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

Amandine Aftalion; Luc Nguyen

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arXiv · 2512.01506

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

Amandine Aftalion·Luc Nguyen

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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.

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Amandine Aftalion, Luc Nguyen. 2025-12-01. Mountain pass for the Ginzburg-Landau energy in a strip: solitons and solitonic vortices. https://arxiv.org/abs/2512.01506

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