Nonlinear stability of mKdV breathers

Breather solutions of the modified Korteweg-de Vries equation are shown to be globally stable in a natural H^2 topology. Our proof introduces a new Lyapunov functional, at the H^2 level, which allows to describe the dynamics of small perturbations, including oscillations induced by the periodicity of the solution, as well as a direct control of the corresponding instability modes. In particular, degenerate directions are controlled using low-regularity conservation laws.Comment: 24 pp., submitte

Asymptotic Stability of high-dimensional Zakharov-Kuznetsov solitons

We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schr\"odinger (NLS) dynamics, are strongly asymptotically stable in the energy space in the physical region. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard. Our proofs follow the ideas by Martel and Martel and Merle, applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.Comment: 61 pages, 10 figures, accepted version including referee comment

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

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)

Description of the inelastic collision of two solitary waves for the BBM equation

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves).Comment: submitted for publication. Corrected typo in Theorem 1.