DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation15814210.1155/2009/158142158142Research ArticleApplication of Symbolic Computation in Nonlinear Differential-Difference EquationsXieFuding1,22WangZhen3JiMin1ZhouYong1Department of Computer ScienceLiaoning Normal UniversityLiaoning, Dalian 116081Chinalnnu.edu.cn2Key Laboratory of Mathematics and Mechanization (KLMM), Academy of Mathematics and Systems ScienceChinese Academy of ScienceBeijing 100080Chinacas.ac.cn3School of Physics and Electronic TechnologyLiaoning Normal UniversityLiaoning, Dalian 116029Chinalnnu.edu.cn20092792009200918032009120920092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A method is proposed to construct closed-form solutions of nonlinear differential-difference equations. For the variety of nonlinearities, this method only deals with such equations which are written in polynomials in function and its derivative. Some closed-form solutions of Hybrid lattice, Discrete mKdV lattice, and modified Volterra lattice are obtained by using the proposed method. The travelling wave solutions of nonlinear differential-difference equations in polynomial in function tanh are included in these solutions. This implies that the proposed method is more powerful than the one introduced by Baldwin et al. The results obtained in this paper show the validity of the proposal.

1. Introduction

Wadati  introduced the following nonlinear differential-difference equation (NDDE): dun(t)dt=(α+βun+γun2)(un-1-un+1), where α, β, and γ0 are constants.

However, (1.1) can be thought as a discrete version of a nonlinear partial differential equation: ut+6αuux+6βu2ux+uxxx=0, which can be solved by the inverse scattering method .

However, (1.1) obviously includes the following famous NDDEs, namely,

(a) hybrid lattice : dun(t)dt=(1+βun+γun2)(un-1-un+1),

(b) discrete mKdV lattice [1, 4]: dun(t)dt=(1+un2)(un+1-un-1),

(c) modified Volterra lattice :   dun(t)dt=un2(un+1-un-1).

The travelling wave solutions of (1.3) in polynomial in function tanh are reported in . Ü. Göktas and W. Hereman  investigated the conservations laws of (1.4). However, (1.5) is a very well studied integrable model. It is a bi-Hamiltonian, possesses a Lax pair, recursion operator, local master-symmetry, infinite hierarchy of higher symmetries, and conservation laws . In this study, searching for the closed-form solutions, especially solitary wave solutions and periodic solutions of (1.4) and (1.5), is considered.

In the theory of lattice-soliton, there existed several classical methods to seek for solutions of NDDEs, such as the inverse scattering method [6, 7], bilinear form [8, 9], symmetries , and numerical methods . As far as we know, little work is being done to find closed-form solutions of NDDEs by using of symbolic computation. Baldwin et al.  recently presented an adaptation of the tanh-method to solve NDDEs. Some analytical (closed-form) solutions of several lattices in polynomial in function tanh have been obtained . Their work may be thought as a breakthrough in solving NDDEs symbolically.

In this study, the method proposed in  where tanh-solutions are only considered is firstly generalized, and then is applied to solve (1.1). As a result, a variety of closed-form solutions of (1.1) have been found in terms of trigonometric and Jacobi elliptic functions. The proposed approach allows us to exactly solve NDDEs with the aid of symbolic computation. The solutions presented here not only cover the known one presented by Baldwin et al., but also introduce new solutions for some NDDEs.

The rest of the paper is organized as follows: inthe following section, an improved method is proposed and how to obtain the solitary wave solutions and periodic solutions to NDDEs is depicted. Section 3 is devoted to illustrating the application of the proposal in exactly solving (1.1). As a result, some new solitary wave solutions and periodic solutions of Hybrid lattice, discrete mKdV lattice, and modified Volterra lattice have been obtained. The final is conclusions.

2. The Improved Method

To solve NDDEs directly, in this section, one would like to describe the improved method and its algorithm. Suppose the NDDE we study in this work is in the following form

P((t),un+p2(k)(t)un+p1(t),un+p2(t),,un+ps(t),un+p1(t),un+p2(t),,un+ps(t),,un+p1(k)(t),un+p2(k)(t),,un+ps(k)(t))=0, where P is a polynomial; un(t) is a dependent variable; t is a continuous variable; the superscript denotes the order of derivative; n, piZ.

To compute the travelling wave solutions of (2.1), we first set un(t)=u(ξn) and

ξn=dn+ct+ξ0, where d and c are constants to be determined later and ξ0 is a constant.

Step 1.

We assume that the travelling wave solutions of (2.1) we are looking for are in the following frame: un(t)=a0+i=1mgi-1(ξn)[aif(ξn)+bjg(ξn)], with f(ξn)=sinh(ξn)cosh  (ξn)+r,g(ξn)=1cosh(ξn)+r, where ai, bj are all constants to be determined later and r a constant.

The degree m in (2.3) can be determined by balancing the highest nonlinear term and the highest-order derivative term in (2.1) as in the continuous case.

The functions f(ξn) and g(ξn) in (2.4) satisfy the following equations: f(ξn)=(1+r2)g2(ξn)-rg(ξn),      g(ξn)=-f(ξn)g(ξn),f2(ξn)=(1-rg(ξn))2-g2(ξn).

However, (2.5) reveals that fj(ξn)  (j2) can be expressed by fl(ξn)gh(ξn)  (l=0,1).

Step 2 (to derive and solve the algebraic system).

The following identity is easily obtained in terms of (2.2): ξn+pi=d(n+pi)+ct+ξ0=ξn+dpi. It says the relationship between ξn+pi and ξn. To find the solutions of NDDEs, one also utilizes the following formulae: sinh(x±y)=sinh(x)cosh(y)±cosh(x)sinh(y),cosh(x±y)  =cosh(x)cosh(y)±sinh(x)sinh(y).

If the ξn in (2.3) is replaced by ξn+pi, then expression of un+pi in sinh(ξn) and cosh(ξn)  will be obtained in terms of (2.7).

With the aid of symbolic computation software Maple 8, we substitute un+pi into (2.1) and apply the rule

sinh2(x)=cosh2(x)-1, to simplify the expression. Clearing the denominator and collecting the coefficients of   coshk(ξn)sinhl(ξn)  (k=0,1,,h;  l=0,1) and setting them to zero, we can obtain the nonlinear algebraic equations. The values of unknowns will be found by using the Wu's method  to solve algebraic system.

Step 3 (construct and test the exact solutions).

Substitute the values obtained in Step 2 along with (2.2) and (2.4) into (2.3), one can find the solutions of (2.1). To assure the correctness of the solutions, it is necessary to substitute them into the original equation.

Remark 2.1.

If we set r=0 in (2.4), the travelling wave solutions of (2.1) in polynomial in tanh and sech will be found. If one further requires that bj=0, then the polynomial travelling wave solutions in tanh will be obtained. In this sense, we can say that this method covers the one given in .

Remark 2.2.

More important, if the hyperbolic functions in (2.4) are replaced by the trigonometric function, that is, f(ξn)=sin(ξn)cos(ξn)+r,g(ξn)=1cos(ξn)+r, the periodic solutions to (2.1) will be obtained. While doing so, it is necessary to modify the formulae (2.7) and (2.8) properly.

Remark 2.3.

We can obviously write (2.4) into the following solitary wave form which have physical relevance: f(ξn)=tanh(ξn)1+rsech(ξn)=1coth(ξn)+rcsch(ξn),g(ξn)=sech(ξn)1+rsech(ξn)=csch(ξn)coth(ξn)+rcsch(ξn).

In what follows, we will apply this method to solve (1.1). As a result, their abundant exact solutions have been derived.

3. Soliton Wave Solutions and Periodic Solutions of (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>)

By balancing the highest nonlinear term un2 and the highest-order derivative term dun(t)/dt in (1.1), we have m+1=2m, that is, m=1.

Therefore, the following formal solutions for (1.1) can be assumed:

un(t)=u(ξn)=a0+a1sinh(ξn)cosh(ξn)+r+b1cosh(ξn)+r, with ξn as (2.2).

Starting from (2.6), we have

un±1(t)=u(ξn±1)=u(ξn±d).

The expressions of un+1(t) and un-1(t) in sinh(ξn) and cosh(ξn) are obtained by (2.7), (3.1), and (3.2). Substituting them and (3.1) into (1.1), clearing the denominator, and setting coefficients of the terms   sinhl(ξn)coshk(ξn)  (k=0,1,,3;  l=0,1) to zero give

a1[2sinh(d)(γra12+βa0r+αr-βb1-2γa0b1+γa02r)+cr]=0,2sinh(d)(2γa0a12r-γa12b1-αb1-βa0b1-γa02b1+βa12r)-b1c=0,a1[c+2sinh(d)cosh(d)(γa12+γa02+βa0+α)+4sinh(d)(γa02r2-γb12+βa0r2+αr2)+2cr2cosh(d)]=0,sinh(d)(βa12r2-2γa0b12-βb12-2γa02rb1-2αrb1+2γa0r2a12-2βa0rb1+γa12b1r)+2a12sinh(d)cosh(d)(2γ  a0+β)-rb1ccosh(d)=0,a1[2sinh(d)cosh(d)(2αr+βb1+2β  a0r+γa0b1+γa02r)+2sinh(d)(-γa12r+2γa0b1+γb12r+γa02r3+βb1r2+αr3+βa0r3+βb1+2γa0r2b1)+2crcosh(d)+cr  cosh2(d)+cr3-cr]=0,2b1sinh(d)(-γb12-γa02r2+γa12-αr2-βa0r2-2γa0rb1-βb1r)+2a12sinh(d)cosh(d)(2γa0r+2γb1+βr)-b1c  cosh2(d)+b1c-b1cr2=0,a1[2sinh(d)cosh(d)(βa0r2-γa12+βb1r+γa02r2+2γa0rb1+αr2+γb12)+2b1sinh(d)(2γb1+βr)+4γa0ra1b1a1ccosh2(d)+cr2-c]=0.

It is difficult to solve this system by hand. Thus, we fall back on symbolic computation software Maple 8 and Wu's method which is a powerful tool to deal with nonlinear algebraic equations. With the aid of them, we can find the solutions to the above system as follows:

Case 1.

a1=0,a0=-β2γr(β2-4αγ)(r2-1)tanh(d/2)2γ(r2-1)b1=±[2r2-1-cosh(d)](β2-4αγ)(r2-1)tanh(d/2)2γ(r2-1),c=[2r2-1-cosh(d)](β2-4αγ)tanh(d/2)2γ(r2-1);

Case 2.

a0=-β2γ,a1=±β2-4αγ tanh(d/2)2γ,b1=±(β2-4αγ)(r2-1)  tanh(d/2)2γ,c=(β2-4αγ)tanh(d/2)γ;

Case 3.

b1=0,r=0,a0=-β2γ,a1=±β2-4αγtanh(d)2γ,c=(β2-4αγ)tanh(d)2γ.

Thus the solitary wave solutions of (1.1) are

un(t)=β2γ(β2-4αγ)(r2-1)  tanh(d/2)2γ(r2-1)(r-2r2-1-cosh(d)cosh(ξn)+r), where ξn=nd+([2r2-1-cosh(d)](β2-4αγ)tanh(d/2)/2γ(r2-1))t+ξ0;

un(t)=-β2γ±β2-4αγ  tanh(d/2)2γsinh(ξn)+r2-1cosh(ξn)+r, where ξn=nd+((β2-4αγ)tanh(d/2)/γ)t+ξ0;

un(t)=-β2γ±β2-4αγ  tanh(d)2γtanh(nd+(β2-4αγ)tanh(d)2γt+ξ0).

Similarly, we can also assume that (1.1) possesses the following form solution:

un(t)=s0+s1sin(ξn)cos(ξn)+r+s2cos(ξn)+r, where si  (i=0,1,2) is a constant to be determined later.

Repeating the above process and properly modifying the formulae (2.7) and (2.8), we can derive the periodic solutions for (1.1) as follows. For brevity, the procedure of seeking for periodic solutions to (1.1) is omitted:

un(t)=-β2γ±(β2-4αγ)(r2-1)tan(d/2)2γ(r2-1)(r+1+cosh(d)-2r2cos(ξn)+r), where ξn=nd+([2r2-1-cos(d)](β2-4αγ)tan(d/2)/2γ(r2-1))t+ξ0;

un(t)=-β2γ±β2-4αγtan(d/2)2γsin(ξn)+r2-1cos(ξn)+r, where ξn=nd+((β2-4αγ)tan(d/2)/γ)t+ξ0;

un(t)=-β2γ±β2-4αγtan(d)2γtan(nd+(β2-4αγ)tan(d)2γt+ξ0).

If we properly set the parameters α, β, and γ, then it is easily to obtain closed-form solutions of Hybrid lattice, discrete mKdV lattice, and modified Volterra lattice, respectively.

In fact, the solution given in  for (1.3) is nothing but the solution (3.9). As far as we know, no works have been reported on the closed form solutions of (1.4) and (1.5). Thus, in this sense, these results are novel.

4. Conclusions

A method is proposed to find the solitary wave solutions and periodic solutions for NDDEs. The new closed-form solutions of Hybrid lattice, discrete mKdV lattice, and modified Volterra lattice have been found by using the proposal and symbolic computation. This study reveals that computer algebra plays an important role in exactly solving NDDEs. Meanwhile, it is worthwhile to point out that the proposed method may be applicable to other NDDEs to seek for their travelling wave solutions.

Acknowledgment

The authors would like to thank the anonymous reviewer for his/her valuable suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 10771092).

WadatiM.Transformation theories for nonlinear discrete systemsProgress of Theoretical Physics Supplement197659366310.1143/PTPS.59.36WadatiM.Wave propagation in nonlinear lattice—IJournal of the Physical Society of Japan197538673680MR037129610.1143/JPSJ.38.673BaldwinD.GöktasÜHeremanW.Symbolic computation of hyperbolic tangent solutions for nonlinear differential-difference equationsComputer Physics Communications2004162320321710.1016/j.cpc.2004.07.0022-s2.0-4444343860GöktasÜHeremanW.Computation of conservation laws for nonlinear latticesPhysica D19981231–44254362-s2.0-0001746868AdlerV. E.SvinolupovS. I.YamilovR. I.Multi-component Volterra and Toda type integrable equationsPhysics Letters A19992541-22436MR168809910.1016/S0375-9601(99)00087-0ZBL0983.37082AblowitzM. J.LadikJ. F.Nonlinear differential-difference equationsJournal of Mathematical Physics19751635986032-s2.0-36749115552TodaM.Theory of Nonlinear Lattices1988Berlin, GermanySpringerHuX.-B.MaW.-X.Application of Hirota's bilinear formalism to the Toeplitz lattice—some special soliton-like solutionsPhysics Letters A20022933-4161165MR188929910.1016/S0375-9601(01)00850-7ZBL0985.35072TamH.-W.HuX.-B.Soliton solutions and Bäcklund transformation for the Kupershmidt five-field lattice: a bilinear approachApplied Mathematics Letters2002158987993MR1925925ZBL1009.37047LouS. Y.Generalized symmetries and W algebras in three-dimensional Toda field theoryPhysical Review Letters1993712540994102MR124968310.1103/PhysRevLett.71.4099ZBL0972.81559ElmerC. E.Van VleckE. S.A variant of Newton's method for the computation of traveling waves of bistable differential-difference equationsJournal of Dynamics and Differential Equations20021434935172-s2.0-4444281421WuW. T.Polynomial Equation-Solving and Its Application, Algorithms and Computation1994Berlin, GermanySpringer