| The nonlinear Schrödinger equation as both a PDE and a dynamical system (2002) | |||||||||||||||||
Abstract | |||||||||||||||||
| Nonlinear dispersive wave equations provide excellent examples of innite dimensional dynamical systems which possess diverse and fascinating phenomena including solitary waves and wave trains, the generation and propagation of oscillations, the formation of singularities, the persistence of homoclinic orbits, the existence of temporally chaotic waves in deterministic systems, dispersive turbulence and the propagation of spatiotemporal chaos. Nonlinear dispersive waves occur throughout physical and natural systems whenever dissipation is weak. Important applications include nonlinear optics and long distance communication devices such as transoceanic optical bers, waves in the atmosphere and the ocean, and turbulence in plasmas. Examples of nonlinear dispersive partial dierential equations include the Korteweg de Vries equation, nonlinear Klein Gordon equations, nonlinear Schrodinger equations, and many others. In this survey article, we choose a class of nonlinear Schrodinger equations (NLS) as prototypal examples, and we use members of this class to illustrate the qualitative phenomena described above. Our viewpoint is one of partial dierential equations on the one hand, and innite dimensional dynamical systems on the other. In particular, we will emphasize global qualitative information about the solutions | |||||||||||||||||
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