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

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.

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 𝑢^′′+𝑓(𝑢)=0,𝑢(0)=𝐴,𝑢′(0)=0. (2.1) We use two trial functions𝑢₁(𝑡)=𝐴cos𝜔₁𝑢𝑡,₂=𝐴cos𝜔₂𝑡, (2.2)(2.3) which are, respectively, the solutions of the following linear oscillator equations:𝑢^′′+𝜔²₁𝑢𝑢=0,_′′+𝜔²₂𝑢=0, (2.4) 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:𝑅₁(𝑡)=−𝐴𝜔²₁cos𝜔₁𝑡+𝑓𝐴cos𝜔₁𝑡,𝑅₂(𝑡)=−𝐴𝜔²₂cos𝜔₂𝑡+𝑓𝐴cos𝜔₂𝑡. (2.5)

According to the ancient Chinese algorithm, an amplitude-frequency formulation for nonlinear oscillators was proposed [18].𝜔²=𝜔²₁𝑅₂(0)−𝜔²₂𝑅₁(0)𝑅₂(0)−𝑅₁(0). (2.6) 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:𝑑𝑢_𝑛(𝑡)𝑢𝑑𝑡=𝑓_𝑛−1,𝑢_𝑛,𝑢_𝑛+1, (3.1) where 𝑢_𝑛=𝑢(𝑛,𝑡) is a dependent variable, and 𝑡 is a continuous variable, and 𝑛,𝑝_𝑖∈𝑍.

Using the basic idea of the ancient Chinese algorithm, we choose two trial functions in the following forms:𝑢_𝑛,1(𝑛,𝑡)=𝑓𝜉_𝑛+𝜔₁𝑡,𝑢_𝑛,2(𝑛,𝑡)=𝑔𝜉_𝑛+𝜔₂𝑡, (3.2) where 𝜉_𝑛=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:𝑢_𝑛,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝜔1𝑡}+𝑒^{−(𝜉𝑛+𝜔1𝑡)}+𝐵,𝜔₁𝑢=1,_𝑛,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝜔𝑡}+𝑒^{−(𝜉𝑛+𝜔𝑡)}+𝐵,𝜔₂=𝜔. (3.3)(3.4) For 𝑢_𝑛,𝑢_𝑛−1, and 𝑢_𝑛+1 should be compatible; then, from (3.3), we have𝑢_𝑛−1,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛−𝑑+𝑡}+𝑒^{−(𝜉𝑛−𝑑+𝑡)},𝑢+𝐵_𝑛+1,1()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝑑+𝑡}+𝑒^{−(𝜉𝑛+𝑑+𝑡)},+𝐵 (3.5) and from (3.4), we have𝑢_𝑛−1,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛−𝑑+𝜔𝑡}+𝑒^{−(𝜉𝑛−𝑑+𝜔𝑡)},𝑢+𝐵_𝑛+1,2()=𝐴𝑛,𝑡𝑒^{𝜉𝑛+𝑑+𝜔𝑡}+𝑒^{−(𝜉𝑛+𝑑+𝜔𝑡)}.+𝐵 (3.6)(3.7)

Solution Procedure
Step 1. Define residual function 𝑅(𝑡)=𝑑𝑢_𝑛(𝑡)𝑢𝑑𝑡−𝑓_𝑛−1,𝑢_𝑛,𝑢_𝑛+1. (3.8) Substituting (3.3)–(3.7) into (3.1), we can obtain, respectively, the residual functions 𝑅₁ and 𝑅₂𝑅₁(𝑡)=𝑑𝑢_𝑛,1(𝑡)𝑢𝑑𝑡−𝑓_𝑛−1,1,𝑢_𝑛,1,𝑢_𝑛+1,1,𝑅₂(𝑡)=𝑑𝑢_𝑛,2(𝑡)𝑢𝑑𝑡−𝑓_𝑛−1,2,𝑢_𝑛,2,𝑢_𝑛+1,2. (3.9)Step 2. Solitary-solution formulation is constructed as follows: 𝜔²=𝜔²₁𝑅₂(0)−𝜔²₂𝑅₁(0)𝑅₂(0)−𝑅₁(0), (3.10) where 𝜔₁=1 and 𝜔₂=𝜔.Step 3. Combining the coefficients of 𝑒^𝜉𝑛 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]𝑑𝑢_𝑛=𝑑𝑡𝛼−𝑢²_𝑛𝑢_𝑛−1−𝑢_𝑛+1, (4.1) where 𝛼≠0.

Substituting (3.3)–(3.7) into (4.1), we obtain, respectively, the residual functions 𝑅₁ and 𝑅₂, and using the solitary-solution formulation, (3.10), we have 𝑒^3𝜉𝑛𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑−𝑒^−3𝜉𝑛𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑+𝑒^2𝜉𝑛𝐵𝜔³𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑−𝑒^−2𝜉𝑛𝐵𝜔³𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑+𝑒^𝜉𝑛(𝜔³𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑)−𝑒^−𝜉𝑛(𝜔³𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑)=0. (4.2) Setting the coefficients of 𝑒^𝑖𝜉𝑛 to be zero, we have 𝜔³+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑−𝜔−𝛼𝑒^−𝑑+𝛼𝑒^𝑑𝐵𝜔=0,₃𝑒^−𝑑+𝜔³𝑒^𝑑−2𝜔²𝛼𝑒^𝑑+2𝜔²𝛼𝑒^−𝑑−𝜔𝑒^𝑑−𝜔𝑒^−𝑑+2𝛼𝑒^𝑑−2𝛼𝑒^−𝑑𝜔=0₃𝐵²−𝜔³+𝜔³𝑒^−2𝑑+𝜔³𝑒^2𝑑+𝜔²𝐴²𝑒^𝑑−𝜔²𝐴²𝑒^−𝑑+𝜔²𝛼𝑒^−𝑑−𝜔²𝛼𝑒^𝑑+𝜔²𝛼𝐵²𝑒^−𝑑,−𝜔²𝛼𝐵²𝑒^𝑑−𝜔𝐵²+𝜔−𝜔𝑒^2𝑑−𝜔𝑒^−2𝑑+𝐴²𝑒^−𝑑−𝐴²𝑒^𝑑+𝛼𝐵²𝑒^𝑑−𝛼𝐵²𝑒^−𝑑+𝛼𝑒^𝑑−𝛼𝑒^−𝑑=0. (4.3) Solving (4.3) simultaneously, we have(𝑑),√𝜔=2𝛼sinh𝐴=2−𝛼sinh(𝑑),𝐵=0. (4.4) We, therefore, obtain the following needed solitary solution:𝑢_𝑛=2√−𝛼sinh(𝑑)𝑒^{𝜉𝑛+2𝛼sinh(𝑑)𝑡}+𝑒^{−(𝜉𝑛+2𝛼sinh(𝑑)𝑡)}=√−𝛼sinh(𝑑)sech2𝛼sinh(𝑑)𝑡+𝑛𝑑+𝜉₀. (4.5)

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.

Acknowledgment

The work is supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).