Research supported by this award will focus on various aspects of nonlinear evolution equations: nonlinear partial differential equations whose solutions evolve in time. The work combines classical theory of differential equations with harmonic analysis and mathematical physics. The principal areas of study include nonlinear dispersive systems arising as approximate models in propagation of nonlinear waves. Particular interest will be paid to determining the lower bound of Sobolev exponents which guarantee local well-posedness for the generalized Korteweg-deVries equation. A similar effort will be applied to the study of nonlinear Schrodinger equations. In both cases preliminary work has already improved known estimates on the exponents. Higher order equations will also be analyzed. Here the nonlinear part of the equations will be restricted to polynomials having no constant term. This class of evolution equation models waves in elastic media. Local results have already been established for the smoothness of solutions, but long-time behavior is not well understood. For instance, fifth order systems do not have solitary wave solutions. Additional work is planned on the Zakharov-Schulman systems which describe interactions of small-amplitude, high frequencey wave with acoustic type waves. Where the linear part is elliptic, the equations have been extensively studied, otherwise no results are known. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.
