Thresholds for breather solutions on the Discrete Nonlinear Schr'odinger Equation with saturable and power nonlinearity

摘要

We consider the question of existence of periodic solutions (called breathersolutions or discrete solitons) for the Discrete Nonlinear Schr\"odingerEquation with saturable and power nonlinearity. Theoretical and numericalresults are proved concerning the existence and nonexistence of periodicsolutions by a variational approach and a fixed point argument. In thevariational approach we are restricted to DNLS lattices with Dirichlet boundaryconditions. It is proved that there exists parameters (frequency ornonlinearity parameters) for which the corresponding minimizers satisfyexplicit upper and lower bounds on the power. The numerical studies performedindicate that these bounds behave as thresholds for the existence of periodicsolutions. The fixed point method considers the case of infinite lattices.Through this method, the existence of a threshold is proved in the case ofsaturable nonlinearity and an explicit theoretical estimate which isindependent on the dimension is given. The numerical studies, testing theefficiency of the bounds derived by both methods, demonstrate that thesethresholds are quite sharp estimates of a threshold value on the power neededfor the the existence of a breather solution. This it justified by theconsideration of limiting cases with respect to the size of the nonlinearityparameters and nonlinearity exponents.