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Gap solitons in periodic discrete nonlinear Schrödinger equations with saturable nonlinearities. (English) Zbl 1197.35273
Summary: The main result of the paper concerns the existence of nontrivial exponentially decaying solutions to periodic stationary discrete nonlinear Schrödinger equations with saturable nonlinearities, provided that zero belongs to a spectral gap of the linear part. The proof is based on the critical point theory in combination with periodic approximations of solutions. As a preliminary step, we prove also the existence of nontrivial periodic solutions with arbitrarily large periods.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
39A12Discrete version of topics in analysis
35C08Soliton solutions of PDE
81Q05Closed and approximate solutions to quantum-mechanical equations
35A15Variational methods (PDE)
35B10Periodic solutions of PDE
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