CAREER: Perturbation Problems in PDE Dynamics

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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.