Mathematical Sciences: Invariant Manifolds and Attracting Sets of Nonlinear Partial Differential Equations

这个数学研究项目的中心主题是非线性偏微分方程的分析。特别强调将描述不变流形和吸引可积非线性方程的摄动集的方法扩展到高维半线性微分方程的情况。还将研究确定最能描述有限维吸引集的全局坐标的方法。这些研究在重要物理问题上的应用包括对描述约瑟夫森结的阻尼和驱动正弦-戈登方程的研究，以及对描述液体中气泡云的欧拉-罗利-普莱塞特方程系统的分析。额外的工作包括研究改变扰动方程空间结构的扰动；特别要讨论这些方程是否存在呼吸解的问题。这些答案对包含解的相空间辛几何的影响也将被考虑。本研究的长期目标包括应用在微扰可积微分方程研究中发展起来的分岔技术，以提高对一般流体方程的理解

The central theme of this mathematical research project is the analysis of nonlinear partial differential equations. Particular emphasis will be placed on extending methods which describe invariant manifolds and attracting sets of perturbations of integrable nonlinear equations to the case of higher dimensional semilinear differential equations. Work will also be done in developing methods for determining global coordinates that best describe the finite dimensional attracting sets. Application of these investigations to problems of physical importance include studies of the damped and driven sine-Gordon equation describing the Josephson junction and the analysis of a system of Euler-Raleigh-Plesset equations describing a cloud of gas bubbles in liquid. Additional work includes studies of perturbations that change the spatial structure of the perturbed equations; in particular the question of existence of breather solutions to these equations will be taken up. The bearing these answers have on the symplectic geometry of the phase space containing the solutions will also be considered. Long term objectives of this research include the application of bifurcation techniques developed in the study of perturbed integrable differential equations to improved understanding of general fluid equations.***