Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime

Abstract

Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mari\\c\{s\} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Mari\\c\{s\}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.