Interacting helical traveling waves for the Gross–Pitaevskii equation
Entity
UAM. Departamento de MatemáticasPublisher
EMS PressDate
2022-05-22Citation
10.4171/AIHPC/32
Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire 39.6 (2023): 1319-1367
ISSN
0294-1449 (print); 1873-1430 (online)DOI
10.4171/AIHPC/32Project
Gobierno de España. PDI2019-110712GB-100Editor's Version
https://doi.org/10.4171/AIHPC/32Subjects
Helices; Traveling Waves Equations; Energy; Gross-Pitaevskii Equation; Anti-Plane Shear; MatemáticasRights
© 2022 Association Publications de l’Institut Henri Poincaré Published by EMS PressAbstract
We consider the three-dimensional Gross–Pitaevskii equation (Equation presented) and construct traveling wave solutions to this equation. These are solutions of the form ψ (t, x) = (x1, x2, x3 - Ct) with a velocity C of order ∊|log ∊| for a small parameter ∊ > 0. We build two different types of solutions. For the first type, the functions u have a zero-set (vortex set) close to a union of n helices for n ≥ 2 and near these helices u has degree 1. For the second type, the functions u have a vortex filament of degree -1 near the vertical axis e3 and n ≥ 4 vortex filaments of degree C1 near helices whose axis is e3. In both cases the helices are at a distance of order 1/(∊√|log ∊|) from the axis and are solutions to the Klein–Majda–Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross–Pitaevskii equation, namely the Ginzburg–Landau equation. To prove the existence of these solutions we use the Lyapunov–Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation
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Google Scholar:Dávila, Juan
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del Pino, Manuel
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Medina de ja Torre, María
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Rodiac, Rémy
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