Inverse Scattering Method

Abstract

The inverse scattering method (ISM) is important because it uses linear techniques to solve the initial value problem for a wide variety of nonlinear wave equations of physical interest and to obtain N-soliton (N = 1, 2, 3,…) solutions. The KdV two-soliton solution was the subject of Mathematica File MF08 where it was animated. The ISM was first discovered and developed by Gardner, Greene, Kruskal and Miura [GGKM67] for the KdV equation. A general formulation of the method by Peter Lax [Lax68] soon followed. This nontrivial formulation is the subject of the next few sections. It is presented to give the reader the flavor of a more advanced topic in nonlinear physics. As you will see, the inverse scattering method derives its name from its close mathematical connection for the KdV case to the quantum mechanical scattering of a particle by a localized potential or tunneling through a barrier.

Lo! thy dread empire, chaos! is restore’d:

Alexander Pope (1688–1744), English Poet