The Nonlinear Schrödinger Equation for the Delta-Comb Potential: Quasi-Classical Chaos and Bifurcations of Periodic Stationary Solutions

DOI

10.1142/S1402925109000145How to use a DOI?

Keywords

Gross–Pitaevskii equation; nonlinear Schrödinger equation; Bose–Einstein condensate; chaos; bifurcations; period doubling; nonlinear Bloch bands; Hamiltonian systems

Abstract

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.

Copyright

© 2009 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - D. Witthaut
AU  - K. Rapedius
AU  - H. J. Korsch
PY  - 2021
DA  - 2021/01/07
TI  - The Nonlinear Schrödinger Equation for the Delta-Comb Potential: Quasi-Classical Chaos and Bifurcations of Periodic Stationary Solutions
JO  - Journal of Nonlinear Mathematical Physics
SP  - 207
EP  - 233
VL  - 16
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000145
DO  - 10.1142/S1402925109000145
ID  - Witthaut2021
ER  -