Effects of competing short- and long-range dispersive interactions on discrete breathers

The discrete nonlinear Schrödinger equation with competing short-range and long-range interactions is considered in spatial dimensions d> or =2. This model equation is derived for a situation of two linearly coupled excitations (independently of dimension), and we analytically and numerically study its properties in 2+1 dimensions. We analyze theoretically and demonstrate numerically the dependence of the discrete breather solutions on the amplitude and range of the interactions. We find that complete suppression of the existence thresholds obtained recently for short-range interactions can be achieved beyond a critical value of the amplitude or of the range of the long-range kernel. For supercritical values of the corresponding parameters, staggered branches of solutions are obtained both in theory as well as in the numerical experiment.