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

We introduce and implement a method to compute ground states of nonlinear Schr{o}dinger equations on metric graphs. Ground states are defined as minimizers of the nonlinear Schr{o}dinger energy 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 diminishing and converges (at least locally) toward the ground state. 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 the ground state. Further experiments as well as detailed explanation of our numerical algorithm will be given in a forthcoming companion paper.

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