Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori

Francisco Javier Martínez Sánchez; David Ruiz; Francisco Javier Martínez Sánchez; David Ruiz

doi:10.3934/mine.2023011

Mathematics in Engineering

2023, Volume 5, Issue 1: 1-14. doi: 10.3934/mine.2023011

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Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori

Francisco Javier Martínez Sánchez ¹,
David Ruiz ^{2
,
,}

1.
Universidad de Jaén, Departamento de Matemáticas, Campus Las Lagunillas, 23071 Jaén, Spain
2.
IMAG, Universidad de Granada, Departamento de Análisis Matemático, 18071 Granada, Spain

Received: 02 November 2021 Revised: 07 January 2022 Accepted: 10 January 2022 Published: 11 February 2022

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.
- Gross-Pitaevskii equation,
- mountain-pass solution
Citation: Francisco Javier Martínez Sánchez, David Ruiz. Existence and nonexistence of traveling waves for the Gross-Pitaevskii equation in tori[J]. Mathematics in Engineering, 2023, 5(1): 1-14. doi: 10.3934/mine.2023011

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Abstract

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.

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