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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.

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