The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, basedâ€¦ (More)

This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R . Unlike the defocusingâ€¦ (More)

Two-dimensional deep water waves and some problems in nonlinear optics can be described by various third order dispersive equations, modifying and generalizing the KdV as well as nonlinearâ€¦ (More)

In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in [6]. At the long-wave limit, we establish that solutions of smallâ€¦ (More)

Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-wayâ€¦ (More)

We prove that the Cauchy problem for the KP-I equation is globally well-posed for initial data which are localized perturbations (of arbitrary size) of a non-localized (i.e. not decaying in allâ€¦ (More)

We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards groundâ€¦ (More)

Fabrice Bethuel, Philippe Gravejat, Jean-Claude Saut, Didier Smets

2009

We establish the orbital stability of the black soliton, or kink solution, v0(x) = th( x âˆš 2 ), to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space.