CAREER: Perturbation Problems in PDE Dynamics

VirginiaDMS-0239389-----------------------------------------------------------------This大学曾崇春教授研究了一些哈密顿或接近哈密顿的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，主要内容包括：a)当奇异摄动存在时，找出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的邻域动力学，如周期轨道、准周期轨道、同宿/异宿轨道。涉及摄动的三类相关问题将被研究：1)具有同宿轨到鞍心的系统的正则哈密顿摄动，例如Sine-Gordon呼吸子的摄动。2)含有最高阶导数的摄动。对于偏微分方程组的这种类型的扰动，系统的性质可能会发生巨大的变化，例如有限速度传播的损失和/或解在时间倒退时的消失。我们将研究耗散扰动和守恒扰动对动力学的影响。3)具有快速振荡的哈密顿运动。应用平均法和形式多尺度分析方法得到了极限波或薛定谔映射。除了证明在不同时间尺度上的收敛，还将研究结构的稳定性，这是以正常椭圆型几何奇异摄动问题的形式。许多著名的发展偏微分方程作为数学模型，是真实世界动力系统的近似。为了更好地理解原始问题，我们不仅需要研究这些偏微分方程，而且还需要研究它们的扰动，这些扰动可能包括小粘性或弱弹性等因素。另一方面，尽管有一些特殊的众所周知的因素，但通常很难研究许多发展偏微分方程的定性性质和时间渐近行为。严格和形式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。这项建议侧重于与波、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、强弥散等。同时，还建议将这项研究的几个方面纳入课程发展。这一努力将包括对现有的一些课程进行应用性改革，并开发一门新的研究生PDE动力学课程，展示PDE的动力系统观点。此外，将通过发展和改进研讨会和垂直整合的工作组，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.