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 (see [4,3,8]) 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.and such that u p is solution to the minimization problemRemark 2. In [4], existence of solutions is obtained for small prescribed speeds c. Instead of using minimization under constraint, the idea there is to introduce, for given c, the Lagrangianwhose critical points are solutions to (TWc), and then to apply a mountain-pass argument.Although it is likely that the solutions obtained in [4] correspond to the solutions obtained in Theorem 1 for large p, we have no proof of this fact.Remark 3. Theorem 1 shows in particular that there exist travelling wave solutions of arbitrary small energy. This suggests that scattering in the energy space is not likely to hold.

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 Schrödinger 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 [46,45] established by formal and numerical arguments.

We study the limit at infinity of the travelling waves of finite energy in the Gross-Pitaevskii equation in dimension larger than two: their uniform convergence to a constant of modulus one and their asymptotic decay. RésuméNous étudions la limite à l'infini des ondes progressives d'énergie finie pour les équations de Gross-Pitaevskii en dimension supérieure ou égale à deux : leur convergence uniforme vers une constante de module un et leur comportement asymptotique.

We pursue our work [5] on the dynamical stability of dark solitons for the one-dimensional Gross-Pitaevskii equation. In this paper, we prove their asymptotic stability under small perturbations in the energy space. In particular, our results do not require smallness in some weighted spaces or a priori spectral assumptions. Our strategy is reminiscent of the one used by Martel and Merle in various works regarding generalized Korteweg-de Vries equations. The important feature of our contribution is related to the fact that while Korteweg-de Vries equations possess unidirectional dispersion, Schrödinger equations do not.