Interacting helical traveling waves for the Gross–Pitaevskii equation

Abstract

Abstract. We consider the 3D Gross-Pitaevskii equation i∂tψ +∆ψ + (1 − |ψ| )ψ = 0 for ψ : R× R → C and construct traveling waves solutions to this equation. These are solutions of the form ψ(t, x) = u(x1, x2, x3 −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 e3 and n ≥ 4 vortex filaments of degree +1 near helices whose axis is e3. In both cases the helices are at a distance of order 1/(ε √