Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves

Research output: Contribution to journalArticle

Abstract

In this article we use one-dimensional nonlinear Schrödinger equations (NLS) to illustrate chaotic and turbulent behavior of nonlinear dispersive waves. It begins with a brief summary of properties of NLS with focusing and defocusing nonlinearities. In this summary we stress the role of the modulational instability in the formation of solitary waves and homoclinic orbits, and in the generation of temporal chaos and of spatiotemporal chaos for the nonlinear waves. Dispersive wave turbulence for a class of one-dimensional NLS equations is then described in detail - emphasizing distinctions between focusing and defocusing cases, the role of spatially localized, coherent structures, and their interaction with resonant waves in setting up the cycles of energy transfer in dispersive wave turbulence through direct and inverse cascades. In the article we underline that these simple NLS models provide precise and demanding tests for the closure theories of dispersive wave turbulence. In the conclusion we emphasize the importance of effective stochastic representations for the prediction of transport and other macroscopic behavior in such deterministic chaotic nonlinear wave systems.

Original languageEnglish (US)
Pages (from-to)4125-4153
Number of pages29
JournalJournal of Mathematical Physics
Volume41
Issue number6
StatePublished - Jun 2000

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Unstable
Nonlinear Equations
nonlinear equations
Turbulence
Nonlinear Waves
turbulence
defocusing
Stochastic Representation
Modulational Instability
Spatiotemporal Chaos
chaos
Coherent Structures
Homoclinic Orbit
Energy Transfer
Solitary Waves
Cascade
Chaos
Closure
Nonlinearity
closures

ASJC Scopus subject areas

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

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Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves. / Cai, David; McLaughlin, David W.

In: Journal of Mathematical Physics, Vol. 41, No. 6, 06.2000, p. 4125-4153.

Research output: Contribution to journalArticle

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