Nonlinear Waves, Solitons, and IST

Abstract

These lectures are written for a wide audience with diverse backgrounds. The subject is approached from a general perspective and overly detailed discussions are avoided. Many of the topics require only a standard background in applied mathematics.

The lectures deal with the following topics: fundamentals of linear and nonlinear wave motion; isospectral flows with associated compatible linear systems including PDE’s in 1+1 and 2+1 dimensions, with remarks on differential-difference and partial difference equations; the Inverse Scattering Transform (IST) for decaying initial data on the infinite line for problems in 1+1 dimensions; IST for 2+1 dimensional problems; remarks on self-dual Yang-Mills equations and their reductions. The first topic is extremely broad, but a brief review provides motivation for the other subjects covered in these lectures.