CAREER: Perturbation Problems in PDE Dynamics

主要研究者：Chongchun C.本项目主要研究非线性波动方程和非线性薛定谔方程等一类哈密顿或近似哈密顿偏微分方程动力学中的各种扰动问题，主要问题包括：a）当存在奇异摄动时，找到正确的形式渐近性及其严格的证明; B）研究具有典型定性性质的特殊轨道（如周期、拟周期、同宿/异宿轨道）邻域内的动力学。本文将研究三类相互关联的扰动问题：1）同宿轨到鞍心的系统的正则Hamilton扰动，例如sine-Gordon呼吸子的扰动。2)包含最高阶导数的扰动。通过对偏微分方程的这种扰动，系统的性质可以显著改变，例如有限速度传播的损失和/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。3)具有快速振荡的汉密尔顿运动。平均方法和形式多尺度分析被应用于推导极限波或薛定谔映射。除了证明在不同时间尺度上的收敛性外，我们还将研究结构稳定性问题，这类问题的形式是通常的椭圆型几何奇异摄动问题。许多著名的演化偏微分方程作为数学模型，都是真实的世界动力系统的近似。为了更好地理解原始问题，我们不仅需要研究这些偏微分方程，但他们的扰动，以及，其中可能包括小粘度或弱弹性等因素，另一方面，虽然有一些特殊的理解，它通常是相当困难的定性性质和时间渐近行为的许多演化偏微分方程的研究。严格的形式渐近分析为研究与这些特殊系统接近的系统的动力学行为提供了有效的途径。这个建议的重点是与波，铁磁性等有关的某些规则和奇异摄动问题，涉及快速振荡，强耗散，强色散等与此同时，还建议将本研究的几个方面纳入课程开发。这项工作将包括一个应用型的改革目前的一些课程和一个新的研究生PDE动力学课程的发展提出了动力系统的观点偏微分方程。此外，将通过发展和改进研讨会和纵向一体化工作组，将研究和教育结合在一起。

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.