Mathematical Sciences: Theory and Applications of Nonlinear Wave Equations

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Walter Craig NSF DMS 9501514 The principal investigator will pursue a research program in four areas of the theory of partial differential equations, all of which are interesting problems in mathematical analysis, and all of them motivated by problems in mathematical physics and applied mathematics. The first topic is the continued development of methods of Kolmogorov, Arnold and Moser (KAM) that are suited to systems with infinitely many degrees of freedom. This includes nonlinear evolution equations such as the nonlinear wave equation, nonlinear Schrodinger equation, versions of the Korteweg deVries equation, and the water waves system of equations for the free surface of a fluid. The intent of the work is to understand the stable motions of these nonlinear evolution equations, and to describe some of the principal invariant structures of the infinite dimensional phase spaces in which they are posed. The second project concerns further the water wave system, and its asymptotic scaling limits in the most physically important scaling regimes. This analysis sets rigorous bounds on the descriptions of waves in free surfaces by modulation theory and by long wave theories described by the Boussinesq and the Korteweg deVries equations. Several new elements of analysis have already been introduced which are useful in numerical modeling of the fluid dynamical problem. The third research project concerns the evolution of singularities of Schrodinger's equation, in both the linear and nonlinear cases. The goal is to understand the location and structure of the singularities of the fundamental solution, which is microlocal information and is related to the classical trajectories of the high energy particles. Finally, the fourth subject concerns the quantum mechanical inverse spectral problem, the object being to understand the spectral transform in the three principal settings of scattering theory, Floquet theory and the theory of random potentia ls, and to quantify the similarities between these settings. As mathematics is the language of the sciences, physical phenomena are understood through the properties of solutions of the equations of physics, which are for the most part partial differential equations. The investigator's principal interests are in the equations which describe conservative phenomena, which are most often Hamiltonian systems. The problems that are addressed in this project proposal are all of central importance in the physical and engineering sciences, and govern a remarkable variety of systems, from fluid motions of the ocean surface to the nonlinear quantum mechanics of semiconductor devices. The goals of all four projects are to understand important aspects of the solutions of these equations, all of which are relevant to the understanding of the systems, and many of them which also present very challenging problems in mathematical analysis. One observation is that it is remarkable that one finds related structure in systems which describe very different phenomena. For example in the first project there is a relation between the description of nonlinear quantum mechanics and wave phenomena in water surfaces; this becomes clear only in the mathematical analysis of the two systems. Furthermore the mathematical analysis has in some cases led to improvements in the modeling of the physical systems, and to the implementation of new computational procedures for predicting wave phenomena, which is a topic relevant to ocean and climate modeling in the Federal strategic interest area of global change and the environment.