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

### Juan Dávila

University of Bath, UK### Manuel del Pino

University of Bath, UK; Universidad de Chile, Santiago, Chile### Maria Medina

Universidad Autónoma de Madrid, Spain### Rémy Rodiac

Université Paris-Saclay, Orsay, France

## Abstract

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

These solutions are $2π/ε$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior, as $ε→0$,

where $W(z)=w(r)e_{iθ}$, $z=re_{iθ}$, is the standard degree $+1$ vortex solution of the planar Ginzburg–Landau equation $ΔW+(1−∣W∣_{2})W=0$ in $R_{2}$ and

Existence of these solutions was previously conjectured by del Pino and Kowalczyk (2008), $f(t)=(f_{1}(t),…,f_{n}(t))$ 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.

## Cite this article

Juan Dávila, Manuel del Pino, Maria Medina, Rémy Rodiac, Interacting helical vortex filaments in the three-dimensional Ginzburg–Landau equation. J. Eur. Math. Soc. 24 (2022), no. 12, pp. 4143–4199

DOI 10.4171/JEMS/1175