Spatial properties of integrable and nonintegrable discrete nonlinear Schrödinger equations

Abstract

We study the spatial properties of a nonlinear discrete Schr\"odinger equation introduced by Cai, Bishop, and Gro/nbech-Jensen [Phys. Rev. Lett. 72, 591 (1994)] that interpolates between the integrable Ablowitz-Ladik equation and the nonintegrable discrete nonlinear Schr\"odinger equation. We focus on the stationary properties of the interpolating equation and analyze the interplay between integrability and nonintegrability by transforming the problem into a dynamical system and investigating its Hamiltonian structure. We find explicit parameter regimes where the corresponding dynamical system has regular trajectories leading to propagating wave solutions. Using the anti-integrable limit, we show the existence of breathers. We also investigate the wave transmission problem through a finite segment of the nonlinear lattice and analyze the regimes of regular wave transmission. By analogy of the nonlinear lattice problem with chaotic scattering systems, we find the chain lengths at which reliable information transmission via amplitude modulation is possible.