Existence and properties of travelling waves for the Gross-Pitaevskii equation

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^N. Unlike the defocusing nonlinear Schr\"odinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts established by formal and numerical arguments

Orbital stability of the black soliton to the Gross-Pitaevskii equation

We establish the orbital stability of the black soliton, or kink solution, \v_0(x) = \th \big(\frac{x}{\sqrt{2}} \big), to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space.Comment: Final version published in Indiana University Mathematics Journa

Travelling waves for the Gross-Pitaevskii equation II

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, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.Comment: Final version accepted for publication in Communications in Mathematical Physics with a few minor corrections and added remark

Travelling-waves for the Gross-Pitaevskii equation

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 ground states for the Kadomtsev-Petviashvili equation (KP I).