The Whitham Equations and Their Solutions

Ohio State University, The, Columbus OH

Investigators

Abstract

NSF Award Abstract - DMS-0103849 Mathematical Sciences: The Whitham Equations and Their Solutions Abstract DMS-0103849 Tian The Principal Investigator will consider a variety of problems concerning the Whitham equations, which describe the macrostructure of nonlinear dispersive oscillations. In particular, the Principal Investigator will study (1) multiphase Whitham equations in one spatial dimension, and (2) Whitham equations in several spatial dimensions. The primary interest of the first project is in the interaction of single-phase oscillations and generation and propagation of double and higher phase oscillations. The basic goal of the second project is to understand how the zero phase Whitham solution develops singularities in several dimensional space. The proposed methods will be both analytical and computational. Results of this project will have broad impact in interdisciplinary work. The multiphase Whitham equations in one spatial variable play an essential role in both zero dispersion limit and modulation theories of nonlinear dispersive oscillations. They also have applications in the transmission of pulses in optical fibers. The Whitham equations in several spatial dimensions are intrinsically connected to Landau-Ginzburg models in topological field theory and the Seiberg-Witten solution in supersymmetric Yang-Mills Theory.