1/2Ψ0 −∆µ3/2Ψµ
+ V (r)µ1/2Ψ0 + V (r)µ
3/2Ψµ
+ σµ3/2Ψ0Ψ0Ψ0 + σµ
5/2Ψ0ΨµΨ0 + σµ
5/2ΨµΨ0Ψ0 + σµ
7/2ΨµΨµΨ0
+ σµ5/2Ψ0Ψ0Ψµ + σµ
7/2Ψ0ΨµΨµ + σµ
7/2ΨµΨ0Ψµ + σµ
9/2ΨµΨµΨµ.
(4.30)
We recall the definition (4.25) for the residual Res(µ1/2Ψ) and with
Res(µ3/2Ψµ) = −i∂tµ3/2Ψµ −∆µ3/2Ψµ + V (r)µ3/2Ψµ + σ
∣∣∣µ3/2Ψµ∣∣∣2 µ3/2Ψµ,
we rewrite (4.30) to
Res(µ1/2Ψ) = Res(µ1/2Ψ0) + Res(µ
3/2Ψµ)
+ 2σµ5/2|Ψ0|2Ψµ + 2σµ7/2|Ψµ|2Ψ0 + σµ5/2ΨµΨ20 + σµ7/2Ψ0Ψ2µ.
(4.31)
In order to sort the nonlinear terms by the power of µ, we use the decompositions µ1/2Ψ0(t, r) =
µ1/2ϕ0(T, r)E(t) and µ
3/2Ψµ(t, r) = µ
3/2ϕµ(T, r)E(t) to obtain the following result:
Res(µ1/2Ψ) = Res(µ1/2Ψ0) + Res(µ
3/2Ψµ)
+ 2σµ5/2|ϕ0(T, r)|2ϕµ(T, r)E(t) + 2σµ7/2|ϕµ(T, r)|2ϕ0(T, r)E(t)
+ σµ5/2ϕµ(T, r)ϕ
2
0(T, r)E(t) + σµ
7/2ϕ0(T, r)ϕ
2
µ(T, r)E(t).
(4.32)
Next, we see that the residual of the additional term µ3/2Ψµ is given by
Res(µ3/2Ψµ) = −i∂tµ3/2Ψµ −∆µ3/2Ψµ + V (r)µ3/2Ψµ + σ|µ3/2Ψµ|2µ3/2Ψµ
= −iµ3/2∂t(ϕµ(T, r)E(t)) + µ3/2 (Lϕµ(T, r))E(t)
+ σµ9/2 |ϕµ(T, r)|2 ϕµ(T, r)E(t)
= −iµ5/2 (∂Tϕµ(T, r))E(t) + µ3/2
(
L−
(
Ên,0 + ωj
))
ϕµ(T, r)E(t)
+ σµ9/2 |ϕµ(T, r)|2 ϕµ(T, r)E(t),
(4.33)
54
and by substituting (4.28) and (4.33) into (4.32), we conclude
Res(µ1/2Ψ) = µ1/2
∑
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )φ̂n,m(x)ψj(y)E(t)
+ σµ3/2
(
1
σ
(
L−
(
Ên,0 + ωj
))
ϕµ(T, r) + |ϕ0(T, r)|2 ϕ0(T, r)
−
∑
m∈Z
β |am(T )|2 am(T )φ̂n,m(x)ψj(y)
)
E(t)
+ σµ5/2
(
− i
σ
∂Tϕµ(T, r) + 2 |ϕ0(T, r)|2 ϕµ(T, r) + ϕµ(T, r)ϕ20(T, r)
)
E(t)
+ σµ7/2
(
2 |ϕµ(T, r)|2 ϕ0(T, r)E(t) + ϕ0(T, r)ϕ2µ(T, r)
)
E(t)
+ σµ9/2 |ϕµ(T, r)|2 ϕµ(T, r)E(t).
(4.34)
As we can see in the equations above, we get additional O(µ3/2) terms in the residual of the
improved approximation µ1/2Ψ. Our next goal is to control the terms of the formal order O(µ3/2)
in (4.34) by using the orthogonal projection Πn,j : L
2(R2) → En,j , where En,j ⊂ L2(R2) is
defined as the direct product of the n-th spectral band En in x-direction and the eigenspace Uj
corresponding to the eigenvalue ωj of the harmonic oscillator problem (4.6). Therefore, we map
the nonlinear term |ϕ0|2ϕ0 onto En,j and its complement E⊥n,j , such that
|ϕ0|2 ϕ0 = Πn,j |ϕ0|2 ϕ0 + (I −Πn,j) |ϕ0|2 ϕ0. (4.35)
Note that this approach is similar to the decomposition used in the one dimensional case, c.f.
[29] and [32]. The following two lemmas are useful to control the projection (I −Πn,j)|ϕ0|2ϕ0.
Lemma 4.4.1. Let ~a(T ) ∈ l1(Z) for a fixed T ∈ R and ϕ0(r) =
∑
m∈Z amφ̂n,m(x)ψj(y) for a
fixed n ∈ N, then ϕ0(r) ∈ En,j ⊂ L2(R2) and also ϕ0(r) ∈ H1,2(R2).
Proof. Since ‖~a‖l2 ≤ ‖~a‖l1 , we have ~a ∈ l2(Z) and it follows directly from spectral theory for the
operator L = Lx + Ly in L
2(R2) that ϕ0 ∈ En,j = En × Uj for a fixed n ∈ N. The second part
of the theorem follows from the fact that φ̂n,m(x) = φ̂n,0(x− 2pim). Hence, we get
‖ϕ0‖2H1,2 = ‖
∑
m∈Z
amφ̂n,mψj‖2H1,2
≤
∑
m∈Z
‖amφ̂n,mψj‖2H1,2
≤
∑
m∈Z
|am|2‖φ̂n,mψj‖2H1,2
= ‖~a‖2l2‖φ̂n,0ψj‖2H1,2
≤ ‖~a‖2l1‖φ̂n,0ψj‖2H1,2 ,
(4.36)
where we used the triangular inequality and the bound (B.15).
55
Lemma 4.4.2. There exists a unique solution ϕ ∈ H1,2(R2) of the inhomogeneous equation(
L−
(
Ên,0 + ωj
))
ϕ = (I −Πn,j) f, (4.37)
for any f ∈ L2(R2), such that 〈ϕ, ϑ〉L2 = 0 for every ϑ ∈ En,j and
‖ϕ‖H1,2 ≤ C‖f‖L2 (4.38)
for some C > 0 uniformly in 0 < µ 1.
Proof. At first, we write En,j = En × Uj and by the band separation property (4.15) of Lemma
4.2.2, we get Ên,0 6∈ σ(Lx)|E⊥n . It is also clear that ωj 6∈ σ(Ly)|U⊥j . With the same argument as
in [29], if ϕ is a solution of(
(Lx − Ên,0) + (Ly − ωj)
)
ϕ = (I −Πn,j) f,
then ϕ ∈ L2(R2) for any f ∈ L2(R2) because of the decomposition
ϕ(r) =
∫
T1
∫ ∞
y′=−∞
∑
n′∈N\{n}
∑
j′∈N\{j}
f˜n′(`, y
′)ψj′(y′)φ˜n′(`, x)
(En′(`)− Ên,0) + (ωj′ − ωj)
dy′d` · ψj(y),
which implies that there is a constant CN > 0 such that
‖ϕ‖2L2 ≤
∫
T1
∫ ∞
y′=−∞
∑
n′∈N\{n}
∑
j′∈N\{j}
∣∣∣f˜n′(`, y′)ψj′(y′)∣∣∣2(
(En′(`)− Ên,0) + (ωj′ − ωj)
)2dy′d`
≤ CN
∫
T1
∫ ∞
y′=−∞
∑
n′∈N\{n}
∑
j′∈N\{j}
∣∣∣f˜n′(`, y′)ψj′(y′)∣∣∣2 dy′d`
≤ CN‖f‖2L2 .
(4.39)
This inequality holds under the non-resonance condition (4.23) stated in Theorem 4.3.1.
Next, we prove the bound (4.38) and initially get
‖ϕ‖2H1,2 ≤ |〈(Lx + I)ϕ,ϕ〉L2 |+ |〈(Ly + I)2ϕ,ϕ〉L2 |+ |〈ϕ,ϕ〉L2 |
≤ |〈Lxϕ,ϕ〉L2 |+ |〈ϕ,ϕ〉L2 |+ |〈Lyϕ,Lyϕ〉L2 |
+ 2|〈Lyϕ,ϕ〉L2 |+ |〈ϕ,ϕ〉L2 |+ |〈ϕ,ϕ〉L2 |
≤ |〈Lϕ,Lϕ〉L2 |+ 3|〈Lϕ,ϕ〉L2 |+ 3|〈ϕ,ϕ〉L2 |.
(4.40)
The last line of (4.40) follows directly from the inequalities
|〈Lxϕ,ϕ〉L2 | ≤ |〈(Lx + Ly)ϕ,ϕ〉L2 |, |〈Lyϕ,ϕ〉L2 | ≤ |〈(Lx + Ly)ϕ,ϕ〉L2 | (4.41)
and
|〈Lyϕ,Lyϕ〉L2 | ≤ |〈(Lx + Ly)ϕ, (Lx + Ly)ϕ〉L2 |, (4.42)
which can be verified by adding the scalar products
|〈Lxϕ,ϕ〉L2 | = ‖∂xϕ‖2L2 + ‖V 1/2x ϕ‖2L2 , |〈Lyϕ,ϕ〉L2 | = ‖∂yϕ‖2L2 + ‖V 1/2y ϕ‖2L2
56
and
|〈Lyϕ,Lxϕ〉L2 | = ‖∂x(∂yϕ)‖2L2 + ‖V 1/2x (∂yϕ)‖2L2 + ‖∂x(V 1/2y ϕ)‖2L2 + ‖V 1/2x (V 1/2y ϕ)‖2L2
to the left hand side of (4.41) and (4.42) respectively. Since Πn,jf ∈ En,j , we obtain that
〈Πn,jf, ϕ〉L2 = 0 and with (4.39), we write
|〈Lϕ,ϕ〉L2 | = |〈(I −Πn,j)f − (Ên,0 + ωj)ϕ,ϕ〉L2 |
≤ |〈f, ϕ〉L2 |+ |Ên,0 + ωj ||〈ϕ,ϕ〉L2 |
≤ ‖f‖L2‖ϕ‖L2 + (E0 + ωj) ‖ϕ‖2L2
≤ C1‖f‖2L2 .
(4.43)
Since ‖(I −Πn,j)f‖L2 ≤ ‖f‖L2 , we also conclude
|〈Lϕ,Lϕ〉L2 | ≤ |〈(I −Πn,j)f, (I −Πn,j)f〉L2 |+ |〈(Ên,0 + ωj)ϕ, (I −Πn,j)f〉L2 |
+ |〈(I −Πn,j)f, (Ên,0 + ωj)ϕ〉L2 |+ |〈(Ên,0 + ωj)ϕ, (Ên,0 + ωj)ϕ〉L2 |
≤ ‖f‖2L2 + 2 (E0 + ωj) ‖ϕ‖L2‖f‖L2 + (E0 + ωj)2 ‖ϕ‖2L2
≤ C2‖f‖2L2 .
(4.44)
Inserting the estimates (4.43) and (4.44) into (4.40) yields to
‖ϕ‖2H1,2 ≤ C2‖f‖2L2 + 3C1‖f‖2L2 + 3CN‖f‖2L2 = C2‖f‖2L2
and thus the bound (4.38) holds. Uniqueness of ϕ follows from the fact that the inverse operator
(I−Πn,j)(L− (Ên,0 +ωj))−1(I−Πn,j) is a continuous map from L2(R2) to H1,2(R2) uniformly
in 0 < µ 1.
Now we return to the computation of Res(µ1/2Ψ) and use Lemma 4.4.2 to define ϕµ ∈ E⊥n,j as
a unique solution of the system(
L−
(
Ên,0 + ωj
))
ϕµ = −σ (I −Πn,j) |ϕ0|2 ϕ0. (4.45)
Using the decomposition (4.35) and the projection (4.45), the O(µ3/2) terms in the residual
(4.34) can be rewritten as
Πn,j |ϕ0(T, r)|2 ϕ0(T, r)−
∑
m∈Z
β |am(T )|2 am(T )φ̂n,m(x)ψj(y)
=
∑
m∈Z
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1(T )am2(T )am3(T )φ̂n,m(x)ψj(y),
(4.46)
where we introduced the integral kernel
κj(m,m1,m2,m3) = 〈φ̂n,m1ψjφ̂n,m2ψjφ̂n,m3ψj , φ̂n,mψj〉L2 . (4.47)
57
As a result, we obtain the residual (4.34) in the form
Res(µ1/2Ψ) = µ1/2
∑
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )φ̂n,m(x)ψj(y)E(t)
+ σµ3/2
∑
m∈Z
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1(T )am2(T )am3(T )
× φ̂n,m(x)ψj(y)E(t)
+ σµ5/2
(
− i
σ
∂Tϕµ(T, r) + 2 |ϕ0(T, r)|2 ϕµ(T, r) + ϕµ(T, r)ϕ20(T, r)
)
E(t)
+ σµ7/2
(
2 |ϕµ(T, r)|2 ϕ0(T, r)E(t) + ϕ0(T, r)ϕ2µ(T, r)
)
E(t)
+ σµ9/2 |ϕµ(T, r)|2 ϕµ(T, r)E(t).
(4.48)
The formal derivation of (4.46) and (4.47) can be found in the second part of Appendix B.2.
4.4.3. Estimates on the error term
The improved approximation leads to the term (4.46), which is of the formal order O(µ3/2) and
does not vanish. However, Lemma 4.4.2 simplified these terms to be an element of the subspace
En,j and this projection is required to control the approximation error.
As in [32], we now want to obtain an evolution equation for the error. Therefore, we write the
solution u of the evolution problem (4.1) as a sum of the improved approximation µ1/2Ψ and the
error term µ3/2R, i.e.,
u(t, r) = µ1/2Ψ(t, r) + µ3/2R(t, r)
= µ1/2Ψ0(t, r) + µ
3/2Ψµ(t, r) + µ
3/2R(t, r)
= µ1/2ϕ0(T, r)E(t) + µ
3/2ϕµ(T, r)E(t) + µ
3/2ρ(t, r)E(t)
= µ1/2ϕ0(T, r)E(t) + µ
3/2 (ϕµ(T, r) + ρ(t, r))E(t).
(4.49)
Substituting this decomposition into (4.1) yields to
i∂tu = i∂t
(
µ1/2Ψ0 + µ
3/2Ψµ + µ
3/2R
)
= −∆
(
µ1/2Ψ0 + µ
3/2Ψµ + µ
3/2R
)
+ V (r)
(
µ1/2Ψ0 + µ
3/2Ψµ + µ
3/2R
)
+ σ
∣∣∣µ1/2Ψ0 + µ3/2Ψµ + µ3/2R∣∣∣2 (µ1/2Ψ0 + µ3/2Ψµ + µ3/2R)
(4.50)
and by rearranging (4.50), we get the evolution problem
i∂tµ
3/2R = i∂t
(
µ1/2Ψ0 + µ
3/2Ψµ
)
−∆
(
µ1/2Ψ0 + µ
3/2Ψµ
)
+ V (r)
(
µ1/2Ψ0 + µ
3/2Ψµ
)
+ σ
∣∣∣µ1/2Ψ0 + µ3/2Ψµ∣∣∣2 (µ1/2Ψ0 + µ3/2Ψµ)
−∆µ3/2R+ V (r)µ3/2R+ σµ5/2 (2|Ψ0|2R+RΨ20)
+ σµ7/2
(
2Ψ0ΨµR+ 2ΨµΨ0R+ 2|R|2Ψ0 + 2RΨ0Ψµ + Ψ0R2
)
+ σµ9/2
(
2|Ψµ|2R+ 2|R|2Ψµ +RΨ2µ + ΨµR2 + |R|2R
)
.
(4.51)
58
As in the derivation for the residual of the improved approximation in Section 4.4.2, we insert
the decompositions Ψ0(t, r) = µ
1/2ϕ0(T, r)E(t),Ψµ(t, r) = µ
3/2ϕµ(T, r)E(t) and R(t, r) =
µ3/2ρ(t, r)E(t) into (4.51) and get
iµ3/2∂t (ρ(t, r)E(t)) = Res(µ
1/2Ψ) + µ3/2 (−∆ + V (r)) ρ(t, r)E(t)
+ σµ5/2
(
2|ϕ0(T, r)|2ρ(t, r) + ρ(t, r)ϕ20(T, r)
)
E(t)
+ σµ7/2
(
2ϕ0(T, r)ϕµ(T, r)ρ(t, r) + 2ϕµ(T, r)ϕ0(T, r)ρ(t, r)
+ 2|ρ(t, r)|2ϕ0(T, r) + 2ρ(t, r)ϕ0(T, r)ϕµ(T, r)
+ϕ0(T, r)ρ
2(t, r)
)
E(t)
+ σµ9/2
(
2|ϕµ(T, r)|2ρ(t, r) + 2|ρ(t, r)|2ϕµ(T, r)
+ ρ(t, r)ϕ2µ(T, r) + ϕµ(T, r)ρ
2(t, r)
+|ρ(t, r)|2ρ(t, r))E(t).
(4.52)
Using (4.48) and the time derivative i∂t(ρ(t, r)E(t)) = (i∂tρ(t, r))E(t) + (Ên,0−ωj)ρ(t, r)E(t),
we summarize the terms in (4.52) by their formal order O(µ) and find
i∂tρ(t, r) =
1
µ
∑
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )φ̂n,m(x)ψj(y)
+
(
−∆ + V (r)−
(
Ên,0 − ωj
))
ρ(t, r)
+ σ
∑
m∈Z
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1(T )am2(T )am3(T )
× φ̂n,m(x)ψj(y)
+ σµ
(
− i
σ
∂Tϕµ(T, r) + 2|ϕ0(T, r)|2 (ϕµ(T, r) + ρ(t, r))
+ϕ20(T, r)
(
ϕµ(T, r) + ρ(t, r)
))
+ σµ2
(
2ϕ0(T, r) |ϕµ(T, r) + ρ(t, r)|2 + ϕ0(T, r) (ϕµ(T, r) + ρ(t, r))2
)
+ σµ3
(
(ϕµ(T, r) + ρ(t, r)) |ϕµ(T, r) + ρ(t, r)|2
)
.
(4.53)
For simplicity, we introduce the sum S(~a) =
∑
m∈Z sm(~a)φ̂n,m(x)ψj(y) with
sm(~a) = =
1
µ2
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )
+ σ
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1(T )am2(T )am3(T )
(4.54)
59
and the nonlinear term given by
N(~a, ρ) = σ
(
− i
σ
∂Tϕµ(T, r) + 2|ϕ0(T, r)|2 (ϕµ(T, r) + ρ(t, r))
+ϕ20(T, r)
(
ϕµ(T, r) + ρ(t, r)
))
+ σµ
(
2ϕ0(T, r) |ϕµ(T, r) + ρ(t, r)|2 + ϕ0(T, r) (ϕµ(T, r) + ρ(t, r))2
)
+ σµ2
(
(ϕµ(T, r) + ρ(t, r)) |ϕµ(T, r) + ρ(t, r)|2
)
.
(4.55)
Thus, with (4.54) and (4.55) the function ρ(t, r) satisfies the evolution equation (4.53) in the
abstract form
i∂tρ =
(
L−
(
Ên,0 − ωj
))
ρ+ µS(~a) + µN(~a, ρ). (4.56)
As in the one-dimensional case, we now prove that the terms S(~a) and N(~a, ρ) of the evolution
problem (4.56) are uniformly bounded in H1,2(R2) for 0 < µ  1. Since the result (4.3.1) is
local in time, we consider a ball of finite radius δ1 > 0 in l
1(Z) denoted by Bδ1(l1(Z)) and a ball
of finite radius δ2 > 0 in H1,2(R2) denoted by Bδ2(H1,2(R2)) to control these terms with the
following lemma, cf. [29, 32].
Lemma 4.4.3. Let ~a(T ), ∂T~a(T ) ∈ Bδ1(l1(Z)) and ρ(t, ·), ρ˜(t, ·) ∈ Bδ2(H1,2(R2)) for fixed
δ1, δ2 > 0. Then, for any 0 < µ 1, there exist µ-independent constants CS , CN ,KN > 0 such
that
‖S(~a)‖H1,2 ≤ CS‖~a‖l1 , (4.57)
‖N(~a, ρ)‖H1,2 ≤ CN (‖~a‖l1 + ‖ρ‖H1,2) , (4.58)
‖N(~a, ρ)−N(~a, ρ˜)‖H1,2 ≤ KN‖ρ− ρ˜‖H1,2 . (4.59)
Proof. At first, we want to prove the bound (4.57) and therefore, we remember the notation
S(~a) =
∑
m∈Z sm(~a)φ̂n,m(x)ψj(y), where the components sm(~a) are defined by (4.54). Using
the triangle inequality∥∥∥∥∥∑
m∈Z
sm(~a)φ̂n,m(x)ψj(y)
∥∥∥∥∥
H1,2
≤ ‖~s(~a)‖l1‖φ̂n,0(x)ψj(y)‖H1,2
and (B.15), it remains to show that ‖~s(~a)‖l1 ≤ Cs‖~a‖l1 with ~s(~a) = {sm(~a)}m∈Z holds for some
Cs > 0. The first term in ~s(~a) is estimated as follows:∥∥∥∥∥∥∥∥
∑
m′∈Z
m′ 6={m−1,m,m+1}
Ên,m′−mam′(T )
∥∥∥∥∥∥∥∥
l1
≤
∑
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
|Ên,m′−m||am′(T )|
≤ K1‖~a‖l1 ,
where
K1 = sup
m∈Z
∑
m′∈Z
m′ 6={m−1,m,m+1}
|Ên,m′−m| =
∑
l∈Z
l 6={−1,0,1}
|Ên,l|.
(4.60)
60
The sum on the right hand side of (4.61) is bounded, because the band function En(`) is analyt-
ically continued along the Riemann surface on ` ∈ T1 by Theorem XIII.95 in [36]. Hence, En(`)
is infinitely often differentiable for ` ∈ T1 and we have En(`) ∈ Hs(T1) for any s > 0, such that
{Ên,l}l∈Z ∈ l1(Z) and K1 <∞. By property (4.17) of Lemma 4.2.2, Ên,l = O(µ2) for all l ≥ 2,
such that K1/µ
2 is uniformly bounded in 0 < µ 1. Estimating the second term in ~s(~a), we get∥∥∥∥∥∥∥∥∥
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1(T )am2(T )am3(T )
∥∥∥∥∥∥∥∥∥
l1
≤
∑
m∈Z
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
|κj(m,m1,m2,m3)||am1(T )||am2(T )||am3(T )|
≤ K2‖~a‖3l1 ,
where
K2 = sup
(m1,m2,m3)∈Z3/
{(m,m,m)}
∑
m∈Z
|κj(m,m1,m2,m3)|.
(4.61)
Now we use the exponential decay (4.20) and the explicit representation (4.21) of ψj for a fixed
j ∈ N to obtain ∑
m∈Z
|φ̂n,m(x)ψj(y)| ≤
∑
m∈Z
|φ̂n,m(x)||ψj(y)|
≤ CnCj
∑
m∈Z
e−ηn|x−2pim|
≤ An
for some An > 0 uniformly in r ∈ R2 and therefore,∑
m∈Z
|κj(m,m1,m2,m3)| ≤ An
∫
R2
|φ̂n,m1(x)ψj(y)||φ̂n,m2(x)ψj(y)||φ̂n,m3(x)ψj(y)|dr
≤ An‖φ̂n,m(x)ψj(y)‖2L4‖φ̂n,m(x)ψj(y)‖L2
(4.62)
uniformly in (m1,m2,m3) ∈ Z3, where the last line of (4.62) follows with φ̂n,m(x) = φ̂n,0(x −
2pim) and the Ho¨lder inequality. Using the embeddings stated in Appendix B.1, we conclude
‖φ̂n,0(x)ψj(y)‖L4 ≤ C˜L‖φ̂n,0(x)ψj(y)‖H1 ≤ C˜L‖φ̂n,0(x)ψj(y)‖H1,2 ≤ C˜L‖φ̂n,0(x)ψj(y)‖H1,2
and then the bound (B.15) yields to K2 < ∞. By property (4.19) of Lemma 4.2.2 and with
ψj = O(1), we get that the integral kernel
κj(m,m1,m2,m3) = O(µ|m1−m|+|m2−m|+|m3−m|+|m2−m1|+|m3−m1|+|m3−m2|),
for every (m1,m2,m3) ∈ Z3, and so K2/µ is uniformly bounded in 0 < µ  1. Thus, we get a
µ-independent constant CS > 0 and the inequality (4.57) is proved.
61
In order to obtain the second bound (4.58), we have to control the norms of the nonlinear term
|ϕ0|2 ϕ0 and of the additional term ϕµ in H1,2(R2). Using the inequality (4.36) from the proof of
Lemma 4.4.1 and the bound (B.15), for a fixed n ∈ N and ‖~a(T )‖ ∈ l1(Z), there exists a Ca > 0
uniformly in 0 < µ 1 such that
‖ϕ0(T, ·)‖H1,2 ≤ Ca‖~a(T )‖l1 .
Moreover, we have shown in Theorem B.1.5 that the space H1,2(R2) is closed under pointwise
multiplication and hence
‖ |ϕ0(T, ·)|2 ϕ0(T, ·)‖H1,2 ≤ C2B‖ϕ0(T, ·)‖3H1,2 ≤ C2BC3a‖~a(T )‖3l1 . (4.63)
Next, we bound the function ϕµ by recalling
ϕµ(T, ·) = −σ(I −Πn,j)(L− (Ên,0 + ωj))−1(I −Πn,j) |ϕ0(T, ·)|2 ϕ0(T, ·)
and σ = ±1. As mentioned in the proof of Lemma 4.4.2, the inverse operator (I − Πn,j)(L −
(Ên,0 + ωj))
−1(I −Πn,j) is a continuous map from L2(R2) to H1,2(R2) uniformly in 0 < µ 1
and therefore, we get
‖ϕµ(T, ·)‖H1,2 ≤ Cϕ‖~a(T )‖3l1 . (4.64)
Now, the bounds (4.63), (4.64) and the fact that both H1,2(R2) and l1(Z) form Banach algebras
with respect to pointwise multiplication lead directly to the estimate (4.58). Since ~a(T ), ∂T~a(T ) ∈
Bδ1(l
1(Z)), the functions ϕ(T, ·), ϕµ(T, ·), ∂Tϕµ(T, ·) ∈ Bδ2(H1,2(R2)) and N(~a, ρ) maps ρ ∈
H1,2(R2) to an element of H1,2(R2). Thus, N(~a, ρ) is uniformly bounded in 0 < µ 1 and the
constant KN > 0 is µ-independent.
The proof of the third bound (4.59) also follows from the explicit expression of N(~a, ρ).
Remark 4.4.4. Note that in our argumentation above, we followed very closely the proof of
Lemma 4 in [32] and Lemma 2.18 in [29], respectively. This makes sense because the sum S(~a)
is defined in the same way as in the one-dimensional case. Also, the nonlinear term N(~a, ρ) is
identical with the one in [32] and can be estimated in two space dimensions with the same ideas
used there.
4.5. Local Existence and uniqueness
In this section, we prove the local existence and uniqueness of solutions of the evolution equation
(4.56). However, we first need to show that the initial-value problem for the dNLS equation (4.4)
is locally well-posed in C1([0, T0], l
1(Z)) for a T0 > 0. One can also find the following lemma as
Theorem 2 in the updated version of [32].
Lemma 4.5.1. Let ~a0 ∈ l1(Z). Then, there exists a T0 > 0 and a unique solution ~a(T ) ∈
C1([0, T0], l
1(Z)) of the dNLS equation (4.4) with the initial data ~a(0) = ~a0.
Proof. By the variation of constant formula, we have
~a(T ) = ~a0 − i
∫ T
0
(α∆d~a(s) + σβΓ(~a(s))) ds,
where (∆d~a)m = am+1 + am−1 and (Γ(~a))m = |am|2am. Since l1(Z) forms a Banach algebra,
the right-hand side of the integral equation maps an element of l1(Z) to an element of l1(Z).
Therefore, there exists a unique solution ~a(T ) ∈ C1([0, T0], l1(Z)) of the integral equation for
sufficiently small T0 > 0.
62
As a consequence of this result, we now obtain the existence of a unique solution of the Cauchy
problem associated with the system (4.56) in H1,2(R).
Lemma 4.5.2. Let ~a(T ) ∈ C1([0, T0], l1(Z)) and ρ(0, ·) = ρ0 ∈ H1,2(R2). Then, there exists a
µ0 > 0 such that, for any µ ∈ (0, µ0), the time evolution problem (4.56) admits a unique solution
ρ(t, ·) ∈ C1([0, T0/µ],H1,2(R2)).
Proof. The operators Lx = −∂2x + Vx(x) and Ly = −∂2y + Vy(y) are self-adjoint in L2(R2) and
thus, by classical semigroup theory, cf. [28], we get
‖e−i(Lx−Ên,0)tρ‖L2 = ‖ρ‖L2 and ‖e−i(Ly−ωj)tρ‖L2 = ‖ρ‖L2
with LxLy = LyLx. Moreover, the commutativity of Lx and e
−i(Lx−Ên,0)t and of Ly and
e−i(Ly−ωj)t holds. Hence, with
e−i(L−(Ên,0+ωj))t = e−i(Lx−Ên,0)te−i(Ly−ωj)t
and the definition (4.5) of the norm of H1,2(R2), we obtain the following equality:
‖e−i(L−(Ên,0+ωj))tρ‖2H1,2 = ‖ (Lx + I)1/2 e−i(Lx−Ên,0)te−i(Ly−ωj)tρ‖2L2
+ ‖ (Ly + I) e−i(Lx−Ên,0)te−i(Ly−ωj)tρ‖2L2
+ ‖e−i(Lx−Ên,0)te−i(Ly−ωj)tρ‖2L2
= ‖e−i(Lx−Ên,0)te−i(Ly−ωj)t (Lx + I)1/2 ρ‖2L2
+ ‖e−i(Lx−Ên,0)te−i(Ly−ωj)t (Ly + I) ρ‖2L2
+ ‖e−i(Lx−Ên,0)te−i(Ly−ωj)tρ‖2L2
= ‖ (Lx + I)1/2 ρ‖2L2 + ‖ (Ly + I) ρ‖2L2 + ‖ρ‖2L2
= ‖ρ‖2H1,2 .
Therefore, the operator e−i(L−(Ên,0+ωj))t forms a unitary group inH1,2(R2) for every ρ ∈ H1,2(R2)
and every t ∈ R. By Duhamel’s principle, we now rewrite the Cauchy problem associated with
the evolution equation (4.56) as the integral equation
ρ(t, ·) = e−i(L−(Ên,0+ωj))tρ0
+ µ
∫ t
0
e−i(L−(Ên,0+ωj))(t−s) (S(~a(µs)) +N(~a(µs), ρ(s, ·))) ds,
where the solution is considered in the subspace Bδ2(H1,2(R2)). For every ρ0 ∈ H1,2(R2), there is
a sufficiently small T0 > 0 such that the right-hand side of the integral equation is a contraction
in the ball Bδ2(H1,2(R2)), where we used the bounds (4.57) and (4.58). The existence of a
unique solution ρ(t, ·) ∈ C1([0, T0/µ],H1,2(R2)) follows from Banach’s fixed point theorem and
(4.59).
4.6. Control on the error bound
Here we complete the proof of Theorem 4.3.1. The proof of the approximation result is based
on an energy estimate for the evolution problem (4.56) and a simple application of Gronwall’s
63
inequality. Although the analysis of this section is similar to the one in the updated version of
[32], we have to transfer the calculations to our two-dimensional setting.
As in the one-dimensional case, our goal is to control the error term µ3/2R = µ3/2ρ(t, r)E(t) for
every t ∈ [0, T0/µ]. The following lemma gives us a bound on the time evolution of ρ(t, r) in the
function space H1,2(R2).
Lemma 4.6.1. Let ρ(t, ·) ∈ C1([0, T0/µ], Bδ2(H1,2(R2)) be a local solution of the time evolution
problem (4.56) for some T0 > 0 and ~a(T ) ∈ C1([0, T0], l1(Z)). Then, for any µ ∈ (0, µ0), there
exists a µ-independent constant CE > 0 such that∣∣∣∣ ddt‖ρ(t, ·)‖2H1,2
∣∣∣∣ ≤ µCE (‖~a‖2l1 + ‖ρ(t, ·)‖2H1,2) . (4.65)
Proof. Using definition (4.5), we write
d
dt
‖ρ‖2H1,2 =
d
dt
〈(Lx + I)1/2ρ, (Lx + I)1/2ρ〉L2 +
d
dt
〈(Ly + I)ρ, (Ly + I)ρ〉L2 +
d
dt
〈ρ, ρ〉L2
= 〈(Lx + I)1/2(∂tρ), (Lx + I)1/2ρ〉L2 + 〈(Ly + I)(∂tρ), (Ly + I)ρ〉L2 + 〈(∂tρ), ρ〉L2
+ 〈(Lx + I)1/2ρ, (Lx + I)1/2(∂tρ)〉L2 + 〈(Ly + I)ρ, (Ly + I)(∂tρ)〉L2 + 〈ρ, (∂tρ)〉L2
and with
∂tρ = −i
(
L−
(
Ên,0 − ωj
))
ρ− iµS(~a)− iµN(~a, ρ),
we obtain by direct calculation the following equation:
d
dt
‖ρ‖2H1,2 = −iµ〈∂xS(~a), ∂xρ〉L2 + iµ〈∂xρ, ∂xS(~a)〉L2
− iµσ〈∂xN(~a, ρ), ∂xρ〉L2 + iµσ〈∂xρ, ∂xN(~a, ρ)〉L2
− iµ〈VxS(~a), ρ〉L2 + iµ〈ρ, VxS(~a)〉L2
− iµσ〈VxN(~a, ρ), ρ〉L2 + iµσ〈ρ, VxN(~a, ρ)〉L2
− iµ〈∂2yS(~a), ∂2yρ〉L2 + iµ〈∂2yρ, ∂2yS(~a)〉L2
− iµσ〈∂2yN(~a, ρ), ∂2yρ〉L2 + iµσ〈∂2yρ, ∂2yN(~a, ρ)〉L2
− iµ〈V 2y S(~a), ρ〉L2 + iµ〈ρ, V 2y S(~a)〉L2
− iµσ〈V 2y N(~a, ρ), ρ〉L2 + iµσ〈ρ, V 2y N(~a, ρ)〉L2
+ iµ〈∂2yS(~a), Vyρ〉L2 − iµ〈∂2yρ, VyS(~a)〉L2
+ iµσ〈∂2yN(~a, ρ), Vyρ〉L2 − iµσ〈∂2yρ, VyN(~a, ρ)〉L2
+ iµ〈VyS(~a), ∂2yρ〉L2 − iµ〈Vyρ, ∂2yS(~a)〉L2
+ iµσ〈VyN(~a, ρ), ∂2yρ〉L2 − iµσ〈Vyρ, ∂2yN(~a, ρ)〉L2
− 2iµ〈∂yS(~a), ∂yρ〉L2 + 2iµ〈∂yρ, ∂yS(~a)〉L2
− 2iµσ〈∂yN(~a, ρ), ∂yρ〉L2 + 2iµσ〈∂yρ, ∂yN(~a, ρ)〉L2
− 2iµ〈VyS(~a), ρ〉L2 + 2iµ〈ρ, VyS(~a)〉L2
− 2iµσ〈VyN(~a, ρ), ρ〉L2 + 2iµσ〈ρ, VyN(~a, ρ)〉L2
− 3iµ〈S(~a), ρ〉L2 + 3iµ〈ρ, S(~a)〉L2
− 3iµσ〈N(~a, ρ), ρ〉L2 + 3iµσ〈ρ,N(~a, ρ)〉L2 .
(4.66)
64
Next, we summarize the terms in (4.66) to
d
dt
‖ρ‖2H1,2 = −iµ〈S(~a), (Lx + I)ρ+ (Ly + I)2ρ+ ρ〉L2
− iµ〈(Lx + I)ρ+ (Ly + I)2ρ+ ρ, S(~a)〉L2
− iµσ〈N(~a, ρ), (Lx + I)ρ+ (Ly + I)2ρ+ ρ〉L2
− iµσ〈(Lx + I)ρ+ (Ly + I)2ρ+ ρ,N(~a, ρ)〉L2
(4.67)
where we applied the operator representation
〈−∂2xρ+ Vxρ+
(
∂2y
)2
ρ+ V 2y ρ− Vy
(
∂2yρ
)− ∂2y (Vyρ)− 2∂2yρ+ 2Vyρ+ 3ρ, ·〉L2
= 〈Lxρ+ L2yρ+ 2Lyρ+ 3ρ, ·〉L2
= 〈(Lx + I)ρ+ (Ly + I)2ρ+ ρ, ·〉L2 .
Then, the Cauchy-Schwarz inequality yields to
d
dt
‖ρ‖2H1,2 ≤ −iµ‖S(~a)‖L2‖ρ‖H1,2 − iµ‖ρ‖H1,2‖S(~a)‖L2
− iµσ‖N(~a, ρ)‖L2‖ρ‖H1,2 − iµσ‖ρ‖H1,2‖N(~a, ρ)‖L2
≤ −2iµ‖S(~a)‖H1,2‖ρ‖H1,2 − 2iµσ‖N(~a, ρ)‖H1,2‖ρ‖H1,2 ,
and by inserting the bounds (4.57) and (4.58), we have∣∣∣∣ ddt‖ρ‖2H1,2
∣∣∣∣ ≤ 2µ‖ρ‖H1,2 (‖S(~a)‖H1,2 + ‖N(~a, ρ)‖H1,2)
≤ 2µ‖ρ‖H1,2 ((CS + CN ) ‖~a‖l1 + CN‖ρ‖H1,2)
≤ 2µ (CS + CN ) ‖~a‖2l1 + 2µ ((CS + CN ) + CN ) ‖ρ‖2H1,2
≤ µCE
(‖~a‖2l1 + ‖ρ‖2H1,2) ,
where we used the estimate ‖ρ‖H1,2‖~a‖l1 ≤ ‖ρ‖2H1,2 + ‖~a‖2l1 .
By using the bound (4.65), we obtain for the evolution problem the following integral formula:
‖ρ(t, ·)‖2H1,2 ≤ ‖ρ(0, ·)‖2H1,2 + µCE
∫ t
0
(‖~a(µs)‖2l1 + ‖ρ(s, ·)‖2H1,2) ds.
Now Gronwall’s inequality finally allows us to estimate the function ρ(t, r) on the time intervall
t ∈ [0, T0/µ] by
sup
t∈[0,T0/µ]
‖ρ(t, ·)‖2H1,2 ≤
(
‖ρ(0, ·)‖2H1,2 + CET0 sup
t∈[0,T0/µ]
‖~a(T )‖2l1
)
eCET0 =: M,
where the constant M > 0 is uniformly bounded in 0 < µ  1. Then, the decomposition
µ3/2R = µ3/2ρ(t, r)E(t) yields to a bound on the error term and with the bound (4.64) on the
function ϕµ(T, r), the proof of Theorem 4.3.1 is completed.
65

A. Appendices to Chapter 3
A.1. Computation of the spectral bands ω(`)
Our goal is to get an explicit representation of the the spectral bands ω(`) of the linear problem
(3.5). Therefore, we restrict our calculations to the eigenvalue problem
−(∂x + i`)2f(`, x) + f(`, x) = ω2(`)f(`, x), x ∈ T2pi, (A.1)
of the Bloch function f(`, ·) = (f0, f+, f−)(`, ·) with the corresponding boundary conditions{
f0(`, pi) = f+(`, pi) = f−(`, pi),
f0(`, 0) = f+(`, 2pi) = f−(`, 2pi)
(A.2)
and {
(∂x + i`)f0(`, pi) = (∂x + i`)f+(`, pi) + (∂x + i`)f−(`, pi),
(∂x + i`)f0(`, 0) = (∂x + i`)f+(`, 2pi) + (∂x + i`)f−(`, 2pi),
(A.3)
which were introduced before in Chapter 3.
First, we want to obtain a solution for the equation (A.1) by using the ansatz
fj(`, x) = Cje
µ(`)x, j ∈ {0,+,−}, x ∈ T2pi,
where Cj ∈ R. This ansatz yields to the following result on the spectral bands ω(`):
−(∂x + i`)2Cjeµ(`)x + Cjeµ(`)x = ω2(`)Cjeµ(`)x
− ((µ(`) + i`)2 − 1)Cjeµ(`)x = ω2(`)Cjeµ(`)x
− ((µ(`) + i`)2 − 1) = ω2(`).
Now we define µ(`) = ±i√ω2(`)− 1− i` and write the solutions fj(`, ·) of (A.1) as
fj(`, x) = Cj,1e
i
(√
ω2(`)−1−`
)
x
+ Cj,2e
−i
(√
ω2(`)−1+`
)
x
, j ∈ {0,+,−}, x ∈ T2pi. (A.4)
Inserting (A.4) into (A.2) and (A.3) changes the continuity conditions at the vertices to the
system 
C0,1e
i
(√
ω2(`)−1−`
)
pi
+ C0,2e
−i
(√
ω2(`)−1+`
)
pi
= C+,1e
i
(√
ω2(`)−1−`
)
pi
+ C+,2e
−i
(√
ω2(`)−1+`
)
pi
,
C0,1e
i
(√
ω2(`)−1−`
)
pi
+ C0,2e
−i
(√
ω2(`)−1+`
)
pi
= C−,1e
i
(√
ω2(`)−1−`
)
pi
+ C−,2e
−i
(√
ω2(`)−1+`
)
pi
,
C0,1 + C0,2 = C+,1e
i
(√
ω2(`)−1−`
)
2pi
+ C+,2e
−i
(√
ω2(`)−1+`
)
2pi
,
C0,1 + C0,2 = C−,1e
i
(√
ω2(`)−1−`
)
2pi
+ C−,2e
−i
(√
ω2(`)−1+`
)
2pi
.
(A.5)
67
The conditions for the continuity of the fluxes at the vertices are thus given by
i
√
ω2(`)− 1 · C0,1ei
(√
ω2(`)−1−`
)
pi − i√ω2(`)− 1 · C0,2e−i(√ω2(`)−1+`)pi
= i
√
ω2(`)− 1 · C+,1ei
(√
ω2(`)−1−`
)
pi − i√ω2(`)− 1 · C+,2e−i(√ω2(`)−1+`)pi
+ i
√
ω2(`)− 1 · C−,1ei
(√
ω2(`)−1−`
)
pi − i√ω2(`)− 1 · C−,2e−i(√ω2(`)−1+`)pi
i
√
ω2(`)− 1 · C0,1 − i
√
ω2(`)− 1 · C0,2
= i
√
ω2(`)− 1 · C+,1ei
(√
ω2(`)−1−`
)
2pi − i√ω2(`)− 1 · C+,2e−i(√ω2(`)−1+`)2pi
+ i
√
ω2(`)− 1 · C−,1ei
(√
ω2(`)−1−`
)
2pi − i√ω2(`)− 1 · C−,2e−i(√ω2(`)−1+`)2pi.
(A.6)
The boundary conditions (A.5) and (A.6) lead to a homogeneous system of linear equations given
by the matrix equation
M · C = 0,
where C = (Cj,1, Cj,2)
T with j ∈ {0,+,−} is the 6-dimensional solution vector of the matrix M .
It is obvious that a homogeneous system has nontrivial solutions if det(M) = 0. We obtain
det(M) = −e−3ipi
(
3`+
√
ω2(`)−1
)
·
(
9e2ipi` − 2e2ipi
(
`+
√
ω2(`)−1
)
− 8e2ipi
(
2`+
√
ω2(`)−1
)
+9e
2ipi
(
`+2
√
ω2(`)−1
)
− 8e2ipi
√
ω2(`)−1
)
·
(
e2ipi
√
ω2(`)−1 − 1
)
· (ω2(`)− 1) , (A.7)
which leaves us to solve the equations for the factors of the product (A.7) separately. While
e
−3ipi
(
3`+
√
ω2(`)−1
)
6= 0 for every ` ∈ T1, we get for the second term
9e2ipi` − 2e2ipi
(
`+
√
ω2(`)−1
)
− 8e2ipi
(
2`+
√
ω2(`)−1
)
+ 9e
2ipi
(
`+2
√
ω2(`)−1
)
− 8e2ipi
√
ω2(`)−1
= 9e2ipi`e2ipi
√
ω2(`)−1
(
e2ipi
√
ω2(`)−1 + e−2ipi
√
ω2(`)−1
)
− 2e2ipi`e2ipi
√
ω2(`)−1
− 8e2ipi`e2ipi
√
ω2(`)−1
(
e2ipi` + e−2ipi`
)
= 2e
2ipi
(
`+
√
ω2(`)−1
) (
9 cos
(
2pi
√
ω2(`)− 1
)
− 8 cos (2pi`)− 1
)
and the solutions ω(`) of the equation
9 cos
(
2pi
√
ω2(`)− 1
)
− 8 cos (2pi`)− 1 = 0
are given by
ω(`) = ±
√(
1
2pi
arccos
(
8
9
cos (2pi`) +
1
9
)
+m
)2
+ 1 (A.8)
for every m ∈ Z. The last two factors on the right hand side of (A.7) lead to the relation
ω(`) = ±
√
m2 + 1, ∀m ∈ Z, (A.9)
which represents the spectral bands corresponding to the eigenvalues of infinite multiplicity men-
tioned in Section 3.2.1. Using (A.8) and (A.9), we obtain the spectral curves shown in Figure
3.1.
68
A.2. Calculations for the derivation of the effective amplitude
equation
In this section, we will show a formal derivation of the amplitude equation (3.27) and its associated
complex conjugate equation. Therefore, we insert the simple approximation ansatz (3.26) given
by
V˜app(t, `) = V˜app,1(t, `) + V˜app,−1(t, `)
= A˜1
(
ε2t,
`− `0
ε
)
E1(t, `) + A˜−1
(
ε2t,
`+ `0
ε
)
E−1(t, `),
(A.10)
with U˜⊥ = 0 into the system (3.22)-(3.23). By using this ansatz, we just have to evaluate the
first equation
∂2t V˜app,1(t, `) = −(ω(m0)(`))2V˜app,1(t, `)−NV (V˜app,1, 0)(t, `). (A.11)
to obtain a NLS equation for the amplitude function A˜1. First, we look at the left-hand side of
(A.11) and with ∂t = ε
2∂T for the slow time T = ε
2T , we have
∂2t V˜app,1(t, `) = −
(
ω(m0)(`0) + ∂`ω
(m0)(`0)(`− `0)
)2
A˜1 (T, ξ)E
1(t, `)
+ 2iε2
(
ω(m0)(`0) + ∂`ω
(m0)(`0)(`− `0)
)
∂T A˜1 (T, ξ)E
1(t, `)
+ ε4∂2T A˜1 (T, ξ)E
1(t, `)
= −
(
ω(m0)(`0)
)2
A˜1 (T, ξ)E
1(t, `)
− 2ε
(
∂`ω
(m0)(`0)
)
ω(m0)(`0)ξA˜1 (T, ξ)E
1(t, `)
+ 2iε2ω(m0)(`0)∂T A˜1 (T, ξ)E
1(t, `)− ε2
(
∂`ω
(m0)(`0)
)2
ξ2A˜1 (T, ξ)E
1(t, `)
+ 2iε3
(
∂`ω
(m0)(`0)
)
ξ∂T A˜1 (T, ξ)E
1(t, `)
+ ε4∂2T A˜1 (T, ξ)E
1(t, `).
(A.12)
where ` = `0+εξ. In order to calculate the right-hand side, we now evolve the parameter ω
(m0)(`)
in terms of `0 and get
ω(m0)(`) = ω(m0)(`0) + ∂`ω
(m0)(`0)(`− `0) + 1
2
∂2`ω
(m0)(`0)(`− `0)2
= ω(m0)(`0) + ε∂`ω
(m0)(`0)ξ +
1
2
ε2∂2`ω
(m0)(`0)ξ
2.
(A.13)
Substituting (A.13) into (A.11) yields to the term
−
(
ω(m0)(`)
)2
V˜app,1(t, `) = −
[(
ω(m0)(`0)
)2
+ 2ε
(
∂`ω
(m0)(`0)
)
ω(m0)(`0)ξ
+ ε2
(
ω(m0)(`0)
(
∂2`ω
(m0)(`0)
)
+
(
∂`ω
(m0)(`0)
)2)
ξ2
+ ε3
(
∂`ω
(m0)(`0)
)(
∂2`ω
(m0)(`0)
)
ξ3
+
1
4
ε4
(
∂2`ω
(m0)(`0)
)2
ξ4
]
A˜1 (T, ξ)E
1(t, `).
(A.14)
69
For the calculation of the nonlinear terms, we recall (3.24) and write
NV (V˜app,1, 0)(t, `) =
∫
T1
∫
T1
β(`, `1, `2, `− `1 − `2)
× V˜app,1(t, `1)V˜app,1(t, `2)V˜app,1(t, `− `1 − `2)d`1d`2.
With the simple approximation ansatz (A.10), we then have in the formal order O(ε2) the con-
volution integrals
NV (V˜app,1, 0)(t, `)
= 3β(`0, `0, `0,−`0)
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜1(T, ξ1)A˜1(T, ξ2)A˜−1(T, ξ − ξ1 − ξ2)dξ1dξ2E1(t, `)
+ 3β(−`0,−`0, `0,−`0)
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜−1(T, ξ1)A˜1(T, ξ2)A˜−1(T, ξ − ξ1 − ξ2)dξ1dξ2E−1(t, `)
+ β(3`0, `0, `0, `0)
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜1(T, ξ1)A˜1(T, ξ2)A˜1(T, ξ − ξ1 − ξ2)dξ1dξ2E3(t, `)
+ β(−3`0,−`0,−`0,−`0)
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜−1(T, ξ1)A˜−1(T, ξ2)A˜−1(T, ξ − ξ1 − ξ2)dξ1dξ2E−3(t, `),
(A.15)
where the integral kernels β((j1 + j2 + j3)`0, j1`0, j2`0, j3`0) with j1,2,3 = ±1 are given by the
definition (3.25) in Section 3.4.2. Comparing the terms (A.12),(A.14) and (A.15), we obtain in
the leading order ε2E1 the NLS equation
2ω(m0)(`0)i∂T A˜1(T, ξ) = −
(
∂2`ω
(m0)(`0)
)
ω(m0)(`0)ξ
2A˜1(T, ξ)
− ν˜
∫ 1
2ε
− 1
2ε
∫ 1
2ε
− 1
2ε
A˜1(T, ξ1)A˜1(T, ξ2)A˜−1(T, ξ − ξ1 − ξ2)dξ1dξ2,
(A.16)
where ν˜ = 3β(`0, `0, `0,−`0). Note that the terms of the formal order ε2E−1 fulfill the complex
conjugate NLS equation associated to (A.16). The remaining nonlinear O(ε2)-terms do not
vanish. In order to eliminate them, we need an improved approximation ansatz which is discussed
in Section 3.5.
70
B. Appendices to Chapter 4
B.1. The function space H1,2
In this section we take a closer look on the function space H1,2(R2) defined in Chapter 4. As
mentioned before in (4.5), the space H1,2(R2) is equipped with the norm
‖u‖2H1,2 = 〈(Lx + I)u, u〉L2 + 〈(Ly + I)2u, u〉L2 + 〈u, u〉L2 (B.1)
For the justification of the DNLS equation in two dimensions, we need H1,2(R2) to satisfy several
properties. The results stated in this section are used in different parts of the proof of Theorem
4.3.1.
First, we will show that H1,2(R2) forms a Banach algebra under pointwise multiplication. There-
fore, we additionally define the space H1,2(R2) and equip it with the squared norm
‖u‖2H1,2 = ‖∂xu‖2L2 + ‖∂2yu‖2L2 + ‖∂yu‖2L2 + ‖u‖2L2 . (B.2)
It is clear that ‖u‖H1 ≤ ‖u‖H1,2 . The following lemmas now allow us to control two more norms
with (B.2), which we need in the proof for the Banach algebra.
Lemma B.1.1. There exists a positive constant CC > 0 such that
‖u‖C0b ≤ CC‖u‖H1,2 (B.3)
for all u ∈ H1,2(R2).
Proof. In this proof we use the Fourier transform F and its inverse denoted by F−1. The
continuous mapping F−1 : L1(R2)→ C0b (R2) leads to
‖u‖C0b ≤ ‖û‖L1
= ‖ûρρ−1‖L1
≤ ‖ûρ‖L2‖ρ−1‖L2 ,
where ρ(k, l) = 1 + |k| + |l|2 and û = F(u). In fact, the Fourier transform F is an isometric
isomorphism from L2(R2) to L2(R2) and by using the weight ρ we get
‖ûρ‖L2 ≤ ‖u‖H1,2 .
71
Hence, it remains to show that ‖ρ−1‖L2 <∞ to obtain the bound (B.3). We find
‖ρ−1‖2L2 =
∫ ∞
l=−∞
∫ ∞
k=−∞
(
1 + |k|+ |l|2)−2 dkdl
≤
∫ 1
l=−1
∫ ∞
k=−∞
(
1 + |k|+ |l|2)−2 dkdl + ∫ ∞
l=−∞
∫ 1
k=−1
(
1 + |k|+ |l|2)−2 dkdl
+ 4
∫ ∞
l=1
∫ ∞
k=1
(
1 + |k|+ |l|2)−2 dkdl
≤ 2
∫ ∞
k=−∞
(1 + |k|)−2 dkdl + 2
∫ ∞
l=−∞
(
1 + |l|2)−2 dkdl + 16 ∫ ∞
l=1
∫ ∞
k=1
(
k + l2
)−2
dkdl
≤ 2C1 + 2C2 − 16
∫ ∞
l=1
[(
k + l2
)−1]∞
k=1
dl
= 2C1 + 2C2 + 16 [arctan(l)]
∞
l=1
= 2C1 + 2C2 + 4pi <∞,
and with CC =
√
2C1 + 2C2 + 4pi the inequality (B.3) is proven.
Remark B.1.2. Note that this embedding just holds in two space dimensions. In the three-
dimensional case, the integral∫ ∞
ly=1
∫ ∞
lz=1
∫ ∞
k=1
(
k + l2y + l
2
z
)−2
dkdlzdly =
∫ ∞
k=1
∫ ∞
r=1
(
k + r2
)−2
rdrdk
is divergent and we cannot bound the norm ‖ρ−1‖L2 of the weight function ρ(k, ly, lz).
Using the same idea as in the proof of Lemma B.1.1, we can also estimate ‖∂yu‖L4 with the
norm of the Sobolev space H1,2 in two space dimensions.
Lemma B.1.3. There exists a positive constant CL > 0 such that
‖∂yu‖L4 ≤ CL‖u‖H1,2 (B.4)
for all u ∈ H1,2(R2).
Proof. The inverse Fourier transform F−1 is a continuous mapping from Lq to Lp with 1/p+1/q =
1 for q ≤ 2 due to the Hausdorff-Young inequality. For p = 4 and q = 4/3, this leads to the
inequality
‖∂yu‖L4 ≤ ‖ilû‖L4/3
= ‖ilûρρ−1‖L4/3
≤ ‖ûρ‖L2‖ilρ−1‖L4 ,
(B.5)
where ρ = 1 + |k| + |l|2 and ilû = F(∂yu). We also applied the generalized Ho¨lder inequality
‖uv‖Lr ≤ ‖u‖Lp‖v‖Lq with 1/p + 1/q = 1/r to obtain the last line of (B.5). The norm of the
72
weight function
‖ilρ−1‖4L4 =
∫ ∞
l=−∞
∫ ∞
k=−∞
|l|4 · (1 + |k|+ |l|2)−4 dkdl
≤
∫ 1
l=−1
∫ ∞
k=−∞
|l|4 · (1 + |k|+ |l|2)−4 dkdl + ∫ ∞
l=−∞
∫ 1
k=−1
|l|4 · (1 + |k|+ |l|2)−4 dkdl
+ 4
∫ ∞
l=1
∫ ∞
k=1
|l|4 · (1 + |k|+ |l|2)−4 dkdl
≤ 2
∫ ∞
k=−∞
(2 + |k|)−4 dkdl + 2
∫ ∞
l=−∞
|l|4 · (1 + |l|2)−4 dkdl
+ 16
∫ ∞
l=1
∫ ∞
k=1
l4 · (k + l2)−4 dkdl
≤ 2C1 + 2C2 − 16
3
∫ ∞
l=1
[
l4 · (k + l2)−3]∞
k=1
dl
= 2C1 + 2C2 +
16
3
∫ ∞
l=1
l4 · (1 + l2)−3 dl <∞
is bounded and with ‖ûρ‖L2 ≤ ‖u‖H1,2 , we complete the proof.
Remark B.1.4. With the same strategy as in the Proof of Lemma B.1.3 using the weight function
ρ =
(
1 + k2 + l2
)1/2
, we get in addition the embedding result ‖u‖L4 ≤ C˜L‖u‖H1 for all u ∈
H1(R2).
With these two embedding results, the following inequality holds.
Theorem B.1.5. The space H1,2(R2) forms a Banach algebra under pointwise multiplication,
such that
‖uv‖H1,2 ≤ CB‖u‖H1,2‖v‖H1,2 (B.6)
for every u, v ∈ H1,2(R2) and some positive constant CB > 0.
Proof. In order to obtain the inequality above, we first show that ‖u‖H1,2 ≤ ‖u‖H1,2 . Expanding
the norm defined in (B.1), we get
‖u‖2H1,2 = 〈Lxu, u〉L2 + 〈L2yu, u〉L2 + 2〈Lyu, u〉L2 + 3〈u, u〉L2 .
Now we look at the occurent terms seperately and we obtain for the quadratic term
〈L2yu, u〉L2 = 〈−∂2yu+ Vyu,−∂2yu+ Vyu〉L2
= ‖∂2yu‖2L2 + ‖Vyu‖2L2 + 〈−∂2yu, Vyu〉L2 + 〈Vyu,−∂2yu〉L2
(B.7)
and for the linear terms
〈Lyu, u〉L2 = 〈−∂2yu+ Vyu, u〉L2 = ‖∂yu‖2L2 + ‖V 1/2y u‖2L2 (B.8)
and
〈Lxu, u〉L2 = 〈−∂2xu+ Vxu, u〉L2 = ‖∂xu‖2L2 + ‖V 1/2x u‖2L2 (B.9)
73
respectively. Using (B.7),(B.8) and (B.9), we can rewrite the norm as follows:
‖u‖2H1,2 = ‖∂2yu‖2L2 + ‖Vyu‖2L2 + 〈−∂2yu, Vyu〉L2 + 〈Vyu,−∂2yu〉L2
+ 2‖∂yu‖2L2 + 2‖V 1/2y u‖2L2 + ‖∂xu‖2L2 + ‖V 1/2x u‖2L2 + 3‖u‖2L2
= ‖u‖2H1,2 + ‖Vyu‖2L2 + 〈−∂2yu, Vyu〉L2 + 〈Vyu,−∂2yu〉L2
+ ‖∂yu‖2L2 + 2‖V 1/2y u‖2L2 + ‖V 1/2x u‖2L2 + 2‖u‖2L2 .
(B.10)
Hence, we simplify the scalar products in (B.10) to
〈−∂2yu, Vyu〉+ 〈Vyu,−∂2yu〉L2 = 〈∂yu, ∂y(Vyu)〉L2 + 〈∂y(Vyu), ∂yu〉L2
= 〈∂yu, (∂yVy)u〉L2 + 〈(∂yVy)u, ∂yu〉L2
+ 2〈∂yu, Vy(∂yu)〉L2
= −〈u, (∂2yVy)u〉L2 − 〈u, (∂yVy)(∂yu)〉L2
+ 〈(∂yVy)u, ∂yu〉L2 + 2‖V 1/2y (∂yu)‖2L2
= −〈u, (∂2yVy)u〉L2 + 2‖V 1/2y (∂yu)‖2L2
= −2‖u‖2L2 + 2‖V 1/2y (∂yu)‖2L2 ,
(B.11)
where Vy, ∂yVy, ∂
2
yVy ∈ R and ∂2yVy = 2. Inserting (B.11) into (B.10), we directly get
‖u‖2H1,2 = ‖u‖2H1,2 + ‖Vyu‖2L2 + 2‖V 1/2y (∂yu)‖2L2
+ ‖∂yu‖2L2 + 2‖V 1/2y u‖2L2 + ‖V 1/2x u‖2L2
(B.12)
and thus the bound
‖u‖2H1,2 ≤ ‖u‖2H1,2 (B.13)
holds.
Now we return to the proof of the Banach algebra and write
‖uv‖2H1,2 = ‖∂2y(uv)‖2L2 + ‖Vy(uv)‖2L2 + 2‖V 1/2y (∂y(uv))‖2L2
+ 2‖∂y(uv)‖2L2 + 2‖V 1/2y (uv)‖2L2 + ‖∂x(uv)‖2L2
+ ‖V 1/2x (uv)‖2L2 + ‖uv‖2L2 .
(B.14)
We look at the particular terms on the right hand side of (B.14) separately:
• The triangle inequality yields to
‖∂2y(uv)‖L2 ≤ ‖(∂2yu)v‖L2 + 2‖(∂yu)(∂yv)‖L2 + ‖u(∂2yv)‖L2
≤ ‖∂2yu‖L2‖v‖C0b + 2‖∂yu‖L4‖∂yv‖L4 + ‖u‖C0b ‖∂
2
yv‖L2
and with the bounds (B.3), (B.4) and (B.13), we obtain
‖∂2y(uv)‖L2 ≤ 2CC‖u‖H1,2‖v‖H1,2 + 2C2L‖u‖H1,2‖v‖H1,2 ≤ C‖u‖H1,2‖v‖H1,2
74
• Using the representation (B.12) and the bound (B.3), we can estimate the term ‖Vy(uv)‖2L2
as follows:
‖Vy(uv)‖L2 ≤ ‖v‖C0b ‖Vyu‖L2
≤ CC‖v‖H1,2‖u‖H1,2
≤ CC‖v‖H1,2‖u‖H1,2 .
• The estimates for the remaining terms follow with the same idea as in the calculations
above.
In total, we get the inequality
‖uv‖2H1,2 ≤ C2B‖u‖2H1,2‖v‖2H1,2
and the function space H1,2(R2) is closed under pointwise multiplication.
In the second part of this section, we want to estimate the norm ‖φ̂n,mψj‖H1,2 . This bound will
help us to control the parameter β and the terms of the nonlinear evolution problem (4.56).
Lemma B.1.6. For every n ∈ N, we get the bound
‖φ̂n,0ψj‖H1,2 ≤ (ωj + 1)2 + (E0 + 1) + 1, (B.15)
and therefore φ̂n,0(x)ψj(y) ∈ H1,2(R2) uniformly in ε > 0.
Proof. The representation (B.1) directly leads to the equation
‖φ̂n,0ψj‖2H1,2 = 〈(Lx + I)φ̂n,0ψj , φ̂n,0ψj〉L2 + 〈(Ly + I)2φ̂n,0ψj , φ̂n,0ψj〉L2
+ 〈φ̂n,0ψj , φ̂n,0ψj〉L2 ,
and since the orthogonality and normalization relation (4.14) holds, we have
〈φ̂n,0ψj , φ̂n,0ψj〉L2 =
∫
R2
φ̂n,0(x)ψj(y)φ̂n,0(x)ψj(y)dr
=
∫ ∞
x=−∞
φ̂n,0(x)φ̂n,0(x)dx ·
∫ ∞
y=−∞
ψj(y)ψj(y)dy
= 1.
Now it remains to calculate the scalar products associated with the one-dimensional differential
operators Lx and Ly. By applying (4.13) and using the band boundedness (4.16), we obtain
〈(Lx + I)φ̂n,0ψj , φ̂n,0ψj〉L2 = 〈Lxφ̂n,0ψj , φ̂n,0ψj〉L2 + 〈φ̂n,0ψj , φ̂n,0ψj〉L2
= 〈
∑
m′∈Z
Ên,−m′ φ̂n,m′ψj , φ̂n,0ψj〉L2 + 1
=
∑
m′∈Z
Ên,m′〈φ̂n,m′ψj , φ̂n,0ψj〉L2 + 1
= Ên,0 + 1
≤ E0 + 1,
75
while the scalar product associated with the quadratic term (Ly + I)
2 is estimated as follows:
〈(Ly + I)2φ̂n,0ψj , φ̂n,0ψj〉L2 = 〈L2yφ̂n,0ψj , φ̂n,0ψj〉L2 + 〈Lyφ̂n,0ψj , φ̂n,0ψj〉L2
+ 〈φ̂n,0ψj , Lyφ̂n,0ψj〉L2 + 〈φ̂n,0ψj , φ̂n,0ψj〉L2
= 〈Lyφ̂n,0ψj , Lyφ̂n,0ψj〉L2 + 2〈Lyφ̂n,0ψj , φ̂n,0ψj〉L2 + 1
= 〈ωjφ̂n,0ψj , ωjφ̂n,0ψj〉L2 + 2〈ωjφ̂n,0ψj , φ̂n,0ψj〉L2 + 1
= ω2j 〈φ̂n,0ψj , φ̂n,0ψj〉L2 + 2ωj〈φ̂n,0ψj , φ̂n,0ψj〉L2 + 1
= ω2j + 2ωj + 1
= (ωj + 1)
2 ,
where Lyψj = ωjψj . In total, we conclude
‖φ̂n,0ψj‖H1,2 ≤ (ωj + 1)2 + (E0 + 1) + 1 <∞.
B.2. Computation of the projection Πn,j
In this section, we will focus on the explicit calculation of the O(µ3/2) terms of the residual
(4.34) by using the orthogonal projection Πn,j introduced in Lemma 4.4.2. First, we recall the
decomposition (4.35) and the differential equation (4.45) to obtain
1
σ
(
L−
(
Ên,0 + ωj
))
ϕµ + |ϕ0|2 ϕ0 −
∑
m∈Z
β |am|2 amφ̂n,mψj
= Πn,j |ϕ0|2 ϕ0 −
∑
m∈Z
β |am|2 amφ̂n,mψj .
(B.16)
In order to further simplify (B.16), we need Πn,j |ϕ0|2 ϕ0 to be represented by an expansion of
the functions φ̂n,m(x)ψj(y) ∈ En,j ⊂ L2(R2) and thus rewrite the projection as follows:
Πn,j |ϕ0|2 ϕ0 =
∑
m∈Z
〈|ϕ0|2 ϕ0, φ̂n,mψj〉L2 · φ̂n,mψj .
Using ϕ0(T, r) =
∑
m∈Z am(T )φ̂n,m(x)ψj(y), the nonlinear term in the scalar product is given
by
|ϕ0|2 ϕ0 =
∑
m1∈Z
am1 φ̂n,m1ψj
 ·
∑
m2∈Z
am2 φ̂n,m2ψj
 ·
∑
m3∈Z
am3 φ̂n,m3ψj

=
∑
(m1,m2,m3)∈Z3
am1am2am3 φ̂n,m1ψjφ̂n,m2ψjφ̂n,m3ψj
and therefore, we get
Πn,j |ϕ0|2 ϕ0 =
∑
m∈Z
∑
(m1,m2,m3)∈Z3
〈φ̂n,m1ψjφ̂n,m2ψjφ̂n,m3ψj , φ̂n,mψj〉L2
× am1am2am3 φ̂n,mψj
=
∑
m∈Z
∑
(m1,m2,m3)∈Z3
κj(m,m1,m2,m3)am1am2am3 φ̂n,mψj
(B.17)
76
where the integral kernel is defined as in (4.47). Inserting (B.17) into (B.16) yields to the equation
Πn,j |ϕ0|2 ϕ0 −
∑
m∈Z
β |am|2 amφ̂n,mψj
=
∑
m∈Z
∑
(m1,m2,m3)∈Z3
κj(m,m1,m2,m3)am1am2am3 φ̂n,mψj
−
∑
m∈Z
κj(0, 0, 0, 0)amamamφ̂n,mψj ,
(B.18)
with β = ‖φ̂n,0ψj‖4L4 = κj(0, 0, 0, 0). Now, we want to summarize the two terms on the left
hand side of (B.18) and thereby rearrange the sum to∑
m∈Z
∑
(m1,m2,m3)∈Z3
κj(m,m1,m2,m3)am1am2am3 φ̂n,mψj
−
∑
m∈Z
κj(0, 0, 0, 0)amamamφ̂n,mψj
=
∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1am2am3 φ̂n,mψj
+
∑
m∈Z
(κj(m,m,m,m)− κj(0, 0, 0, 0)) amamamφ̂n,mψj .
Since φ̂n,m(x) = φ̂n,0(x−2pim), the difference of the integral kernels κj(m,m,m,m)−κj(0, 0, 0, 0) =
0 and we obtain ∑
(m1,m2,m3)∈Z3/
{(m,m,m)}
κj(m,m1,m2,m3)am1am2am3 φ̂n,mψj
for the terms (B.16) of the formal order O(µ3/2) of the residual Res(µ1/2Ψ).
77

Bibliography
[1] M.J. Ablowitz, Ch.W. Curtis, and Yi Zhu, “On tight-binding approximations in optical lat-
tices”, Stud. Appl. Math. 129 (2012), 362-388.
[2] R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, “Constrained energy minimization and
orbital stability for the NLS equation on a star graph”, Ann. Inst. H. Poincare´ AN 31 (2014),
1289-1310.
[3] R. Adami, C. Cacciapuoti, D. Finco, and D. Noja, “Variational properties and orbital stability
of standing waves for NLS equation on a star graph”, J. Differential Equations 257 (2014)
3738-3777.
[4] R. Adami, E. Serra, and P. Tilli, “NLS ground states on graphs”, Calc. Var. PDEs 54 (2015),
743-761.
[5] R. Adami, E. Serra, and P. Tilli, “Threshold phenomena and existence results for NLS ground
state on metric graphs”, J. Funct. Anal. 271 (2016), 201-223.
[6] A. Aftalion and B. Helffer, “On mathematical models for Bose-Einstein condensates in optical
lattices”, Rev. Math. Phys. 21 (2009), 229-278.
[7] G. Alfimov, P. Kevrekidis, V. Konotop, and M. Salerno, “Wannier functions analysis of the
nonlinear Schro¨dinger equation with a periodic potential”, Phys. Rev. E 66 (2002), 046608.
[8] D. Bambusi and A. Sacchetti, “Exponential times in the one-dimensional Gross-Pitaevskii
equation with multiple well potential”, Comm. Math. Phys. 275 (2007), 1-36.
[9] G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, Mathematical Surveys
and Monographs 186 (Providence, RI: AMS, 2013).
[10] K. Busch, G. Schneider, L. Tkeshelashvili, and H. Uecker, “Justification of the nonlinear
Schro¨dinger equation in spatially periodic media”, Z. Angew. Math. Phys. 57 (2006), 905-
939.
[11] T. Cazenave, Semilinear Schro¨dinger equations, Courant Lecture Notes in Mathematics 10,
New York University, Courant Institute of Mathematical Sciences, New York; American
Mathematical Society, Providence, RI, 2003.
[12] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantenmechanik. Band 1, (de Gruyter, Berlin,
2007).
[13] T. Dohnal, D. Pelinovsky, and G. Schneider, “Coupled-mode equations and gap solitons in
a two-dimensional nonlinear elliptic problem with a separable periodic potential”, J. Nonlin.
Sci. 19 (2009), 95-131.
79
[14] M.S.P. Eastham, The spectral theory of periodic differential equations, Texts in Mathematics,
Edinburgh-London: Scottish Academic Press. X, 130 p. (1973).
[15] Th. Gallay, G. Schneider, and H. Uecker, “Stable transport of information near essentially
unstable localized structures”, Discrete Contin. Dyn. Syst.Ser. B 4 (2004), 349-390.
[16] I.M. Gelfand, “Expansion in eigenfunctions of an equation with periodic coefficients”, Dokl.
Akad. Nauk. SSSR 73 (1950), 1117-1120.
[17] S. Gilg, D. Pelinovsky, and G. Schneider, “Validity of the NLS approximation for periodic
quantum graphs”, Nonlinear Differ. Equ. Appl. 23, 63 (2016).
[18] S. Gnutzmann and U. Smilansky, “Quantum graphs: applications to quantum chaos and
universal spectral statistics”, Adv. Phys. 55 (2006), 527-625.
[19] P. Kirrmann, G. Schneider, and A. Mielke, “The validity of modulation equations for extended
systems with cubic nonlinearities”, Proc. Roy. Soc. Edinburgh A 122 (1992), 85-91.
[20] E. Korotyaev and I. Lobanov, “Schro¨dinger operators on zigzag nanotubes”, Ann. Henri
Poincare 8 (2007), 1151-1176.
[21] P. Kuchment and O. Post, “On the spectra of carbon nano-structures”, Commun. Math.
Phys. 275 (2007), 805-826.
[22] E. Lieb, R. Seiringer, J. Solovej, and J. Yngvason, “The Mathematics of the Bose Gas and
its Condensation”, Oberwolfach Seminars 34, (Birkha¨user, Basel, 2005).
[23] A. Maluckov, G. Gligoric´, L. Hadzˇievski, B. Malomed, and T. Pfau, “Stable periodic density
waves in Dipolar Bose-Einstein Condensates Trapped in Optical Lattices”, Phys. Rev. Lett.
108 (2012), 140402.
[24] A. Mun˜oz Mateo and V. Delgado, “Accurate one-dimensional effective description of realistic
matter-wave gap solitons”, Journal of Physics A 47 (2014), 245202.
[25] H. Niikuni, “Decisiveness of the spectral gaps of periodic Schro¨dinger operators on the
dumbbell-like metric graph”, Opuscula Math. 35 (2015), 199-234.
[26] D.Noja, Nonlinear Schro¨dinger equation on graphs: recent results and open problems, Phil.
Trans. R. Soc. A, 372 (2014), 20130002 (20 pages).
[27] W. Nolting, Grundkurs: Theoretische Physik. 5. Quantenmechanik, Teil 1, (Zimmermann-
Neufang, Ulmen, 1992).
[28] A. Pazy, Semigroups of linear operators and applications to partial differential equations,
Applied Mathematical Sciences 44 (Springer-Verlag, New York, 1983).
[29] D.E. Pelinovsky, Localization in periodic potentials: from Schro¨dinger operators to the Gross-
Pitaevskii equation, LMS Lecture Note Series 390 (Cambridge University Press, Cambridge,
2011).
[30] D. Pelinovsky, “Survey on global existence in the nonlinear Dirac equations in one dimension”,
in Harmonic Analysis and Nonlinear Partial Differential Equations (Editors T. Ozawa and M.
Sugimoto) RIMS Kokyuroku Bessatsu, B 26 (2011), 37-50.
80
[31] D. Pelinovsky and G. Schneider, “Justification of the coupled-mode approximation for a
nonlinear elliptic problem with a periodic potential”, Appl. Anal. 86, 1017-1036 (2007)
[32] D. Pelinovsky and G. Schneider, “Bounds on the tight-binding approximation for the Gross-
Pitaevskii equation with a periodic potential”, J. Diff. Eqs. 248 (2010), 837-849.
See also updated version, arXiv: 0711.2694v1.
[33] D. Pelinovsky and G. Schneider, “Bifurcations of standing localized waves on periodic
graphs”, Annales H. Poincare´ (2016), to appear, arXiv: 1603.05463.
[34] D. Pelinovsky, G. Schneider, and R. MacKay, “Justification of the lattice equation for a
nonlinear problem with a periodic potential”, Comm. Math. Phys. 284 (2008), 803-831.
[35] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, (Oxford University Press, Oxford,
2003).
[36] M. Reed and B. Simon, Methods of modern mathematical physics. III. Scattering theory,
(Academic Press, New York-London, 1979).
[37] G. Schneider, “Validity and limitation of the Newell-Whitehead equation”, Math. Nachr. 176
(1995), 249-263.
[38] G. Schneider, D.A. Sunny, and D. Zimmermann, “The NLS approximation makes wrong
predictions for the water wave problem in case of small surface tension and spatially periodic
boundary conditions”, J. Dynam. Differ. Eqs. 27(3) (2015), 1077-1099.
[39] G. Schneider and H. Uecker, “Nonlinear coupled mode dynamics in hyperbolic and parabolic
periodically structured spatially extended systems”, Asymp. Anal. 28, 163-180 (2001)
[40] G. Staffilani, “On the growth of high Sobolv norms of solutions for KdV and Schro¨dinger
equations”, Duke Math. J. 86 (1997), 109-142.
[41] A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute Bose-Einstein
condensates”, Phys. Rev. Lett. 86 (2001), 2353-2356.
[42] H. Uecker, D. Grieser, Z. Sobirov, D. Babajanov, and D. Matrasulov, “Soliton transport in
tubular networks: Transmission at vertices in the shrinking limit”, Phys. Rev. E 91 (2015),
023209.
81