Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori
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Gross-Pitaevskii equationMountain-pass solution
Sánchez, F. J. M., & Ruiz, D. Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori. Mathematics in Engineering, 5(1): 1–14. [DOI:10.3934/mine.2023011]
SponsorshipFEDER-MINECO Grant PGC2018-096422-B-I00; Andalucia (FQM-116); Spanish Ministry of Science and Innovation (MICINN), through the IMAG-Maria de Maeztu Excellence Grant CEX2020-001105-M/AEI/10.13039/501100011033
In this paper we consider traveling waves for the Gross-Pitaevskii equation which are T-periodic in each variable. We prove that if T is large enough, there exists a solution as a global minimizer of the corresponding action functional. In the subsonic case, we can use variational methods to prove the existence of a mountain-pass solution. Moreover, we show that for small T the problem admits only constant solutions.