Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs

Christophe Besse, Romain Duboscq, Stefan Le Coz

Published 2020-06-08Version 1

We introduce and implement a method to compute ground states of nonlinearSchr{\"o}dinger equations on metric graphs.Ground states are defined as minimizers of the nonlinear Schr{\"o}dingerenergy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flowprojected 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 diminishingand converges (at least locally) toward the ground state. We thenestablish the link between the continuous flow and its discretezedversion. We conclude by conducting a series of numericalexperiments in model situations showing the good performance of the discrete flow tocompute the ground state. Further experiments as well as detailledexplananation of our numerical algorithm will be given in aforthcoming companion paper.