CAREER: Perturbation Problems in PDE Dynamics

PI: Chongchun C. Zeng， University of virginia adms -0239389-----------------------------------------------------------------本项目主要研究一类哈密顿或近哈密顿偏微分方程动力学中的各种摄动问题，如非线性波动方程和非线性薛定谔方程等。主要问题包括：a)当存在奇异扰动时，找到正确的形式渐近性及其严格证明；B)研究具有典型定性性质的特殊轨道的邻域动力学，如周期、准周期、同斜/异斜轨道。本文将研究三种涉及微扰的相关问题：1)同斜轨道到鞍中心系统的正则哈密顿微扰，如正弦戈登呼吸子的微扰。2)含有最高阶导数的微扰。当这种扰动作用于偏微分方程时，系统的性质会发生巨大的变化，例如失去有限速度传播和/或解在时间上向后消失。将研究耗散扰动和保守扰动对动力学的影响。具有快速振荡的哈密顿运动。用平均法和形式化多尺度分析推导了极限波或薛定谔映射。除了证明在各种时间尺度上的收敛性外，还将以通常椭圆型几何奇异摄动问题的形式研究结构的稳定性。许多著名的进化偏微分方程，作为数学模型，是真实世界动力系统的近似。为了更好地理解原来的问题，我们不仅需要研究这些偏微分方程，还需要研究它们的摄动，其中可能包括粘度小或弹性弱等因素。另一方面，虽然有一些特殊的偏微分方程被很好地理解，但研究许多进化偏微分方程的定性性质和时间渐近行为通常是相当困难的。严格和形式化的渐近分析为研究接近这些特殊系统的动力学提供了有效的方法。本文主要研究与波、铁磁性等有关的正则和奇异微扰问题，涉及快速振荡、强耗散、强色散等问题。与此同时，还建议将本研究的几个方面纳入课程开发。这项工作将包括对一些现有课程进行应用改革，并开发新的研究生PDE动力学课程，为PDE提供动力系统的观点。此外，研究和教育将通过发展和改进讨论会和纵向一体化工作组而交织在一起。

PI: Chongchun C. Zeng, University of VirginiaDMS-0239389-----------------------------------------------------------------This project is concerned with various perturbation problems in the dynamics of some Hamiltonian or near Hamiltonian PDEs, such as nonlinear wave equations and nonlinear Schroedinger equations, etc. The main issues include: a) finding correct formal asymptotics andtheir rigorous justification when singular perturbations are present; b) studying the dynamics in neighborhoods of special orbits representing typical qualitative properties, such as periodic, quasi-periodic, homoclinic/heteroclinic orbits. Three types of interrelated problems involving perturbations will be investigated: 1) Regular Hamiltonian perturbations of systems with homoclinic orbits to saddle-centers, e.g. perturbations of sine-Gordon breathers. 2) Perturbations containing highest order derivatives. With this type of perturbations to PDEs, the nature of the systems can be changed dramatically, e.g. loss of finite speed propagation and/or disappearance of solutions backward in time. Impact of both dissipative and conservative perturbations on the dynamics will be studied. 3) Hamiltonian motions with fast oscillations. Averaging method and formal multi-scale analysis are applied in deriving the limiting wave or Schroedinger maps. In addition to justifying the convergence on various time scales, the structural stability will be studied, which are in the form of normally elliptic type geometric singular perturbation problems.Many well-known evolutionary PDEs, as mathematical models, are approximations of real world dynamical systems. In order to have better understanding of the original problems, we need to study not only these PDEs, but their perturbations as well, which may include factors like small viscosity or weak elasticity, etc. On the other hand, while there are some special well-understood ones, it is usually rather difficult to study the qualitative properties and temporal asymptotic behavior of many evolutionary PDEs. Rigorous and formal asymptotic analyses provide effective ways to study the dynamics of systems close to those special ones. This proposal focuses on certain regular and singular perturbation problems related to waves, ferromagnetism, etc., involving rapid oscillations, strong dissipation, strong dispersion etc. In conjunction with this, it is also proposed to incorporate several aspects of this research into curriculum development. This effort will include an application-minded reform of some current courses and the development of a new graduate PDE dynamics course presenting the dynamical system point of view for PDEs. In addition, research and education will be woven together through the development and improvement of seminars and vertically integrated work groups.