16 - A General nth Order Spectral Transform

Summary

INTRODUCTION.

The spectral transform was first used by Gardner et al. (1967) to solve the Korteweg-de Vrles equation

Their method was to solve the Schrödinger equation

and extract the “spectral data”, S from this solution. Then they discovered that if the “potential” q(x) evolved with respect to t according to (1.1) each item of spectral data evolved as the solution of a first order linear ordinary differential equation with t as independent variable. Thus they were able to find S at a later time and by using the Gel'fand- Levitan-Marchenko equation to reconstruct q(x) at this later time.

For some years it was thought that this was a freak result until Zakharov and Shabat (1971) solved the nonlinear Schrödinger equation

in a similar way using the spectral problem

This was followed by the famous “AKNS” paper (1974) which firmly established the spectral transform (or inverse scattering method) as a very powerful weapon in the armoury of mathematicians and physicists who were studying the “soliton” (Bullough and Caudrey, 1980).

Matrix generalizations of (1.2) and (1.4) greatly extended their scope (Wadati, 1980; Calogero and Degasperis, 1980), but it was 1980 before a third order spectral transform was discovered (Kaup, 1980; Caudrey, 1980). Higher order ones have been found since then (Mikhailov, 1981).

This paper describes a fairly general n^th order spectral transform (Caudrey, 1982) and is organized as follows. Section 2 describes the spectral problem and shows how it can be solved. Sections 3 and 4 give the spectral transform and its inverse. Section 5 shows that the original spectral transform of Gardner et al. is just a special case of this general transform (as is that of Zakharov and Shabat). Finally, Section 6 shows how a system of nonlinear Klein-Gordon equations can be solved.

THE SPECTRAL PROBLEM.

A general n^thorder spectral problem can be written in the form of a set of first order equations.