Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Generalised Fourier transform and perturbations to soliton equations

Pages: 579 - 595, Volume 12, Issue 3, October 2009      doi:10.3934/dcdsb.2009.12.579

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Georgi Grahovski - School of Electronic Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland (email)
Rossen Ivanov - School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland (email)

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.

Keywords:  Inverse Scattering Method, Soliton Perturbations, KdV equation, Camassa-Holm equation, Ostrovsky equation.
Mathematics Subject Classification:  Primary: 37K15, 37K40, 37K55; Secondary: 35P10, 35P25, 35P30.

Received: May 2009;      Revised: June 2009;      Available Online: July 2009.