Thresholds for breather solutions on the Discrete Nonlinear Schrödinger Equation with saturable and power nonlinearity

We consider the question of existence of nontrivial periodic solutions (called also sometimes, as breather solutions or discrete solitons) for the Discrete Nonlinear Schrödinger Equation with saturable and power nonlinearity. Theoretical and numerical results are presented, concerning the existence and nonexistence of periodic solutions. The existence results obtained via constrained minimization problems or by a fixed point argument, refer to the DNLS equation, in multidimensional lattices, supplemented with Dirichlet boundary conditions, and sometimes also in infinite lattices. Explicit upper and lower bounds on the total power of the periodic solutions are derived, and numerical studies are performed which test their efficiency. The bounds and thresholds derived for the DNLS equation with saturable and power nonlinearity, are also discussed in comparison with the existing results on the existence of excitation thresholds for periodic solutions, and their dependence on the dimension of the lattice.