391 research outputs found
Inelastic interaction of nearly equal solitons for the BBM equation
This paper is concerned with the interaction of two solitons of nearly equal
speeds for the (BBM) equation. This work is an extension of the results
obtained in arXiv:0910.3204 by the same authors, addressing the same question
for the quartic (gKdV) equation. First, we prove that the two solitons are
preserved by the interaction and that for all time they are separated by a
large distance, as in the case of the integrable (KdV) equation in this regime.
Second, we prove that the collision is not perfectly elastic, except in the
integrable case (i.e. in the limiting case of the (KdV) equation)
Space-modulated Stability and Averaged Dynamics
In this brief note we give a brief overview of the comprehensive theory,
recently obtained by the author jointly with Johnson, Noble and Zumbrun, that
describes the nonlinear dynamics about spectrally stable periodic waves of
parabolic systems and announce parallel results for the linearized dynamics
near cnoidal waves of the Korteweg-de Vries equation. The latter are expected
to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees
partielles", Roscoff 201
Generalised Fourier Transform and Perturbations to Soliton Equations
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.Comment: 20 pages, no figures, to appear in: Discrete and Continuous Dynamical
Systems
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