08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access Approximation of a two‐dimensional Gross-Pitaevskii equation with a periodic potential in the tight‐binding limit(2024) Gilg, Steffen; Schneider, GuidoThe Gross-Pitaevskii (GP) equation is a model for the description of the dynamics of Bose-Einstein condensates. Here, we consider the GP equation in a two‐dimensional setting with an external periodic potential in the x‐direction and a harmonic oscillator potential in the y‐direction in the so‐called tight‐binding limit. We prove error estimates which show that in this limit the original system can be approximated by a discrete nonlinear Schrödinger equation. The paper is a first attempt to generalize the results from [19] obtained in the one‐dimensional setting to higher space dimensions and more general interaction potentials. Such a generalization is a non‐trivial task due to the oscillations in the external periodic potential which become singular in the tight‐binding limit and cause some irregularity of the solutions which are harder to handle in higher space dimensions. To overcome these difficulties, we work in anisotropic Sobolev spaces. Moreover, additional non‐resonance conditions have to be satisfied in the two‐dimensional case.Item Open Access The validity of modulation equations for extended systems with cubic nonlinearities(1992) Kirrmann, Pius; Schneider, Guido; Mielke, AlexanderModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems wuh no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type. As an example for the dIssipative case. we find that the Ginzburg-Landau equation is the modulation equation for the Swift-Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equatton is shown to describe the modulation of wave packets in the Sine-Gordon equation.Item Open Access Nonlinear dynamics of modulated waves on graphene like quantum graphs(2022) Gilg, Steffen; Schneider, Guido; Uecker, HannesWe consider cubic Klein-Gordon equations on infinite two‐dimensional periodic metric graphs having for instance the form of graphene. At non‐Dirac points of the spectrum, with a multiple scaling expansion Nonlinear Schrödinger (NLS) equations can be derived in order to describe slow modulations in time and space of traveling wave packets. Here we justify this reduction by proving error estimates between solutions of the cubic Klein–Gordon equations and the associated NLS approximations. Moreover, we discuss the validity of the modulation equations appearing by the same procedure at the Dirac points of the spectrum.Item Open Access The KdV approximation for a system with unstable resonances(2019) Schneider, GuidoThe KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper, we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. By working in spaces of analytic functions, the approach will allow us to handle more complicated systems without a detailed discussion of the resonances and without finding a suitable energy.Item Open Access Interchanging space and time in nonlinear optics modeling and dispersion management models(2022) Fukuizumi, Reika; Schneider, GuidoInterchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so-called dispersion management equation. It is the purpose of this paper to answer the question whether the dispersion management equation or other modulation equations are more than phenomenological models in this situation. Using Floquet theory we prove that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation and NLS like modulation equations with constant coefficients can be derived and justified through error estimates under the assumption that rather strong stability and non-resonance conditions hold. This is the first NLS approximation result documented for time-periodic dispersive systems. We explain that the failure of these conditions allows us to prove that these modulation equations in general make wrong predictions. The reasons for this failure and the behavior of the system for a fiber periodicity much larger than the wave length of light shows that interchanging the role of space and time for glass fibers with a periodic structure leads to unwanted phenomena.Item Open Access A robust way to justify the derivative NLS approximation(2023) Heß, Max; Schneider, GuidoThe derivative nonlinear Schrödinger (DNLS) equation can be derived as an amplitude equation via multiple scaling perturbation analysis for the description of the slowly varying envelope of an underlying oscillating and traveling wave packet in dispersive wave systems. It appears in the degenerated situation when the cubic coefficient of the similarly derived NLS equation vanishes. It is the purpose of this paper to prove that the DNLS approximation makes correct predictions about the dynamics of the original system under rather weak assumptions on the original dispersive wave system if we assume that the initial conditions of the DNLS equation are analytic in a strip of the complex plane. The method is presented for a Klein-Gordon model with a cubic nonlinearity.