Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation

A. Calini, N. M. Ercolani, D. W. McLaughlin, C. M. Schober

Research output: Contribution to journalArticle

Abstract

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrödinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the NLS. Using heuristic Mel'nikov methods, the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments. In a neighborhood of the persistent homoclinic orbits, the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study.

Original languageEnglish (US)
Pages (from-to)227-260
Number of pages34
JournalPhysica D: Nonlinear Phenomena
Volume89
Issue number3-4
StatePublished - 1996

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Homoclinic Orbit
Nonlinear equations
Chaos theory
nonlinear equations
chaos
Chaos
Nonlinear Equations
Orbits
Perturbed System
orbits
Discretization
Hamiltonians
Horseshoe
Numerical Instability
Chaotic Behavior
Waveform
dynamical systems
Irregular
Numerical Study
Dynamical systems

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. / Calini, A.; Ercolani, N. M.; McLaughlin, D. W.; Schober, C. M.

In: Physica D: Nonlinear Phenomena, Vol. 89, No. 3-4, 1996, p. 227-260.

Research output: Contribution to journalArticle

Calini, A. ; Ercolani, N. M. ; McLaughlin, D. W. ; Schober, C. M. / Mel'nikov analysis of numerically induced chaos in the nonlinear Schrödinger equation. In: Physica D: Nonlinear Phenomena. 1996 ; Vol. 89, No. 3-4. pp. 227-260.
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