Mountain pass for the Ginzburg-Landau energy in a strip: solitons and solitonic vortices

TL;DR

This paper investigates the critical points of the Ginzburg-Landau energy in an infinite strip with phase imprinting, identifying conditions for soliton and vortex solutions and their dependence on the strip width.

Contribution

It characterizes the existence and nature of soliton and solitonic vortex solutions in a strip geometry, revealing how the domain width influences their formation and stability.

Findings

Existence of a critical width for soliton solutions

Identification of vortex solutions above the critical width

Dependence of solution type on domain geometry and symmetry

Abstract

Motivated by recent experiments, we study critical points of the Ginzburg-Landau energy in an infinite strip where phase imprinting is applied to half of the domain. We prove that there is a critical width of the cross section below which the soliton solution is a mountain pass solution and the minimizer within the subspace of odd functions. Above the critical width, we find that the mountain pass solution is a vortex with a solitonic behaviour in the infinite direction, called a solitonic vortex. Moreover, depending on the width, we prove that the minimizer in a space with some symmetries can display one or several solitonic vortices. While the problem shares some similarities with the analysis of stability and minimality of the Ginzburg-Landau vortex of degree one in a disk or the whole plane, the change in geometry introduces subtle analytical differences. Extensions to the case of an infinite cylinder in 3D are also given.