Nonlinear Waves and Inverse Scattering

Abstract

Abstract : We are continuing our research in the development of the Inverse Scattering Transform (IST). IST is a method which allow's one to solve nonlinear wave equations by solving certain related direct and inverse scattering problems. Research is really two pronged. It is necessary for us to understand and effectively solve both classical and new direct and inverse scattering problems. We use these results to find solutions to nonlinear wave equations much like one uses Fourier analysis for linear problems. Moreover the nonlinear wave equations arise naturally in physical problems. We are particularly interested in fluid dynamical applications. Although original discoveries employed IST in one spatial dimension, we have developed effective procedures to carry forth the method for multidimensional problems. In one spatial dimension solving the inverse scattering problem requires one to solve a vector Riemann-Hilbert boundary value problem. In multidimensions we have shown that the DBAR method is essential to solve the increase problem. In a special case it reduces to a Riemann-Hilbert problem (sometimes a nonlocal Riemann-Hilbert problem). The method applies to the multidimensional Schrodinger scattering problem (i.e. the Helmboltz equation) higher order scalar differential operators, multidimensional first order systems and even discrete equations i.e. difference equations. The multidimensional DBAR method is an extremely powerful method to analyze and solve inverse scattering problems. The DBAR method to solve a variety of novel and important inverse scattering problems in multidimensions.