Mathematical Sciences: Waves in Nonlinear Dispersive and Hyperbolic Systems

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This ongoing research program on nonlinear waves will focus on the following three topics: 1. Functional path integrals. 2. Exact modulation. 3. Three wave resonant interactions. The first two topics are concerned with the study of some relationships among equations with the Painleve property, equations solvable by the Inverse Scattering Transform method, the theory of wave modulation and representation of solution of partial differential equations by path integrals. The last topic is concerned with the study of weakly nonlinear interactions of waves in strictly hyperbolic systems. The mathematical theory of nonlinear waves is a mathema- tical model of physical wave propagation. The conclusions deduced from the theory are generally useful to our under- standing of many natural and technological phenomena. The present research program on nonlinear waves is aimed toward a better understanding of resonance and instability which are causes for failures in structural and fluid systems. The research results are expected to contribute to better designs for many such engineereing systems.