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Heteroclinic standing waves in defocussing DNLS equations -- Variational approach via energy minimization

We study heteroclinic standing waves (dark solitons) in discrete nonlinear Schrödinger equations with defocussing nonlinearity. Our main result is a quite elementary existence proof for waves with monotone and odd profile, and relies on minimizing an appropriately defined energy functional. We also study the continuum limit and the numerical approximation of standing waves.