Scattering and Inverse Scattering

Abstract

For a given potential, the scattering method has been commonly used to find the wave functions in quantum mechanics. An inverse process of this scattering is to find the potential from known scattering data. Such a process is called the inverse scattering method. If the potential satisfies a nonlinear evolution equation (the differential equation u _t = E[u], where E is a nonlinear time independent operator), sometimes there exists a linear operator whose potential is u(x,t) such that the spectrum of the linear operator is independent of time t. Hence the inverse scattering method can generate solutions to the nonlinear evolution equation by solving linear problems. This remarkable method that solves nonlinear evolution equations was invented by Kruskal, Greene, Gardner and Miura (1967), and it was first applied to find soliton solutions of an initial value problem for the Korteweg-de Vries equation. Later it was applied to other nonlinear evolution equations, such as the cubic-nonlinear Schrödinger equation, the sine-Gordon equation, the Ginzburg-Landau equation, and the Yang-Mills equations, etc. Many historical papers on this aspect can be found in the book edited by (1984) which is a collection of reprints.