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 $u_{\epsilon }(z,t)$, $\epsilon >0$, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation

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

where $W(z)=w(r)e^{i\theta }$, $z=r{e}^{i\theta }$, 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), $\mathbf{f}(t)=(f_1(t),\dots ,f_n(t))$ 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.