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Global existence and compact attractors for the discrete nonlinear Schrödinger equation. (English) Zbl 1084.35092
Summary: We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove the existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
35B41 Attractors
78A60 Lasers, masers, optical bistability, nonlinear optics
92D20 Protein sequences, DNA sequences
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