Dynamical Systems with Infinitely Many Degrees of Freedom in Mathematical Physics

9501226 Wayne教授的研究将集中在数学物理中出现的各种偏微分方程。他将尝试使用动力系统理论的思想来理解这些系统相空间的几何结构，以及这种结构如何影响这些方程的解。他将学习两大类方程——哈密顿偏微分方程和耗散方程。在第一种情况下，他将关注扩展工具的应用，如Kolmogorov-Arnold-Moser理论，它在有限维哈密顿动力系统的背景下被证明是有用的，到无限维的背景下，也在理解结果解的稳定性。在耗散方程的背景下，他将应用不变流形定理和重整化群来研究这些方程的长期渐近性，特别是在线性化方程具有连续谱的情况下。韦恩的研究将集中在数学物理中出现的微分方程解的行为。这些方程的解可能以非常复杂的方式表现出来，韦恩教授将开发新的技术来理解它们的行为。在更简单的例子中，这种理解的关键往往是阐明与这些方程的解相关的某些关键几何结构，韦恩教授的研究将集中在发现这些结构的类似物是否存在于他的应用中出现的更复杂、无限维度的环境中。他将要研究的各种方程出现在波在流体表面的传播、晶体的振动和化学反应中。* * *

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. ***