# Solitary-Solution Formulation for Differential-Difference Equations Using an Ancient Chinese Algorithm

Authors: Ji-Huan He; S. K. Elagan; Guo-Cheng Wu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper applies an ancient Chinese algorithm to differential-difference equations, and a solitary-solution formulation is obtained. The discrete mKdV lattice equation is used as an example to elucidate the solution procedure.


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## Body

## 1. Introduction

Discrete nonlinear lattices have been the focus of considerable attention in various branches of science. Many differential equations on the nanoscales are invalid, but the problems arising can be well modeled by differential-difference equations [1–5]. As is well known, there are many physically interesting problems such as charge fluctuations in net work, ladder type, electric circuits, phenomena in crystals, and molecular chains, which all can be modelled by differential-difference equations [6–9].Recently many analytical methods were proposed to solve differential-difference equations, such as the exp-function method [10–12], the variational iteration method [13–15], the homotopy perturbation method [16–19], and the parameterized perturbation method [17]. A complete review on various analytical methods is available in [18, 19]. In this paper we will apply an ancient Chinese algorithm to the discussed problem.


## 2. Amplitude-Frequency Formulation for Nonlinear Oscillators

The ancient Chinese algorithm was first applied to nonlinear oscillators in 2006 [18], where a simple frequency formulation is given. Consider a generalized nonlinear oscillator in the form
(2.1)u′′+f(u)=0,u(0)=A,u′(0)=0.
We use two trial functions(2.2)u1(t)=Acos⁡ω1t,(2.3)u2=Acos⁡ω2t,
which are, respectively, the solutions of the following linear oscillator equations:(2.4)u′′+ω12u=0,u′′+ω22u=0,
where ω1 and ω2 are trial frequencies which can be chosen freely; for example, we can set ω1=1 and ω2=2 or ω1=1 and ω2=ω, where ω is assumed to be the frequency of the nonlinear oscillator.Substituting (2.2) and (2.3) into, respectively, (2.1), we obtain the following residuals:(2.5)R1(t)=-Aω12cos⁡ω1t+f(Acos⁡ω1t),R2(t)=-Aω22cos⁡ω2t+f(Acos⁡ω2t).According to the ancient Chinese algorithm, an amplitude-frequency formulation for nonlinear oscillators was proposed [18].(2.6)ω2=ω12R2(0)-ω22R1(0)R2(0)-R1(0).
Other modifications of the frequency formulation are shown in [20–23]. The frequency formulation is now widely used to solve various nonlinear oscillators [20–25], and the basic idea of the Chinese algorithm can also be used to solve other nonlinear problems [26–29].


## 3. Solitary-Solution Formulation for Differential-Difference Equations

Suppose the differential-difference equation we discuss in this paper is in the following nonlinear polynomial form:(3.1)dun(t)dt=f(un-1,un,un+1),
where un=u(n,t) is a dependent variable, and t is a continuous variable, and n,pi∈Z.Using the basic idea of the ancient Chinese algorithm, we choose two trial functions in the following forms:(3.2)un,1(n,t)=f(ξn+ω1t),un,2(n,t)=g(ξn+ω2t),
where ξn=nd+ξ0,ξ0 is arbitrary, and f and g are known functions. If a periodic solution is searched for, f and g must be periodic functions; if a solitary solution is solved, f and g must be of solitary structures. In this paper a bell solitary solution of a differential-difference equation is considered, and trial functions are chosen as follows:(3.3)un,1(n,t)=Aeξn+ω1t+e-(ξn+ω1t)+B,ω1=1,(3.4)un,2(n,t)=Aeξn+ωt+e-(ξn+ωt)+B,ω2=ω.
For un,un-1, and un+1 should be compatible; then, from (3.3), we have(3.5)un-1,1(n,t)=Aeξn-d+t+e-(ξn-d+t)+B,un+1,1(n,t)=Aeξn+d+t+e-(ξn+d+t)+B,
and from (3.4), we have(3.6)un-1,2(n,t)=Aeξn-d+ωt+e-(ξn-d+ωt)+B,(3.7)un+1,2(n,t)=Aeξn+d+ωt+e-(ξn+d+ωt)+B.Solution Procedure
Step 1.
Define residual function(3.8)R̃(t)=dun(t)dt-f(un-1,un,un+1).
Substituting (3.3)–(3.7) into (3.1), we can obtain, respectively, the residual functions R̃1 and R̃2(3.9)R̃1(t)=dun,1(t)dt-f(un-1,1,un,1,un+1,1),R̃2(t)=dun,2(t)dt-f(un-1,2,un,2,un+1,2).Step 2.
Solitary-solution formulation is constructed as follows:(3.10)ω2=ω12R̃2(0)-ω22R̃1(0)R̃2(0)-R̃1(0),
where ω1=1 and ω2=ω.Step 3.
Combining the coefficients ofeξn in (3.10), and setting them to be zero, we can solve the algebraic equations to find the values of ω,A, and B. Finally an explicit solution is obtained.


## 4. Application in Discrete mKdV Lattice

The famous mKdV lattice equation reads [10](4.1)dundt=(α-un2)(un-1-un+1),
where α≠0.Substituting (3.3)–(3.7) into (4.1), we obtain, respectively, the residual functions R̃1 and R̃2, and using the solitary-solution formulation, (3.10), we have
(4.2)e3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)-e-3ξn(ω3+ω2αe-d-ω2αed-ω-αe-d+αed)+e2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)-e-2ξnB(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)+eξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)-e-ξn(ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d)=0.
Setting the coefficients of eiξn to be zero, we have
(4.3)ω3+ω2αe-d-ω2αed-ω-αe-d+αed=0,B(ω3e-d+ω3ed-2ω2αed+2ω2αe-d-ωed-ωe-d+2αed-2αe-d)=0ω3B2-ω3+ω3e-2d+ω3e2d+ω2A2ed-ω2A2e-d+ω2αe-d-ω2αed+ω2αB2e-d,-ω2αB2ed-ωB2+ω-ωe2d-ωe-2d+A2e-d-A2ed+αB2ed-αB2e-d+αed-αe-d=0.
Solving (4.3) simultaneously, we have(4.4)ω=2αsinh⁡(d),A=2-αsinh⁡(d),B=0.
We, therefore, obtain the following needed solitary solution:(4.5)un=2-αsinh⁡(d)eξn+2αsinh⁡(d)t+e-(ξn+2αsinh⁡(d)t)=-αsinh⁡(d)sech{2αsinh⁡(d)t+nd+ξ0}.


## 5. Conclusions

Though there are many analytical methods, such as the exp-function method, the variational iteration method, and the homotopy perturbation method, for differential-difference equations, this paper suggests an effective and simple approach to such problems using the basic idea of the ancient mathematics, and the simple formulation can be used routinely by followers to various differential-difference equations.



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*Source: 861438-2012-03-21.xml*