Interacting helical traveling waves for the Gross-Pitaevskii equation

We consider the 3D Gross-Pitaevskii equation begin{equation} onumber ipartial_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 $varepsilon|logvarepsilon|$ 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 $ngeq 2$ 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 $ngeq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(varepsilonsqrt{|log varepsilon|)}$ 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.