Generalised Fourier Transform and Perturbations to Soliton Equations

TL;DR

This paper reviews the Generalised Fourier Transform (GFT) as a tool for analyzing small perturbations in soliton equations, illustrating its application to KdV, Ostrovsky, and Camassa-Holm equations.

Contribution

It introduces the GFT framework for soliton perturbation analysis and demonstrates its effectiveness on several integrable hierarchies, including KdV and Camassa-Holm.

Findings

GFT provides a natural basis for soliton perturbation analysis.

Application to Ostrovsky and Camassa-Holm equations shows its versatility.

Perturbation theory can modify soliton parameters to account for small disturbances.

Abstract

A brief survey of the theory of soliton perturbations is presented. The focus is on the usefulness of the so-called Generalised Fourier Transform (GFT). This is a method that involves expansions over the complete basis of `squared olutions` of the spectral problem, associated to the soliton equation. The Inverse Scattering Transform for the corresponding hierarchy of soliton equations can be viewed as a GFT where the expansions of the solutions have generalised Fourier coefficients given by the scattering data. The GFT provides a natural setting for the analysis of small perturbations to an integrable equation: starting from a purely soliton solution one can `modify` the soliton parameters such as to incorporate the changes caused by the perturbation. As illustrative examples the perturbed equations of the KdV hierarchy, in particular the Ostrovsky equation, followed by the perturbation theory for the Camassa- Holm hierarchy are presented.