Mathematical Sciences: Nonlinear Wave Equations of Mathematical Physics

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9401377 Craig This award support mathematical research in three areas which related the theory of partial differential equations and mathematical physics. The focus is principally on conservative evolution equations. Examples include nonlinear wave equations, Schrodinger equations, the Korteweg deVries equation and versions of the Boussinesq system. These problems can all be written as Hamiltonian systems, with infinitely many degrees of freedom. The analogy with dynamical systems can be pursued with the goal of describing the important features of solutions and the principal structures of the phase space in which they are posed. The projects include the continuation of development of the theory of Kolmogorov, Arnold and Moser to systems with infinitely many degrees of freedom, in particular, to the above nonlinear equations. The intent of this work is to understand the stable motions of nonlinear wave equations, either in neighborhoods of equilibrium points, or for perturbations of completely integrable systems. A second project concerns the evolution of singularities of solutions of the Schrodinger equation, in both linear and nonlinear cases. This requires microlocal analysis techniques in a novel setting in which there is an interdependence of properties in the spatial and Fourier domains. The final project involves work on the free surface problem of water wave, which is important in mathematical fluid dynamics as well as partial differential equations. Work will address steady free surface problems and nonstationary problems, in both two and three dimensional settings. Partial differential equations form a basis for mathematicalmodeling 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 me thods for approximation of solutions and estimates on the accuracy of these approximations. ***