Unique and nonunique minimizing travelling waves for some Ginzburg–Landau type equations
[Unicité et non unicité d’ondes progressives minimisantes pour des équations de type Ginzburg–Landau]
Eliot Pacherie1
1 CNRS and CY Cergy University UFR Sciences et Techniques, AGM - Département Mathématiques, Bureau E 505 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise, France
Journées équations aux dérivées partielles (2024), Exposé no. 9, 10 p.

This is a summary of an upcoming work in collaboration with Mihai Maris. We study the Schrödinger equation with a nonzero condition at infinity for some nonlinearities, for which the existence of travelling waves has been proven in [22] as solutions of a minimization problem. We show that for some of them, we do not have uniqueness of this minimizer, and we study how these travelling changes when we change the speed and/or the nonlinearity. We also discuss some examples where we have the uniqueness of travelling waves that are solutions to minimisation problems, as well as questions related to the stability of these objects.

Ceci est un résumé d’un travail à venir en collaboration avec Mihai Maris. On étudie l’équation de Schrödinger avec une condition non triviale à l’infini pour des nonlinéarités génériques, où l’éxistence d’ondes progressives a été prouvée dans [22] comme solutions d’un problème de minimisation. On démontre que pour certaines d’entre elles, il n’y a pas unicité du minimiseur, et on étudie comment ses ondes progressives évoluent quand on modifie la vitesse et/ou la nonlinéarité. On discutera aussi d’exemples où il y a unicité du minimiseur, ainsi que des questions autour de la stabilité de ces objets.

Publié le :
DOI : 10.5802/jedp.690
Classification : 35C07, 35B35, 35J15, 35C08, 35B32
Keywords: Travelling waves, minimization, bifurcation, stability
Mots-clés : Ondes progressives, minimisation, bifurcation, stabilité

Eliot Pacherie 1

1 CNRS and CY Cergy University UFR Sciences et Techniques, AGM - Département Mathématiques, Bureau E 505 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise, France 
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Eliot Pacherie. Unique and nonunique minimizing travelling waves for some Ginzburg–Landau type equations. Journées équations aux dérivées partielles (2024), Exposé no. 9, 10 p.. doi: 10.5802/jedp.690

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