12 - Solitons and the Inverse Scattering Transform

Summary

The KdV equation, (8.66), and the nonlinear Schrödinger equation (NLS), which we will derive in subsection 12.2.1, are nonlinear partial differential equations. Although there is no general analytical method for solving such equations, they are members of a family of exceptional cases where the solution can be deduced from that of a related linear equation. Such partial differential equations are said to be integrable, other notable examples being the intermediate long wave equation, the Benjamin–Ono equation, the Kadomtsev–Petviashvili equation, the Klein–Gordon equation, the sine–Gordon equation, the Boussinesq equation, the N-wave interaction equations, the Toda lattice equations, the self-dual Yang–Mills equations, and various generalisations of the nonlinear Schrödinger equation (for a review see Ablowitz and Clarkson (1991)). These include partial differential equations in one and two spatial dimensions, integro-differential equations and differential–difference equations that arise in models of physical situations ranging from fluid and solid mechanics to particle physics. (The sine–Gordon equation arises as a simplified nonlinear field equation in particle physics, and its soliton solutions are often known as Skyrmions after Tony Skyrme, Professor of Mathematical Physics at the University of Birmingham from 1964 to 1987 (see, for example, Perring and Skyrme (1962)).)

We will not consider the deep underlying reasons for the integrability of these various systems, and restrict ourselves to showing how to solve the KdV equation and the NLS equation using the inverse scattering transform.