ADM-Padé technique for the nonlinear lattice equations

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Abstract

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.

Introduction

The study of differential–difference equations (DDEs) has received considerable attention in recent years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [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], [17], [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 byunt=un(un+1-un-1)-vn+vn-1,vnt=vn(un+2-un-1),which were found in the study of lattice analogues of W-algebras by Belov and Chaltikian [5]. BC lattice equations have rich mathematical structures. Belov and Chaltikian established the bi-Hamiltonian structure of this system [5]. Sahadevan and his co-worker not only derived a sequence of conserved densities and generalized symmetries [6] but also obtained (2×2) matrix recursion operator [7]. In Ref. [8], Hu and Zhu derived a Bäcklund transformation, nonlinear superposition formula and obtained multi-soliton solutions of BC lattice. Secondly, we consider the nonlinear self-dual network which is one of the typical integrable nonlinear lattice system and describes the propagation of electrical signals in a cascade of four-terminal nonlinear LC self-dual circuits. The nonlinear self-dual network equations (SDNEs) readunt=(1+γun2)(vn-vn+1),vnt=(1+γvn2)(un-1-un),where γ=±1,un and vn are the voltage and current in the nth capacitance and inductance of the network, respectively. Zhang et al. [15] achieved a series of bright solitons, dark solitons, and kink solitons besides the known bright soliton and kink soliton by the real exponential approach. We solve the two nonlinear lattice equations using ADM-Padé technique converting truncated series solutions of ADM into diagonal Padé approximants. The solitary solutions are obtained with higher accuracy and faster convergence rate than using ADM alone. Comparisons are made between numerical solutions and exact solutions to illustrate the validity and the great potential of the technique.

The paper is organized as follows. In next section, ADM-Padé technique for the differential–difference equations is outlined. In Section 3, Belov–Chaltikian lattice is studied. In Section 4, the nonlinear self-dual network equations are studied. Finally, some discussions and conclusions are given.

Section snippets

The description of ADM for solving the DDEs

For the purposes of the illustration of the decomposition method, we consider a system of nonlinear differential–difference equations as follows:Li(ui(n,t))+Ri(u1(n,t),u1(n+1,t),u1(n-1,t),,uk(n,t),uk(n+1,t),uk(n-1,t),)+Ni(u1(n,t),u1(n+1,t),u1(n-1,t),,uk(n,t),uk(n+1,t),uk(n-1,t),)=gi,i=1,2,,k,where ui(n,t) is the unknown function with respect to the discrete spatial variable n and the temporal variable t,Li is the highest-order derivative which is assumed to be invertible, Ri is the remaind

The soliton solution of the Belov–Chaltikian lattice

Consider Eq. (1) with the initial conditions:u(n,0)=f1,v(n,0)=f2.We rewrite Eq. (1) in operator form:Lt(un)=unun+1-unun-1-vn+vn-1,Lt(vn)=vnun+2-vnun-1,where Lt is a first-order differential operator and Lt-1 is a integrate operator defined byLt-10t(·)dt.Operating Lt-1 on both sides of Eq. (25) and using the initial conditions, we obtainun=f1+Lt-1(unun+1-unun-1)-Lt-1vn+Lt-1vn-1,vn=f2+Lt-1(vnun+2-vnun-1).Therefore,un=f1+Lt-1(M(un,un+1)-N(un,un-1))-Lt-1vn+Lt-1vn-1,vn=f2+Lt-1(P(vn,un+2)-Q(vn,un-1)

The soliton solution of the nonlinear self-dual network equations

Consider Eq. (2) with the initial conditionsu(n,0)=f1,v(n,0)=f2.We rewrite Eq. (2) in operator form:Lt(un)=γun2vn-γun2vn+1+vn-vn+1,Lt(vn)=γvn2un-1-γvn2un+un-1-un,where Lt is a first-order differential operator and Lt-1 is a integrate operator defined byLt-10t(·)dt.Operating Lt-1 on both sides of Eq. (43) and using the initial conditions, we obtainun=f1+γLt-1(un2vn-un2vn+1)+Lt-1vn-Lt-1vn+1,vn=f2+γLt-1(vn2un-1-vn2un)+Lt-1un-1-Lt-1un.Therefore,un=f1+γLt-1(M(un,vn)-N(un,vn+1))+Lt-1vn-Lt-1vn+1,vn=f

Discussion and conclusions

In this paper, we solve Belov–Chaltikian lattice and the nonlinear self-dual network equations with appropriate initial conditions using ADM-Padé technique which gives the approximate solution with higher accuracy and faster convergence rate than using ADM [25], [26], [27], [28], [29], [30], [31] alone. From the good results, we can conclude that the application of Padé approximants to Adomian’s series solution greatly improve the convergence domain and accuracy of the solution. Comparisons are

Acknowledgements

Thanks to the anonymous reviewers for their valuable comments. This work was supported by the National Key Basic Research Project of China (2004CB318000), the National Science Foundation of China (10771072), the National Natural Science Foundation of China (10735030) and Shanghai Leading Academic Discipline Project (B412).

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