This project will study multi-scale wave dynamics in problems governed by several nonlinear, time-dependent partial differential equations such as the Euler-Poisson system and Euler equations. These equations model a variety of physical processes involving multiple length scales, requiring significantly new mathematical and numerical approaches. Specific topics include the scale transition, pattern formation as well as the time regularity of solutions in terms of fundamental physical scales. Mathematical analysis will include justification of scaling limits, stability of wave patterns, and numerical analysis of schemes for computing solutions of these problems.
This research has many applications including plasma sheath formation, high frequency wave propagation and dispersive wave dynamics in a variety of physical processes. The proposed work is expected to have a significant impact not only on these areas, but also to other fields dealing with multi-scale wave dynamics including, but not restricted to geophysics, materials science and mobile telecommunication technologies. The goal is to develop new mathematical tools and computational methods that will significantly advance the state of the art. The theory will be used and driven by identified practical applications. It is also expected that this study will be of fundamental importance to the general theory of nonlinear partial differential equations.
