Generalised Fourier transform and perturbations to soliton equations

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.

Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom

“…The perturbed KdV equation can be treated within the framework of the inverse scattering approach for the KdV equation. The basics of this approach are outlined, for example, in [17,29,30]. If the initial condition is a pure soliton solution for the KdV equation, due to the perturbation waves of radiation will appear and will decrease the energy of the initial soliton.…”

On the Coriolis Effect for Internal Ocean Waves

“…The Ostrovsky equation itself is Hamiltonian and possesses three conservation laws, however it is not bi-Hamiltonian and it is not integrable by the Inverse Scattering Method [2]. Solutions from perturbations of the KdV solitons can be derived in principle, although this is technically difficult, see for example [8] and the references therein. Various other aspects of the equation have been studied extensively by now in numerous works, see for example [9,13] and the references therein.…”

Stability of Solitary Waves and Wave-Breaking Phenomena for the Two-Component Camassa-Holm System

“…Now come to the proof of (4.9). Given x ∈ R. Set 26) with q(t, x) established in (4.3). With these notations and M, γ described in (4.13) the first equation of (4.1) gives 27) for t ∈ [0, T ) with…”

“…A rigorous justification of the derivation of the CH equation as an approach to the governing equations for water waves was recently provided by Constantin and Lannes [16]. Equation (1.2) is integrable, the inverse spectral approach being presented in [17] and [11]; while the inverse scattering approach was obtained in [14] (see also [26]). If A = 0, (1.2) possesses solutions of the form u(t, x) = ce −|x−ct| , c = 0, called peaked solitons or 'peakons' [6,7,16], as these waves form replicate a feature characteristic of the traveling wave solutions to the governing equations for water waves of largest possible amplitude [9,12].…”