Interacting helical traveling waves for the Gross-Pitaevskii equation

Abstract

We consider the 3D Gross-Pitaevskii equation i∂tψ + ∆ψ + (1 − |ψ| 2 )ψ = 0 for ψ : R × R 3 → C and construct traveling waves solutions to this equation. These are solutions of the form ψ(t, x) = u(x 1 , x 2 , x 3 − Ct) with a velocity C of order ε| log ε| for a small parameter ε > 0. We build two different types of solutions. For the first type, the functions u have a zero-set (vortex set) close to an 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 e 3 and n ≥ 4 vortex filaments of degree +1 near helices whose axis is e 3 . In both cases the helices are at a distance of order 1/(ε | log ε|) 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.

The helical vortex filaments of Ginzburg-Landau system in ${\mathbb R}^3$

“…In classical fluid and Bose-Einstein condensates, experiments and numerical methods show that the possible existence of arbitrary number of interacting helical vortex filaments, see [5,22]. The reader can also refer to [13,14,24,32,37] for the constructions of solutions with helical vortex structure, involving Schrödinger map equation, Gross-Pitaevskii equation and Euler equation.…”

“…Hence, much more delicate analysis will be provided to capture fine estimates. This will be bone by adopting the ideas and approaches of Fourier decomposition to the error and nonlinear terms introduced in [12,13] to handle this problem, see Sections 3.2 and 4.2. In fact, a crucial linear theory which gives more precise estimates for the perturbation term ψ will be established, see Proposition 4.2 in Section 4.2.…”

“…In order to get around the technical difficulty, we are led to using the idea doing more precise decompositions and estimates for the perturbation ψ. This was first introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in [12] and [13] to construct vortex helical filaments of classical (single complex component) Ginzburg-Landau equation and Gross-Pitaevskii equation.…”