Open AccessExistence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori
Author(s) -
Francisco Javier Martínez Sánchez,
David Ruiz
Publication year - 2022
Publication title -
mathematics in engineering
Language(s) - English
Resource type - Journals
ISSN - 2640-3501
DOI - 10.3934/mine.2023011
Subject(s) - torus , traveling wave , gross–pitaevskii equation , mathematics , action (physics) , variable (mathematics) , constant (computer programming) , mathematical analysis , mathematical physics , physics , geometry , nonlinear system , quantum mechanics , computer science , programming language
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.