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Mathematics > Analysis of PDEs

arXiv:1812.08713 (math)
[Submitted on 20 Dec 2018 (v1), last revised 23 Sep 2019 (this version, v3)]

Title:Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

Authors:André de Laire, Pierre Mennuni
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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.
Comments: 48 pages, 11 figures
Subjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
MSC classes: 35Q55, 35J20, 35A15, 35C07, 35B35, 37K05, 35C08, 35Q53, 82D50
Cite as: arXiv:1812.08713 [math.AP]
  (or arXiv:1812.08713v3 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1812.08713

Submission history

From: André de Laire [view email]
[v1] Thu, 20 Dec 2018 17:34:23 UTC (785 KB)
[v2] Tue, 16 Jul 2019 14:06:07 UTC (786 KB)
[v3] Mon, 23 Sep 2019 15:21:39 UTC (785 KB)
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