Generalised Fourier transform and perturbations to soliton equations

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October 2009, 12(3): 579-595. doi: 10.3934/dcdsb.2009.12.579

Generalised Fourier transform and perturbations to soliton equations

Georgi Grahovski ^1, and Rossen Ivanov ^2,

1.	School of Electronic Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland
2.	School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland

Received May 2009 Revised June 2009 Published July 2009

Abstract
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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 solutions'' 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.

Keywords: Ostrovsky equation., Inverse Scattering Method, KdV equation, Soliton Perturbations, Camassa-Holm equation.

Mathematics Subject Classification: Primary: 37K15, 37K40, 37K55; Secondary: 35P10, 35P25, 35P3.

Citation: Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

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