Mathematical theory of water waves and plasmas

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The intellectual purpose of the project is to study mathematical models of waves that occur in the most fundamental theories of fluids, of plasmas, of semiconductors and of other branches of physical science. The physical properties of exact water waves with vorticity (eddies), such as precise models of ocean waves, will be studied. In particular, the pressure beneath the surface, the paths of individual particles, and the existence and location of stagnation will be investigated. The effects of stratification, due to salinity and temperature variations, on the water waves will be explored. Energy conserving waves of marginal stability, which occur in many scientific theories, will be emphasized. In particular, electric and magnetic effects on the stability of systems of charged particles, such as in fusion reactors or in the solar wind, will be studied in the context of kinetic theory. Moreover, the stability of states in the hydrodynamic model of semiconductors will be studied. Methods of mathematical analysis will be the primary tools employed in the investigations. High-performance numerical computations will also be employed. The development of personnel, including graduate and undergraduate students and postdoctoral fellows, who are trained in the precise mathematical analysis of applied scientific problems, will be an important outgrowth of the project. Furthermore, such research work leads to the development of new pedagogical approaches to the teaching of modern mathematical ideas and technical advances to science and engineering students. The rigorous mathematics makes it possible both to perform stable numerical computations in, and to understand the qualitative features of, plasma waves, mechanical vibrations and many other physical phenomena. The existence of certain kinds of exact waves and their stability or instability has an impact on our understanding of natural phenomena. The research on water waves may improve our understanding of ocean waves and currents, ship safety, the effect of salinity on the formation of whirlpools, and the effect of wind on the formation of rogue waves. It could illuminate how particulate matter moves through the ocean. The research on plasmas could explain which configurations in astrophysical plasma such as the solar wind are stable and therefore likely to be seen. The semiconductor analysis could improve our understanding of miniature semiconductor devices and thus influence their design.