Hamiltonian partial differential equations and dynamical systems, and their applications

我的研究项目包括分析偏微分方程组(PDE)的解，这些偏微分方程组描述了在广泛的物理现象中的波的演变。我致力于研究海浪动力学问题、哈密顿力学技术及其在涡旋动力学中的应用、非线性薛定谔方程(包括与光子学有关的方程)以及许多其他关于非线性波浪的问题。我的研究主要是理论分析，*有助于对偏微分方程的理解。然而，在过去，它对其他学科产生了影响，在这些学科中，进步有助于更有效的数值模拟或对实验更深层次的概念性理解。我经常与物理学家合作，在与波浪现象有关的领域开展项目。*这项建议是为了研究物理科学中出现的特定偏微分方程组的解的性质，这些偏微分方程组描述了自由表面水波、非线性光学和连续介质力学等波动现象。这包括色散演化方程，如非线性薛定谔方程和Klein-Gordon方程，欧拉方程的涡丝演化，以及水波问题。我还计划继续我之前对不可压缩的Navier-Stokes方程的分析。*本程序有三个主要目标：*(1)从相空间分析的观点处理哈密顿偏微分方程组，将哈密顿系统的精细分析技术，如KAM理论、范式和Nekhoroshev稳定性，扩展到它们自然地摆在其中的适当无限维环境中的偏微分方程组的动力学。*(2)研究几种情况下的自由表面水波，并利用它们的哈密顿偏微分方程组的性质来描述非线性相互作用、孤立波碰撞、和渐近标度制度，如使用齐次化技术来描述可变水深上的波。*(3)考虑Navier-Stokes方程正则性理论的各个方面，包括频谱行为上界的分支，以及最近奇异集上的微局部下界。*我总是让博士后研究员和研究生参与我的研究项目。此外，我打算解决对其他科学领域具有潜在价值的问题，我计划继续寻求与非数学物理学家的有趣合作，就像我过去成功追求的那样。**

My research program involves the analysis of solutions of partial differential equations (PDEs) that describe wave evolution in a broad spectrum of physical phenomena. I am working on problems of ocean wave dynamics, techniques of Hamiltonian mechanics and their applications to vortex dynamics, nonlinear Schrodinger equations including ones relevant to photonics, and many other problems in nonlinear waves. My research mostly involves theoretical analysis,***contributing to the understanding of PDEs. However in the past it has had an impact on other disciplines, where advances lend themselves to more efficient numerical simulations or to a deeper conceptual understanding of experiments. And I often collaborate with physical scientists on projects in areas having to do with wave phenomena. *** *This proposal is for research on the properties of solutions of specific PDEs that arise in the physical sciences, which describe wave phenomena such as free surface water waves, nonlinear optics, and continuum mechanics. This includes dispersive evolution equations such as the nonlinear Schrodinger and Klein - Gordon equations, vortex filament evolution for the Euler equations, and problems of water waves. I also plan to continue my prior analysis of the incompressible Navier - Stokes equations.*** ***This program has three principal objectives: ***(1) To address Hamiltonian PDEs from the point of view of a phase space analysis, extending the refined analytic techniques of Hamiltonian systems, such as KAM theory, normal forms, and Nekhoroshev stability, to the dynamics of PDEs in the appropriate infinite dimensional setting in which they are naturally posed.***(2) To study free surface water waves in several settings, and to use their property of a Hamiltonian PDE to describe nonlinear interactions, solitary wave collisions, and asymptotic scaling regimes such as using homogenization techniques to describe waves over a variable bathymetry.***(3) To consider aspects of regularity theory for the Navier - Stokes equations, including the ramifications of upper bounds on spectral behavior, and the recent microlocal lower bounds on the singular set.***I always involve postdoctoral fellows and graduate students in my research projects. In addition, I am intending to address problems that have a potential value to other areas of science, and I plan to continue to seek out interesting collaborations with non-mathematician physical scientists, as I have successfully pursued in the past.**