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 Item Open Access From short-range to contact interactions in two-dimensional many-body quantum systems(2022) Griesemer, Marcel; Hofacker, MichaelQuantum systems composed of N distinct particles in R2with two-body contact interactions of TMS type are shown to arise as limits-in the norm resolvent sense-of Schrödinger operators with suitably rescaled pair potentials.Item Open Access From short-range to contact interactions in the 1d Bose gas(2020) Griesemer, Marcel; Hofacker, Michael; Linden, UlrichFor a system of N bosons in one space dimension with two-body δ-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.Item Open Access 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 Item Open Access 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.