The helical vortex filaments of Ginzburg-Landau system in ${\mathbb{R}}^3$

Abstract

We consider the following coupled Ginzburg-Landau system in ${\mathbb{R}}^3$ \begin{align*} \begin{cases} -ε^2 Δw^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0, \\[3mm] -ε^2 Δw^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^-=0, \end{cases} \end{align*} where $w=(w^{+},w^{-})\in {\mathbb{C}}^2$ and the constant coefficients satisfy $$A_{+},A_{-}>0,B^2<A_{+}{A}_{-},t^{\pm }>0,{t^{+}}^2+{t^{-}}^2=1.$$ If $B<0$, then for every $\epsilon$ small enough, we construct a family of entire solutions $w_{\epsilon }(\tilde{z},t)\in {\mathbb{C}}^2$ in the cylindrical coordinates $(\tilde{z},t)\in {\mathbb{R}}^2\times \mathbb{R}$ for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in {\tt arXiv:1901.02807}. These solutions are $2\pi$-periodic in $t$ and have multiple interacting vortex helices. The main results are the extensions of the phenomena of interacting helical vortex filaments for the classical (single) Ginzburg-Landau equation in ${\mathbb{R}}^3$ which has been studied in {\tt arXiv:1901.02807}. Our results negatively answer the Gibbons conjecture \cite{Gibbons conjecture} for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.