Helicoidal vortex filaments for the Ginzburg-Landau equations in 3D
We construct new entire solutions $u:\mathbb{R}^3\rightarrow \mathbb{C}$ of the Ginzburg-Landau equations \begin{equation*}\nonumber \Delta u+(1-|u|^2)u=0 \text{ in } \mathbb{R}^3. \end{equation*} For every $n\geq 2$ we are able to exhibit a family of solutions that possess $n$ vortex filaments clustering near the vertical $e_3$-axis. All these solutions are screw-symmetric and their vortex filaments are close to some curves $(f_i(t),t)$ for $t\in \mathbb{R}$, $1\leq i \leq n$, where $f_i:\mathbb{R} \rightarrow \mathbb{C}$ are periodic solutions of the logarithmic $n$-body problem also called $n$-vortex problem. We build these solutions by applying the Lyapunov-Schmidt method.