Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Amandine Aftalion, Luc Nguyen

TL;DR

This work analyzes the Ginzburg–Landau energy in an infinite strip under phase imprinting, revealing a width-dependent transition between soliton and solitonic vortex behaviors in 2D and extending the framework to 3D cylinders. It develops a variational approach, employing minimization under symmetry and mountain-pass techniques to classify critical points and their energies, with precise thresholds involving d = sqrt{2}π/2 and d = sqrt{2} j'_{1,1}. The results explain how solitons can be stable or unstable depending on geometry and how solitonic vortices and multi-vortex configurations arise, connecting to experimental observations of snake instability and vortex formation. The paper also discusses extensions, including radial and higher-dimensional cross-sections and a candidate vortex-ring construction, along with open questions about compactness and ring-like states.

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.

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

TL;DR

This work analyzes the Ginzburg–Landau energy in an infinite strip under phase imprinting, revealing a width-dependent transition between soliton and solitonic vortex behaviors in 2D and extending the framework to 3D cylinders. It develops a variational approach, employing minimization under symmetry and mountain-pass techniques to classify critical points and their energies, with precise thresholds involving d = sqrt{2}π/2 and d = sqrt{2} j'_{1,1}. The results explain how solitons can be stable or unstable depending on geometry and how solitonic vortices and multi-vortex configurations arise, connecting to experimental observations of snake instability and vortex formation. The paper also discusses extensions, including radial and higher-dimensional cross-sections and a candidate vortex-ring construction, along with open questions about compactness and ring-like states.

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.

Paper Structure

This paper contains 13 sections, 27 theorems, 225 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. The case $N = 2$
  5. The case $N = 3$
  6. $2D$ equilibrium configurations
  7. Minimizing ${\mycal F}_d$ in ${\mycal A}_d$
  8. Minimizing ${\mycal F}_d$ in other subspaces of ${\mycal A}_d^{\rm x}$
  9. Mountain pass characterization of $2D$ soliton and solitonic vortex
  10. $3D$ equilibrium configurations
  11. Minimization under symmetry assumptions
  12. Estimate of $3D$ mountain-pass energy in ${\mycal A}_d^{\rm x}$
  13. Candidate vortex ring solution

Key Result

Theorem 1.1

Let $N = 2$.

Figures (1)

  • Figure 1: A $2D$ domain $\Sigma$ for which one can use the proof of Theorem \ref{['Thm1Ext2']} to construct critical point on the domain ${\mathbb R} \times d\Sigma$ where vortex lines on the central plane $x = 0$ can be arranged at the three solid lines or the three dashed lines.

Theorems & Definitions (64)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 2.1
  • ...and 54 more