Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime
MetadataShow full item record
Johns Hopkins University Press
Published Version:American Journal of Mathematics, Volume 145, Number 1, February 2023, pp. 109-149[https://doi.org/10.1353/ajm.2023.0002]
SponsorshipMinistry of Education, Universities and Research (MIUR); Spanish Government PGC2018-096422-B-I00; J. Andalucia FQM-116; Ministry of Science and Innovation, Spain (MICINN) Spanish Government CEX2020-001105-M/AEI/10.13039/501100011033
Abstract. In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved 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 Maris¸ in 2013. However, such result is valid only in dimension 3 and higher. In this paper we 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 Maris¸. The planar case turns out to be more intricate and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions.