Asymptotic analysis based on the inverse scattering method (Regularity and Singularity for Partial Differential Equations with Conservation Laws)

Abstract

This is a survey article concerning the asymptotic study of integrable systems. Technical details are often omitted in favor of informal explanations of fundamental ideas. First we briefly recall the inverse scattering method. Our presentation is based on Riemann-Hilbert problems (a kind of boundary value problems on the complex plane) rather than the Gelfand-Levian-Marchenko integral equations. We explain the method of nonlinear steepest descent, which is a powerful tool for analyzing the long-time behavior of integrable systems. Two examples are given: the defocusing integrable nonlinear Schr odinger equation (NLS) and its discrete version (IDNLS) due to Ablowitz-Ladik.