Homoclinic orbits and chaos in discretized perturbed NLS systems. I: Homoclinic orbits
type: Item
instance of: scholarly article
title: Homoclinic orbits and chaos in discretized perturbed NLS systems. I: Homoclinic orbits
zbMATH Open document ID: 0870.35093
DOI: 10.1007/BF02678088
published in: Journal of Nonlinear Science
publication date: 1997-08-27T00:00:00Z
review text: The authors study the \(N\)-particle dynamical system \[ iq_{n} = (1/h^{2}) [ q_{n+1} - 2q_{n} + q_{n-1} ] + |q_{n}|^{2}(q_{n+1} + q_{n-1}) \] \[ -2\omega^{2}q_{n} + i\epsilon [ -\alpha q_{n} + (\beta / h^{2}) (q_{n+1} - 2q_{n} + q_{n-1}) + \Gamma ], \quad q_{n+N} = q_{n}, q_{N-n} = q_{n}, \] where \(i = \sqrt{-1}\), which is a finite difference discretization of a nonlinear Schrödinger equation. The existence of homoclinic orbits in the \(2(M+1)\)-dimensional finite difference approximation is established for any finite \(M\), where \( M = N/2 \;(N \text{ even})\), or \(M = (N-1)/2 \;(N \text{ odd})\). A symbol dynamics studied in other papers is given in the references.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 1016994
zbMATH Keywords: discrete nonlinear Schrödinger equation
zbMATH Keywords: spectral theory
zbMATH Keywords: persistent invariant manifolds
zbMATH Keywords: Fenchel fibers
zbMATH Keywords: Melnikov analysis
zbMATH Keywords: homoclinic orbits
author: Y. Charles Li
author: David W. McLaughlin
reviewed by: Plamen Yalamov
MaRDI profile type: Publication
cites work: A Nonlinear Difference Scheme and Inverse Scattering
cites work: Correlations between Chaos in a Perturbed Sine-Gordon Equation and a Truncated Model System
cites work: Q5658861
cites work: Q4771890
cites work: Q4141818
cites work: Geometric singular perturbation theory for ordinary differential equations
cites work: Orbits homoclinic to resonances: The Hamiltonian case
cites work: Invariant manifolds
cites work: Q4847945
cites work: The stable, center-stable, center, center-unstable, unstable manifolds
cites work: Orbits homoclinic to resonances, with an applications to chaos in a model of the forced and damped sine-Gordon equation
cites work: Morse and Melnikov functions for NLS PDE's
cites work: Q4870176
cites work: Q4880398
description: scientific article; zbMATH DE number 1016994