Mathematical Sciences: Coherence and Chaos in PDE's and Nonlinear Wave Propagation

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This project involves research on nonlinear dynamical systems exhibiting wave phenomena. It has two main goals: (1) to understand to what extent the chaotic behavior in partial differential equations is finite dimensional; (2) to study a variety of propagation phenomena. Under goal (1) the investigations will focus on two kinds of systems described by partial differential equations: nearly integrable systems described by forced damped nonlinear Schrodinger equation and by sine-Gordon equation. They are natural extensions into PDE's of forced damped pendulum, a problem used as a prototype for studying chaos in ordinary differential equations. These equations have a natural choice of a basis, with which the notion of a nonlinear power spectrum can be described explicitly for the first time. The near integrability of these equations provides sufficient analytical control to yield precise, detailed theoretical information about the qualitative behavior. The far from integrable systems include flows in finite geometries, turbulent wakes and boundary layers. Here the issue is to find an optimal decomposition of the flow field for which the dimension of the subspace of modes which dominate the dynamics is minimal. Under goal (2), the propagation of nonlinear waves will be studied in a variety of contexts, trying to answer questions such as the existence of Anderson localization in nonlinear waves propagating in a random medium, theoretical foundations of nonlinear optics, and mathematical theory of singularities of envelope equations. This research is part of a larger effort in the nonlinear analysis of dynamical systems which has been developed in this country in the last ten years. Studies like this are important for a deep understanding of a number of nonlinear phenomena in the existing dynamical systems encountered in technology.