Generalised Fourier Transform and Perturbations to Soliton Equations

Authors

Georgi Grahovski, Technological University DublinFollow
Rossen Ivanov, Technological University DublinFollow

Document Type

Article

Rights

This item is available under a Creative Commons License for non-commercial use only

Disciplines

Applied mathematics

Publication Details

Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 3, 579--595.

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 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.

DOI

https://doi.org/DOI: 10.3934/dcdsb.2009.12.579

Recommended Citation

Grahovski, G. & Ivanov, R. (2009). Generalised Fourier transform and perturbations to soliton equations. Discrete and Continuous Dynamical Systems - Series B, vol. 12, no. 3, pg. 579--595. doi:10.3934/dcdsb.2009.12.579

Funder

INTAS grant No 05-1000008-7883