The helical vortex filaments of Ginzburg-Landau system in ${\mathbb R}^3$ (2207.11927v1)
Abstract: We consider the following coupled Ginzburg-Landau system in ${\mathbb R}3$ \begin{align*} \begin{cases} -\epsilon2 \Delta w+ +\Big[A_+\big(|w+|2-{t+}2\big)+B\big(|w-|2-{t-}2\big)\Big]w+=0, \[3mm] -\epsilon2 \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 constant coefficients satisfy $$ A_+, A_->0,\quad B2<A_+A_-, \quad t\pm >0, \quad {t+}2+{ t-}2=1. $$ 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. D\'avila, 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.