Many physical problems have multiple temporal and spatial scales that pose tremendous difficulties for mathematical analysis, and modeling as well as numerical simulation. The main objective of this research project is to develop analytic and numerical tools to study multi-scale wave dynamics in problems governed by a selected set of partial differential equations, including Schroedinger equations and Fokker-Planck equations, arising in different applications. These equations provide a detailed and accurate description of relevant physical processes in different scales.

Our research aims to understand not only individual mathematical models but also their effective descriptions when physical parameters vary. We will consider some pressing issues from two primary target areas: semi-classical dynamics in Schrroedinger equations, and kinetic description for polymers. Our approach is to determine the solution structure, predict scaling dynamics and numerically capture the rich phenomena at physically relevant scales, and to further use them to pursue new applications in important emerging areas of science, technology and industry.