Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2009-07-12
Discrete and Continuous Dynamical Systems B, v. 12 (2009), no. 3, 579 - 595
Nonlinear Sciences
Exactly Solvable and Integrable Systems
20 pages, no figures, to appear in: Discrete and Continuous Dynamical Systems B
Scientific paper
10.3934/dcdsb.2009.12.579
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.
Grahovski Georgi G.
Ivanov Rossen I.
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