The purpose of this project is to study a variety of problems concerning the Whitham equations, which describe the macroscopic structure of nonlinear dispersive oscillations. In particular, the proposers will study (1) Whitham equations for the KdV hierarchy, (2) Large time behavior of the Whitham solution for the defocusing nonlinear Schroedinger equation, and (3) Whitham dynamics in equilibrium measures and random matrices. The primary goal of the first project is to understand the effect of the hyperbolic degeneracy of the Whitham equations on the nonlinear dispersive oscillations. The basic interest of the second project is in the large time behavior of the oscillations generated by the defocusing nonlinear Schroedinger equation. The purpose of the third project is to study phase transition phenomena in equilibrium measures and random matrices. The proposed methods will be bothanalytical and computational.
Many wave dynamics in nature undergo dispersive processes, while the dissipative or diffusive mechanisms are negligible. Examples include magneto-hydrodynamic waves in plasmas and various nonlinear waves in optics. When the dispersive term is small, there appear regions in space-time which are filled with small scale oscillations. Such phenomena have been observed as collisionless shock in plasmas and optical shocks in optical fibers. A general description of these nonlinear dispersive oscillations has not been accomplished. The main purpose of this proposal is to develop a precise mathematical analysis to study those oscillations. The basic equation to describe those phenomena is so-called the Whitham equations, which also play an essential role in both zero dispersion limit and modulation theories of nonlinear dispersive oscillations. Nonlinear dispersive waves also have physical applications in the transmission and compression of pulses in optical fibers. The Whitham dynamics in equilibrium measures and random matrices have been found to be intrinsically connected to orthogonal polynomials in approximation theory and the Seiberg-Witten solution of suppersymmetric Yang-Mills theory in high-energy physics. The program combines research and education by involving graduate students, postdocs and faculty in the area of Applied Mathematics and Applied Sciences.
