Gradient flow approach to the calculation of stationary states on nonlinear quantum graphs

Besse, Christophe; Duboscq, Romain; Le Coz, Stefan

Annales Henri Lebesgue, Tome 5 (2022), pp. 387-428.

Nous présentons et mettons en œuvre une méthode de calcul des états stationnaires d’équations de Schrödinger non linéaires sur les graphes métriques. Les états stationnaires sont obtenus comme des minimiseurs locaux de l’énergie de Schrödinger non linéaire à masse fixe. Notre méthode est basée sur un flot gradient normalisé pour l’énergie (c’est-à-dire un flot gradient projeté sur une sphère de masse fixe) adapté au contexte des graphes quantiques non linéaires. Nous prouvons d’abord que, au niveau continu, le flot gradient normalisé est bien posé, préserve la masse, diminue l’énergie et converge (au moins localement) vers des états stationnaires. Nous établissons ensuite le lien entre le flot continu et sa version discrétisée. Nous concluons en menant une série d’expériences numériques dans des situations modèles montrant la bonne performance du flot discret pour calculer les états stationnaires. D’autres expériences ainsi qu’une explication détaillée de notre algorithme sont présentées dans un article complémentaire.

We introduce and implement a method to compute stationary states of nonlinear Schrödinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schrödinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

Classification : 35Q55, 35R02, 65M06
Mots clés : normalized gradient flow, ground states, stationary states, quantum graphs, nonlinear Schrödinger equation

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