Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

André de Laire; Pierre Mennuni

doi:10.3934/dcds.2020026

Article Contents

2020, Volume 40, Issue 1: 635-682. Doi: 10.3934/dcds.2020026

This issue Previous Article Global stability of Keller–Segel systems in critical Lebesgue spaces Next Article

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

André de Laire and
Pierre Mennuni

Université de Lille, CNRS, Inria, UMR 8524, Laboratoire Paul Painlevé, F-59000 Lille, France

Received: July 2019

Revised: July 2019

Published: October 2019

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Abstract

We consider a nonlocal family of Gross–Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.

Keywords:

Mathematics Subject Classification: 35Q55, 35J20, 35C07, 35B35, 37K05, 35C08, 35Q53.

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Figure 1. Curve $ E_ \text{min} $ and solitons in the case $ {\cal W} = \delta_0 $

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Figure 2. Curve $ E_ \text{min} $ and solitons for the potential in (7.1), with $ \alpha = 0.05 $ and $ \beta = 0.15 $

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Figure 3. Curve $ E_ \text{min} $ and solitons for the potential in (7.2), with $ \alpha = 0.8 $

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Figure 4. Curve $ E_ \text{min} $ and solitons for the potential in (7.3), with $ \sigma = 10 $

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Figure 5. Dispersion curve associated with potential (7.4), with $ a = -36 $, $ b = 2687 $, $ c = 30 $. Here $ \xi_m\sim 0.33 $ and $ \xi_r\sim 0.53 $

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Figure 6. Curves $ E_ \text{min} $ and solitons for the potential in (7.4), with $ a = -36 $, $ b = 2687 $, $ c = 30 $

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