Mathematical Sciences: Generation and Propagation of Dispersive Waves in Integrable Systems:

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This project continues studies in near-integrable systems, concentrating on four main problems: solution of the small-dispersion KdV equation, the Toda shock problem, the small-dispersion long-time limit of the nonlinear Schrodinger equation, and relations of inverse scattering theory to nearly integrable evolution equations. Integrable systems of partial differential equations often have solutions whose properties can be exactly established. This is quite unlike the usual case for nonlinear partial differential equations. What is important here is that often the properties remain when the system is perturbed. Thus one can tell a great deal about otherwise intractable problems. Such problems arise in nonlinear optics (for example, in the design of long-distance communication networks using fiber optics), in the motion of waves in nonlinear acoustic media, and in flows around thin wings.