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

Abstract

If B < 0, then for every ǫ small enough, we construct a family of entire solutions wǫ(z, t) ∈ C 2 in the cylindrical coordinates (z, t) ∈ R 2 × R for this system via the approach introduced by J. Dávila, M. del Pino, M. Medina and R. Rodiac in 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 R 3 which has been studied in arXiv:1901.02807. Our results negatively answer the Gibbons conjecture [10] for the Allen-Cahn equation in Ginzburg-Landau system version, which is an extension of the question originally proposed by H. Brezis.