Nonlinear Wave Equations and Hamiltonian Systems

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9706273 Craig The proposed research is on a class of partial differential equations which are unified by their character as wave equations and by the fact that they can each be written in the form of a Hamiltonian system. Important examples of this class are the linear and nonlinear Schroedinger equations, the nonlinear wave and Klein-Gordon equations, the Korteweg deVries equation, and the Euler equations governing free surface problems in fluid dynamics. The first project of the investigator is to consider issues of existence and regularity of the initial value problem for these equations, and it will continue a study of KAM tori and other constructions of the principal features of the infinite dimensional phase space in which these equations are posed. The second project is an analysis of dispersive linear equations, principally the Schroedinger equation, addressing details of the local and microlocal regularity of the fundamental solution and its relationship to the global behavior of bicharacteristics of the principal symbol. The third research project includes a special focus on the mathematical analysis of free surface problems of fluid dynamics. The nonlinear differential equations studied in this project occur in mathematical physics and engineering in a remarkable variety of settings, from cosmology to fluid dynamics of the ocean to nonlinear optics. Work in this field addresses some of the basic characteristics of solutions of these equations, which is a contribution to our knowledge of the physical world, and which can contribute in nontrivial ways to developments in advanced technology. This project also specifically pursues the purely mathematical goal of developing the beautiful analogy between the theory of classical mechanics of last century and the modern theory of partial differential equations.