A discrete Adomian decomposition method for the discrete nonlinear Schrödinger equation

In this work we want to describe a discrete version of the well–known Adomian decomposition method (ADM) applied to the discrete nonlinear Schrödinger equation (NLS). The ADM was introduced by Adomian [3], [11] in the early 1980s to solve nonlinear ordinary and partial differential equations. The discrete nonlinear Schrödinger equation is omnipresent [8] in applied sciences, e.g. describing the propagation of electromagnetic waves in glass fibers, one– dimensional arrays of coupled optical waveguides [6] and light–induced photonic crystal lattices [5]. Moreover, it is used to describe Bose–Einstein condensates in optical lattices [10] and it is an established model for optical pulse propagation in various doped fibers [7]. Specifically, we will consider the two most common discrete versions of the standard cubic NLS equation that arise from different spatial discretizations. These discrete nonlinear Schrödinger equations (DNLS) are also called lattice NLS equation [9, Chapter 5.2.2]. For an extension of the ADM to the fully discrete NLS, i.e. after a discretization in time we refer the reader to [4]. The nonlinear cubic Schrödinger equation (NLS) is a typical dispersive nonlinear partial differential equation that plays a key role in a variety of areas in mathematical physics. It describes the spatio–temporal evolution of the complex field u = u(x, t) ∈ C and has the general form