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June  2008, 21(2): 445-475. doi: 10.3934/dcds.2008.21.445

Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity

J. Cuevas 1, , J. C. Eilbeck 2, and  N. I. Karachalios 3,

1. 

Departamento de Fisica Aplicada I, Escuela Universitaria Politénica, C/ Virgen de Africa, 7, University of Sevilla, 41011 Sevilla, Spain
2. 

Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, United Kingdom
3. 

Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece

Received  January 2007 Revised  December 2007 Published  March 2008

We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schrödinger 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 fixed 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 fixed point method considers the case of infinite 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 efficiency 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 justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.
Keywords: variational methods., breather solutions, lattice differential equations, saturable nonlinearity, Discrete Nonlinear Schrödinger Equation.
Mathematics Subject Classification: Primary: 37L60, 35Q55. Secondary: 47J3.
Citation: J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445
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