Stationary waves in a nonlinear periodic medium: Strong resonances and localized structures. I. The discrete model.

Abstract

The stationary-wave equation in a periodic, nonlinear medium is studied as a nonintegrable dynamical system. Two main properties are shown: (i) There exist four types of ``localized solutions'' which correspond to the four strong resonances of the dynamical system, one of which is the so-called ``gap soliton'' already described by Mills and Trullinger [Phys. Rev. B 36, 947 (1987)]. (ii) These solutions are close to analytic ones; however, they are weakly chaotic, and this stochasticity is responsible for observable physical effects. In particular, it assigns a maximum spatial extension to the localized structures. Near the bifurcation giving rise to a localized solution, the onset of stochasticity is shown to be a critical phenomenon, whose critical exponent is evaluated. In this paper we consider a discrete model. In the following paper, we consider a continuous model.