Stability Theory of Nonlinear Waves

Brown University, Providence RI

Investigators

Abstract

NSF Award Abstract - DMS-0071838 Mathematical Sciences: Stability Theory of Nonlinear Waves Abstract 0071838 Strauss This project studies mathematical models of various kinds of waves that occur in the theories of plasmas, of semiconductors, of fluids, of mechanical vibrations and of other branches of physical science. The rigorous mathematics makes it possible both to make stable numerical computations in, and to understand the qualitative features of, such physical phenomena. Stability and instability phenomena are investigated, particularly in the kinetic theory of charged particles. Numerical simulation of semiconducting materials provides the rationale for a study of hybrid quantum-kinetic modeling. Stability problems for a variety of other kinds of waves, such as solitary waves in fluids and equilibria in solids, are also explored. Energy conserving waves of marginal stability, which occur in many of these scientific theories, are emphasized. Methods of mathematical analysis are the primary tool employed in the investigations. One purpose of this project is to analyze the structure of semiconducting materials that are used to manufacture computer chips. As chips become smaller, quantum effects become increasingly important. Since simulation of the quantum mechanics of an entire device is computationally infeasible, this project investigates hybrid models combining macroscopic kinetic theory for the bulk of a device with quantum descriptions in localized regions. Another goal of the project is to study the stability of physical plasmas, important in the shielding effect of the earth's ionosphere as well as in fusion reactors. The development of personnel, including graduate and undergraduate students, who are trained in the precise mathematical analysis of applied scientific problems, is an important outgrowth of the project.