Interacting helical traveling waves for the Gross–Pitaevskii equation

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 of form $\psi(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ order $\varepsilon|\log\varepsilon|$ small parameter $\varepsilon>0$. build two different types solutions. For first type, functions $u$ have zero-set (vortex set) close an union $n$ helices $n\geq 2$ near these has degree 1. second vortex filament $-1$ vertical axis $e_3$ 4$ filaments $+1$ whose is $e_3$. In both cases at distance $1/(\varepsilon\sqrt{|\log \varepsilon|)}$ from Klein-Majda-Damodaran system, supposed describe evolution nearly parallel in ideal fluids. Analogous been constructed recently by authors stationary equation, namely Ginzburg-Landau To prove existence we use Lyapunov-Schmidt method subtle separation between even odd Fourier modes error suitable approximation.