Interacting helical traveling waves for the Gross–Pitaevskii equation

Abstract

We consider the three-dimensional Gross–Pitaevskii equation

and construct traveling wave solutions to this equation. These are solutions of the form $\psi (t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $\epsilon |\log \epsilon |$ for a small parameter $\epsilon >0$. 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 $n\ge 2$ 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 $n\ge 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(\epsilon \sqrt{|\log \epsilon |})$ 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.