Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs
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We introduce and implement a method to compute ground states of nonlinearSchrödinger equations on metric graphs.Ground states are defined as minimizers of the nonlinear Schrödingerenergy 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.
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