Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation

Abstract

For each given $n\ge 2$, we construct a family of entire solutions $u_\varepsilon (z,t)$, $\varepsilon > 0$, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation

These solutions are $2\pi/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior, as $\varepsilon\to 0$,

where $W(z) =w(r) e^{i\theta}$, $z= re^{i\theta}$, is the standard degree $+1$ vortex solution of the planar Ginzburg–Landau equation $\Delta W+(1-|{W}|^2)W=0$ in $\mathbb{R}^2$ and

Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), $\mathbf{f}(t) = (f_1(t),\ldots, f_n(t))$ being a rotating equilibrium point for the renormalized energy of vortex filaments derived there,

corresponding to that of a planar logarithmic $n$-body problem. The modulus of these solutions converges to $1$ as $|{z}|$ goes to infinity uniformly in $(t)$, and the solutions have nontrivial dependence on $t$, thus negatively answering the Ginzburg–Landau analogue of the Gibbons conjecture for the Allen–Cahn equation, a question originally formulated by H. Brezis.