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

In this article we use one-dimensional nonlinear Schrodinger 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 t...

[1]  Jalal Shatah,et al.  PERSISTENT HOMOCLINIC ORBITS FOR A PERTURBED NONLINEAR SCHRODINGER EQUATION , 1996 .

[2]  David W. McLaughlin,et al.  Whiskered Tori for Integrable Pde’s: Chaotic Behavior in Near Integrable Pde’s , 1995 .

[3]  Ablowitz,et al.  Numerically induced chaos in the nonlinear Schrödinger equation. , 1989, Physical review letters.

[4]  David W. McLaughlin,et al.  Morse and Melnikov functions for NLS Pde's , 1994 .

[5]  Y. Li.  Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrödinger Equations , 1999 .

[6]  Craig L. Zirbel,et al.  A mean-field statistical theory for the nonlinear Schrödinger equation , 1999, chao-dyn/9904030.

[7]  G. Haller,et al.  Homoclinic jumping in the perturbed nonlinear Schrödinger equation , 1999 .

[8]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[9]  Richard E. Blahut,et al.  Principles and practice of information theory , 1987 .

[10]  B. Herbst,et al.  The nonlinear Schro¨dinger equation: asymmetric perturbations traveling waves and chaotic structures , 1997 .

[11]  Andrew J. Majda,et al.  A one-dimensional model for dispersive wave turbulence , 1997 .

[12]  M. G. Forest,et al.  Chaotic transport and integrable instabilities in a near-integrable, many-particle, Hamiltonian lattice , 1993 .

[13]  M. Cross,et al.  Pattern formation outside of equilibrium , 1993 .

[14]  T. Brooke Benjamin,et al.  The disintegration of wave trains on deep water Part 1. Theory , 1967, Journal of Fluid Mechanics.

[15]  David Cai,et al.  Chapter 12 The nonlinear Schrödinger equation as both a PDE and a dynamical system , 2002 .

[16]  Annalisa Calini,et al.  Mel'nikov analysis of numerically induced chaos in the nonlinear Schro¨dinger equation , 1996 .

[17]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[18]  Tudor S. Ratiu,et al.  The Geometry of Hamiltonian Systems , 1991 .

[19]  S. Smale.  Differentiable dynamical systems , 1967 .

[20]  Dynamical systems and probabilistic methods in partial differential equations , 1996 .

[21]  Stephen Wiggins.  Global Bifurcations and Chaos: Analytical Methods , 1988 .

[22]  Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation , 1981 .

[23]  I. Procaccia,et al.  SCENARIO FOR THE ONSET OF SPACE-TIME CHAOS , 1998 .

[24]  D. McLaughlin,et al.  A quasi-periodic route to chaos in a near-integrable pde , 1986 .

[25]  Patrick Tabeling,et al.  Turbulence : a tentative dictionary , 1994 .

[26]  Vanden Eijnden E,et al.  Models for stochastic climate prediction. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[27]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[28]  D. McLaughlin,et al.  SPATIOTEMPORAL CHAOS AND EFFECTIVE STOCHASTIC DYNAMICS FOR A NEAR-INTEGRABLE NONLINEAR SYSTEM , 1999 .

[29]  A. A. Galeev,et al.  Basic plasma physics II , 1983 .

[30]  David W. McLaughlin,et al.  Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation , 1990 .

[31]  Jordan,et al.  Self-organization in nonlinear wave turbulence , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[33]  G. Haller,et al.  Chaos near resonance , 1999 .

[34]  A J Majda,et al.  Spectral bifurcations in dispersive wave turbulence. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[35]  C. S. Gardner,et al.  Method for solving the Korteweg-deVries equation , 1967 .

[36]  N. Tonouchi,et al.  Enhancement of cellulose production by expression of sucrose synthase in Acetobacter xylinum. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Stéphane Zaleski,et al.  A stochastic model for the large scale dynamics of some fluctuating interfaces , 1989 .

[38]  Krug,et al.  Dynamic scaling and crossover analysis for the Kuramoto-Sivashinsky equation. , 1992, Physical review. A, Atomic, molecular, and optical physics.