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 wave s. It begins with a brief summary of properties of NLS with focusing and de focusing nonlinearities. In this summary we stress the role of the modulati onal 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 s tructures, and their interaction with resonant waves in setting up the cycl es of energy transfer in dispersive wave turbulence through direct and inve rse cascades. In the article we underline that these simple NLS models prov ide precise and demanding tests for the closure theories of dispersive wave turbulence. In the conclusion we emphasize the importance of effective sto chastic representations for the prediction of transport and other macroscop ic behavior in such deterministic chaotic nonlinear wave systems. (C) 2000 American Institute of Physics. [S0022-2488(00)01606-6].