Dynamical Structures in 2D Lattices

Abstract

Very little is known about nonlinear entities in dimension higher than one, their formation and dynamics are of great interest in various branches of science1. Two dimensional structures2–3(domain walls, vortices) play an important role in the material properties and they become crucial in nonlinear physics involved in the problem of absorbates deposited on crystal surfaces. In this paper we focus on the formation of localized states mediated by modulational instability in a two-dimensional (2D) lattice. The paper is organized as follows. In section 2 we introduce our model which is a two-dimensional non-dissipative Frenkel-Kontorova model with additional nonlinear interactions. In section 3 the discrete equations governing the dynamics of the lattice are reduced to a two-dimensional nonlinear Schrodinger equation in the low amplitude and semi discrete limit. Then, in section 4, the modulational instability conditions of this equation are calculated. Section 5 deals with numerical simulations, we investigate the role played by modulational instability on the evolution into localized states of an initial plane wave, with low amplitude, propagating on the lattice.