Breathers and multibreathers in a periodically driven damped discrete nonlinear Schrödinger equation.

Abstract

We study an integrable discretization of the nonlinear Schrödinger equation (NLS) under the effects of damping and periodic driving, from the point of view of spatially localized solutions oscillating in time with the driver's frequency. We locate the equilibrium states of the discretized (DNLS) system in the plane of its dissipation gamma and forcing amplitude H parameters and use a shooting algorithm to construct the desired solutions psi(n)(t)=phi(n) exp(it) as homoclinic orbits of a four-dimensional symplectic map in the complex phi(n),phi(n+1) space, for -infinity<n<infinity. We derive, in the gamma=0 case, closed form expressions for two fundamental such solutions having a single hump in n, psi(n)+, and psi(n)-, and determine analytically their threshold of existence in the (gamma,H) plane using Mel'nikov's theory. Then, we demonstrate numerically that above this threshold a remarkable variety of multihump structures appear, whose complexity in terms of their spatial extrema grows with increasing H. All these solutions are numerically found to be unstable in time, except for psi(n)-, which is seen to be stable over a certain region in the (gamma,H) plane. In the continuum limit our results are in close agreement with recent studies on the NLS equation. From a more general perspective, we view these DNLS multihump solutions as homoclinic orbits of a higher-dimensional map thereby providing a possible mechanism for explaining the occurrence of similar structures called discrete (multi-) breathers found in a wide variety of one-dimensional nonlinear lattices.