21st Century COE Program : Mathematics of Nonlinear Structure via Singularity

Michiyuki WATANABE : COE Lecture Series

Title: “The Inverse Scattering Method” An Introduction

In 1967 Gardner, Greene, Kruskal and Miura discoverd a method of rigorous solution for the Korteweg-de Vries (KdV) equation. This method (GGKM-method) reduces the solution of the Cauchy problem for the nonlinear equation to the solution of an inverse scattering problem for an associated linear eigenvalue equation. Later, through various generalizations, the GGKM-method has been known as “The Inverse Scattering Method” (also called the method of inverse scattering transform). The aim of this lecture is to present “The Inverse Scattering Method” to graduate students who have never heard about it. Following topics will be touched upon in this lecture.
  • Solution for the Korteweg-de Vries equation by the inverse scattering method
  • Direct and inverse scattering for the Schrödinger equation on the line
  • Zakharov-Shabat system and Non-linear Schrödinger equation
  • On the inverse scattering transform of multidimensional nonlinear equation 〜Solution for the Davey-Stewartson equation〜
21st Century COE Program : Mathematics of Nonlinear Structure via Singularity