On the solitary waves for anisotropic nonlinear Schrödinger models on the plane

The focusing anisotropic nonlinear Schrödinger equation i u t - ∂ xx u + ( - ∂ yy ) s u = | u | p - 2 u in R × R 2 is considered for 0 < s < 1 and p > 2 . Here the equation is of anisotropy, it means that dispersion of solutions along x -axis and y -axis is different. We show that while loc...

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Bibliographic Details
Published in:European journal of mathematics 2023-09, Vol.9 (3), Article 55
Main Authors: Gou, Tianxiang, Hajaiej, Hichem, Stefanov, Atanas G.
Format: Article
Language:English
Subjects:
Algebraic Geometry
Anisotropy
Mathematics
Mathematics and Statistics
Research Article
Schrodinger equation
Solitary waves
Uniqueness
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Summary:The focusing anisotropic nonlinear Schrödinger equation i u t - ∂ xx u + ( - ∂ yy ) s u = | u | p - 2 u in R × R 2 is considered for 0 < s < 1 and p > 2 . Here the equation is of anisotropy, it means that dispersion of solutions along x -axis and y -axis is different. We show that while localized time-periodic waves, that are solutions in the form u = e - i ω t ϕ , do not exist in the range , they do exist in the complementary range 2 < p < p s . We construct them variationally and establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration is easily extendable to higher dimensional case. We also show uniqueness of these waves under a natural weak non-degeneracy assumption. This assumption is actually removed for s close to 1, implying uniqueness for the waves in the full range of parameters.
ISSN:2199-675X
2199-6768
DOI:10.1007/s40879-023-00647-8