2022/07/25 by Lipeng Duan, Qi Gao, Duan, Lipeng +3
Computer Science · Mathematics · #Nonlinear Dynamics and Pattern Formation #Advanced Mathematical Physics Problems #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2207.11927
We consider the following coupled Ginzburg-Landau system in \mathbb R3 \begincases -ε2 Δw+ +[A+(|w+|2-t+2)+B(|w-|2-t-2)]w+=0,
-ε2 Δw- +[A-(|w-|2-t-2)+B(|w+|2-t+2)]w-=0, \endcases where w=(w+, w-)∈ ℂ2 and the constant coefficients satisfy A+, A-gt;0, B20, t+2+ t-2=1. If B<0, then for every ε small enough, we construct a family of entire solutions wε(z, t)∈ ℂ2 in the cylindrical coordinates (z, t)∈ ℝ2 × 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 2π-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 ℝ3 which has been studied in \tt arXiv:1901.02807. Our results negatively answer the Gibbons conjecture \citeGibbons conjecture for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.