PHYSICA D ELSEYIER Physica D I 13 ( 1998) 326-330 The dynamics of coupled perturbed discretized NLS equations Vassilios M. Rothos *, Tassos C. Bountis Cmtrrfiw Rrseurch and Applications of Nonlinrur S~VCIII,S. Depurtmrnt of’Muthrmdcs. Univrrsityv of Putrus, GR 265 00. Patrus, Grew Abstract We study the dynamics of two coupled perturbed discretized NLS equations with periodic boundary conditions. The unperturbed system consists of two integrable discrete NLS equations, for which the corresponding Lax pairs are known. We describe here the homoclinic orbits of the coupled system and give new results for a class of non-integrable perturbations through a Mel’nikov type approach, using the spectral analysis of Lax’s operators and Fenichel’s theory. Copyright 0 1998 Elsevier Science B.V. PACS: 03; 42; 63 Kqwvrds: Coupled discrete Nonlinear Schr6dinger equations: Mel’nikov theory 1. Introduction In recent years, there has been considerable work on proving the existence of chaotic behavior in deterministic dynamical systems. In finite-dimensional systems there exist well-developed techniques to show the existence of chaos, among which Mel’nikov’s method is one of the most popular and widely used [ 1,2]. In this paper we study a system of discretized NLS equations following the approach of [3] and establish. via Mel’nikov’s theory, the transversal intersection of stable and unstable manifolds of coupled perturbed system. We consider two periodically forced and damped focusing coupled perturbed NLS equations, which we take to be of the form: iq, = Y.~,~ + 2(lql’ - w:)q + ie(C - uy - blpl’q). ip, = psX + 2(lpl’ - ~22)~ + ic(D - ap - hlql’p), (1.1) with periodic and even boundary conditions q(x + L. I) = q(x. I). p(x + L. t) = p(x. t), q(x. 0) = 4(-x, 0). pk. 0) = p(-x. 0). where WI, 02, C, D are constants and t is small. The unperturbed system (E = 0) consists of two integrable NLS equations and admits a discretization which was pointed out to be integrable for arbitrary mesh-size by Ablowitz * Corresponding author. 0167-2789/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved PII SO167-2789(97)002SS-6