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We consider the following coupled Ginzburg-Landau system in ${\mathbb R}^3$\begin{align*} \begin{cases} -\epsilon^2 \Delta w^++\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+=0,\\[3mm] -\epsilon^2 \Delta w^-+\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 constantcoefficients satisfy $$ A_+, A_->0,\quad B^2<A_+A_-, \quad t^\pm >0, \quad{t^+}^2+{ t^-}^2=1. $$ If $B<0$, then for every $\epsilon$ small enough, weconstruct 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. D\'avila,M. del Pino, M. Medina and R. Rodiac in {\tt arXiv:1901.02807}. These solutionsare $2\pi$-periodic in $t$ and have multiple interacting vortex helices. Themain results are the extensions of the phenomena of interacting helical vortexfilaments for the classical (single) Ginzburg-Landau equation in $\mathbb{R}^3$which has been studied in {\tt arXiv:1901.02807}. Our results negatively answerthe Gibbons conjecture \cite{Gibbons conjecture} for the Allen-Cahn equation inGinzburg-Landau system version, which is an extension of the questionoriginally proposed by H. Brezis.

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