Interacting helical traveling waves for the Gross–Pitaevskii equation

Abstract

We consider the 3D Gross-Pitaevskii equation \begin{equation}\nonumber i\partial_t \psi +\Delta \psi+(1-|\psi|^2)\psi=0 \text{ for } \psi:\mathbb{R}\times \mathbb{R}^3 \rightarrow \mathbb{C} \end{equation} and construct traveling waves 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 $\varepsilon|\log\varepsilon|$ for a small parameter $\varepsilon>0$. 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 $n\geq 2$ 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 $n\geq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(\varepsilon\sqrt{|\log \varepsilon|)}$ 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.