Thresholds for breather solutions of the Discrete Nonlinear SchrÄodinger Equation with saturable and power nonlinearity¤

We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear SchrÄodinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a ¯xed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The ¯xed point method considers the case of in¯nite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the e±ciency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justi¯ed by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.