On the stability of time-harmonic localized states in a disordered nonlinear medium

Jared C. Bronski, David W. McLaughlin, Michael J. Shelley

Research output: Contribution to journalArticle

Abstract

We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrödinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrödinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley.

Original languageEnglish (US)
Pages (from-to)1077-1115
Number of pages39
JournalJournal of Statistical Physics
Volume88
Issue number5-6
StatePublished - Sep 1997

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Harmonic
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eigenvalues
defocusing
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Unstable
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Argand diagram
nonlinear equations
Nonlinear Equations
simulation
Limiting
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Simulation

Keywords

  • Nonlinear schrödinger equation
  • Random media

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics

Cite this

On the stability of time-harmonic localized states in a disordered nonlinear medium. / Bronski, Jared C.; McLaughlin, David W.; Shelley, Michael J.

In: Journal of Statistical Physics, Vol. 88, No. 5-6, 09.1997, p. 1077-1115.

Research output: Contribution to journalArticle

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