ADM-Padé technique for the nonlinear lattice equations

Keywords: Adomian decomposition method Padé approximants Belov–Chaltikian lattice The nonlinear self-dual network equations Solitary solution a b s t r a c t ADM-Padé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. We solve two nonlinear lattice equations using the technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM alone. Bell-shaped solitary solution of Belov–Chaltikian (BC) lattice and kink-shaped solitary solution of the nonlinear self-dual network equations (SDNEs) are presented. Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the technique. The study of differential–difference equations (DDEs) has received considerable attention in recent years [1–15]. The DDEs play an important role in modelling complicated physical phenomena (particle vibrations in lattices, current flow in electrical networks, pulses in biological chains, etc.). ADM-Padé technique, which is a combination of Adomian decomposition method (ADM) [16–18] and Padé approximants [19,20], has been used to solve DDEs and PDEs by various researchers. Abassy [21] solved Burgers and good Boussinesq equations. Basto [22] approximated the theoretical solution of the Burgers equation. Wazwaz solved the Thomas–Fermi equation [23] and approximated Volterra's population model [24]. Wang [14] derived the solitary solution of the discrete hybrid equation. In this paper, we solve two nonlinear lattice equations using ADM-Padé technique. Firstly, we consider Belov–Chaltikian (BC) lattice defined by