Inverse scattering transform numerical schemes for nonlinear evolution equations and the method of lines

Abstract

In 1975, Ablowitz and Ladik derived differential difference equations that have as limiting forms the nonlinear Schrödinger, Korteweg-de Vries, modified Korteweg-de Vries, nonlinear self-dual network and Toda lattice equations. In 1992 and 1993, Taha derived differential difference equations for the higher nonlinear Schrödinger, Korteweg-de Vries, and modified Korteweg-de Vries equations. These difference equations have a number of special properties. They are constructed by methods related to the inverse scattering transform (IST) and can be used as numerical schemes for their associated nonlinear evolution equations. They maintain many of the important properties of their original partial differential equations such as infinite numbers of conservation laws and solvability by IST. Numerical experiments have shown that these schemes compare very favorably with other known numerical methods. In this paper, a survey and a method of derivation of these IST numerical schemes and an implementation of these schemes by the method of lines will be presented.