Dynamical Systems with Infinitely Many Degrees of Freedom in Mathematical Physics

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9501226 Wayne Professor Wayne's research will focus on a variety of partial differential equations which arise in mathematical physics. He will attempt to use ideas from dynamical systems theory to understand the geometrical structure of the phase space of these systems, and how this structure influences the solutions of these equations. He will study two main classes of equations -- Hamiltonian partial differential equations, and dissipative equations. In the first context, he will be concerned with extending the applications of tools like the Kolmogorov-Arnold-Moser theory which have proven useful in the context of finite dimensional Hamiltonian dynamical systems to this infinite dimensional context and also in understanding the stability of the resulting solutions. In the context of dissipative equations he will apply both invariant manifold theorems and the renormalization group to study the long-time asymptotics of such equations, particularly in cases in which the linearized equation has continuous spectrum. %%% Wayne's research will focus on the behavior of solutions of differential equations which arise in mathematical physics. These equations have solutions which may behave in a very complicated fashion, and Professor Wayne will develop new techniques for understanding their behavior. In simpler examples the key to this understanding has often been the elucidation of certain key geometrical structures related to the solutions of these equations, and Professor Wayne's research will concentrate on discovering whether analogues of these structures exist in the more complicated, infinite dimensional contexts that arise in his applications. The sorts of equations which he will study arise in the propagation of waves on the surface of a fluid, the vibrations of crystals, and chemical reactions. ***