Topics Associated with Nonlinear Evolution Equations and Inverse Scattering in Multidimensions

摘要

The Inverse Scattering Transform (I.S.T.) is a method to solve certain nonlinear evolution equations. There has been wide ranging interest in this method for many reasons. A surprisingly large number of physically interesting nonlinear equations can be solved via IST; there are many applications in physics including: surface waves, internal waves, lattice dynamics, plasma physics, nonlinear optics, particle physics and relativity. Mathematically speaking, the field is also quite rich, with nontrivial results in the areas of analysis, group theory, algebra, differential and algebraic geometry being used by various researchers. From the authors point of view, IST allows us to solve the Cauchy problem for these nonlinear systems. This document concentrates on questions in infinite space. All of the nonlinear equations discussed below arise as the compatibility condition of certain linear equations, one of which is identified as a scattering (direct and inverse scattering is required) problem and the other(s) serves to fix the time evolution of the scattering data.