2019
DOI: 10.48550/arxiv.1911.03944
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Coercivity for travelling waves in the Gross-Pitaevskii equation in $\mathbb{R}^2$ for small speed

Abstract: In the previous paper [4], we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space., the decay in space being critical, to use a Hardy type inequality. We emphasize that here ψ is compactly supported away from 0, it is in particular needed since ψ 1,D… Show more

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“…Later on, traveling waves solutions to a similar equation with a vortex ring, the Schrödinger map equation, were constructed by a perturbation method in [33], see also [34,40]. We refer to [2,3,4,36,11,35,1,13,12] for more on finite energy solutions. Associated to the helix, there exist infinite energy solutions to (1.1).…”
Section: Introductionmentioning
confidence: 99%
“…Later on, traveling waves solutions to a similar equation with a vortex ring, the Schrödinger map equation, were constructed by a perturbation method in [33], see also [34,40]. We refer to [2,3,4,36,11,35,1,13,12] for more on finite energy solutions. Associated to the helix, there exist infinite energy solutions to (1.1).…”
Section: Introductionmentioning
confidence: 99%