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

• Published in: arXiv.org
• Main Authors: Duan, Lipeng, Gao, Qi, Yang, Jun
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.07.2022
• Subjects: Cylindrical coordinates; Filaments; Helices; Landau-Ginzburg equations; Vortex filaments
• ISSN: 2331-8422

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 constant coefficients 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, 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.