Program in Nonlinear Waves, Kinetic Theory and Hamiltonian Partial Differential Equations-Fields Institute, Spg 04

University Of Arizona, Tucson AZ

Investigators

Abstract

Abstract Award: DMS-0352061 Principal Investigator: Nicholas M. Ercolani This proposal is for support of junior US based mathematicians to participate in scientific activities at the Fields Institute during the thematic program semester in Nonlinear Waves, Kinetic Theory and Hamiltonian Partial Differential Equations (PDE) that will take place during the Spring semester of 2004. The focus of the semester will concern areas of research on PDE that are motivated by nonlinear wave theory, kinetic theory, and Hamiltonian systems. Hamiltonian PDE form a class of linear and nonlinear partial differential equations which share the property that they can be written in the form of a Hamiltonian system with infinitely many degrees of freedom, using various and sometimes nonclassical symplectic structures. Principal examples include nonlinear wave equations, nonlinear Schroedinger equations and Euler's equations for water waves. The analogy with dynamical systems raises a number of basic questions, which form a part of the motivation for this semester of focus on Hamiltonian PDE. Of particular note is the connection that has been established between the theory of kinetic equations coming from statistical mechanics, and the nonlinear systems of PDE which arise in their macroscopic limits. The paradigm is the fluid dynamical scaling limit of the Boltzmann equation, but there are numerous emerging areas of relevance for this analysis, including the beginnings of a mathematically rigorous foundation for the theory of nonlinear wave turbulence. The organizers are expecting this thematic program to be a very dynamic focus of research on Partial Differential Equations that model a variety of physical phenomena, such as planetary motions, lasers and optical fiber systems, ocean waves and fluid turbulence. The character of the program is broadly international, and it represents an opportunity for exposure for young mathematicians. The short course series, the four workshops and the symposia are especially appropriate for participation by developing research mathematicians or physicists early in their career. The organizing committee believes that the program will engender further interaction and lasting collaboration among participants from various disciplines. Additionally, the participation of women and under-represented minorities will be actively encouraged.