Interacting helical traveling waves for the Gross–Pitaevskii equation
Juan Dávila
University of Bath, United KingdomManuel del Pino
University of Bath, United Kingdom; Universidad de Chile, Santiago, ChileMaria Medina
Universidad Autónoma de Madrid, SpainRémy Rodiac
University of Paris-Saclay, Orsay, France
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Abstract
We consider the three-dimensional Gross–Pitaevskii equation
and construct traveling wave solutions to this equation. These are solutions of the form with a velocity of order for a small parameter . We build two different types of solutions. For the first type, the functions have a zero-set (vortex set) close to a union of helices for and near these helices has degree . For the second type, the functions have a vortex filament of degree near the vertical axis and vortex filaments of degree near helices whose axis is . In both cases the helices are at a distance of order 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.
Cite this article
Juan Dávila, Manuel del Pino, Maria Medina, Rémy Rodiac, Interacting helical traveling waves for the Gross–Pitaevskii equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 39 (2022), no. 6, pp. 1319–1367
DOI 10.4171/AIHPC/32