The purpose of this project is to study mathematical models of nonlinear waves that occur in the most fundamental theories of fluids, of plasmas, of semiconductors and of other branches of physical science. In particular, the mathematical construction of exact periodic and solitary traveling water waves with vorticity will be proven, both with and without surface tension. Stability and instability phenomena will be investigated for motions of ideal fluids and physical plasmas. Electric and magnetic effects on the stability of charged particles will be studied in the context of the kinetic theory of plasmas. Stability problems for a variety of other kinds of waves will also be explored. Energy conserving waves of marginal stability, which occur in many of these scientific theories, will be emphasized. Methods of mathematical analysis are the primary tool employed in the investigations. The rigorous mathematics makes it feasible to make stable numerical computations and to understand the qualitative features of the nonlinear waves.
Nonlinear waves are encountered everywhere in the natural world. In this project we study several kinds of such nonlinear waves that are well described by mathematics, but where the mathematical equations are extremely difficult to solve. One of our objectives is to study water waves that may occur in the ocean, and to understand how they can form eddies or whirlpools and how they can become turbulent. The knowledge of specific kinds of solutions to the mathematical equations has an impact on our fundamental understanding of ocean waves and currents. Another objective is to study semiconducting materials from which computing devices are manufactured. We will analyze the basic equations that describe how the electrons can move inside and at the edges of the material. A third objective is to understand the behavior of charged particles in the vicinity of the earth's magnetic field, which affect satellite communications and the health of astronauts. A fourth objective of the project is the training of graduate and undergraduate students and postdoctoral fellows in the precise mathematical analysis of applied scientific problems such as nonlinear waves.
