Application of Symbolic Computation in Nonlinear Differential-Difference Equations

Abstract

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 [1] introduced the following nonlinear differential-difference equation (NDDE): 𝑑𝑢_𝑛(𝑡)𝑑𝑡=(𝛼+𝛽𝑢_𝑛+𝛾𝑢²_𝑛)(𝑢_𝑛−1−𝑢_𝑛+1), (1.1) where 𝛼, 𝛽, and 𝛾≠0 are constants.

However, (1.1) can be thought as a discrete version of a nonlinear partial differential equation: 𝑢_𝑡+6𝛼𝑢𝑢_𝑥+6𝛽𝑢²𝑢_𝑥+𝑢_𝑥𝑥𝑥=0, (1.2) which can be solved by the inverse scattering method [2].

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

(a) hybrid lattice [3]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=(1+𝛽𝑢_𝑛+𝛾𝑢²_𝑛)(𝑢_𝑛−1−𝑢_𝑛+1), (1.3) (b) discrete mKdV lattice [1, 4]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=(1+𝑢²_𝑛)(𝑢_𝑛+1−𝑢_𝑛−1), (1.4)(c) modified Volterra lattice [5]: 𝑑𝑢_𝑛(𝑡)𝑑𝑡=𝑢²_𝑛(𝑢_𝑛+1−𝑢_𝑛−1). (1.5)

The travelling wave solutions of (1.3) in polynomial in function tanh are reported in [3]. Ü. Göktas and W. Hereman [4] 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 [5]. 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 [10], and numerical methods [11]. 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. [3] 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 [3]. Their work may be thought as a breakthrough in solving NDDEs symbolically.

In this study, the method proposed in [3] 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.[3], 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(𝑢_𝑛+𝑝1(𝑡),𝑢_𝑛+𝑝2(𝑡),…,𝑢_{𝑛+𝑝𝑠}(𝑡),𝑢^′_𝑛+𝑝1(𝑡),𝑢^′_𝑛+𝑝2(𝑡),…,𝑢^′_{𝑛+𝑝𝑠}𝑢(𝑡),…,_{(𝑘)𝑛+𝑝1}(𝑡),𝑢^(𝑘)_𝑛+𝑝2(𝑡),…,𝑢^(𝑘)_{𝑛+𝑝𝑠}(𝑡))=0, (2.1) where 𝑃 is a polynomial; 𝑢_𝑛(𝑡) is a dependent variable; 𝑡 is a continuous variable; the superscript denotes the order of derivative; 𝑛, 𝑝_𝑖∈𝑍.

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

𝜉_𝑛=𝑑𝑛+𝑐𝑡+𝜉₀, (2.2) 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: 𝑢_𝑛(𝑡)=𝑎₀+^𝑚_𝑖=1𝑔^𝑖−1(𝜉_𝑛)[𝑎_𝑖𝑓(𝜉_𝑛)+𝑏_𝑗𝑔(𝜉_𝑛)], (2.3) with 𝑓(𝜉_𝑛)=sinh(𝜉_𝑛)cosh(𝜉_𝑛,)+𝑟𝑔(𝜉_𝑛1)=cosh(𝜉_𝑛,)+𝑟 (2.4) where 𝑎_𝑖, 𝑏_𝑗 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(𝜉_𝑛) and 𝑔(𝜉_𝑛) in (2.4) satisfy the following equations: 𝑓^′(𝜉_𝑛)=(1+𝑟²)𝑔²(𝜉_𝑛)−𝑟𝑔(𝜉_𝑛),𝑔^′(𝜉_𝑛)=−𝑓(𝜉_𝑛)𝑔(𝜉_𝑛𝑓),₂(𝜉_𝑛)=(1−𝑟𝑔(𝜉_𝑛))²−𝑔²(𝜉_𝑛). (2.5)

However, (2.5) reveals that 𝑓^𝑗(𝜉_𝑛)(𝑗≥2) can be expressed by 𝑓^𝑙(𝜉_𝑛)𝑔^ℎ(𝜉_𝑛)(𝑙=0,1).

Step 2 (to derive and solve the algebraic system). The following identity is easily obtained in terms of (2.2): 𝜉_{𝑛+𝑝𝑖}=𝑑(𝑛+𝑝_𝑖)+𝑐𝑡+𝜉₀=𝜉_𝑛+𝑑𝑝_𝑖. (2.6) It says the relationship between 𝜉_{𝑛+𝑝𝑖} and 𝜉_𝑛. 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). (2.7)

If the 𝜉_𝑛 in (2.3) is replaced by 𝜉_{𝑛+𝑝𝑖}, then expression of 𝑢_{𝑛+𝑝𝑖} in sinh(𝜉_𝑛) and cosh(𝜉_𝑛)will be obtained in terms of (2.7).

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

sinh²(𝑥)=cosh²(𝑥)−1, (2.8) to simplify the expression. Clearing the denominator and collecting the coefficients of cosh^𝑘(𝜉_𝑛)sinh^𝑙(𝜉_𝑛)(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 [12] 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 𝑏_𝑗=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 [3].

Remark 2.2. More important, if the hyperbolic functions in (2.4) are replaced by the trigonometric function, that is, 𝑓(𝜉_𝑛)=sin(𝜉_𝑛)cos(𝜉_𝑛,)+𝑟𝑔(𝜉_𝑛1)=cos(𝜉_𝑛,)+𝑟 (2.9) 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(𝜉_𝑛)=tanh(𝜉_𝑛)1+𝑟sech(𝜉_𝑛)=1coth(𝜉_𝑛)+rcsch(𝜉_𝑛),𝑔(𝜉_𝑛)=sech(𝜉_𝑛)1+𝑟sech(𝜉_𝑛)=csch(𝜉_𝑛)coth(𝜉_𝑛)+rcsch(𝜉_𝑛). (2.10)
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 (1.1)

By balancing the highest nonlinear term 𝑢²_𝑛 and the highest-order derivative term 𝑑𝑢_𝑛(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:

𝑢_𝑛(𝑡)=𝑢(𝜉_𝑛)=𝑎₀+𝑎₁sinh(𝜉_𝑛)cosh(𝜉_𝑛+𝑏)+𝑟cosh(𝜉¹_𝑛,)+𝑟 (3.1) with 𝜉_𝑛 as (2.2).

Starting from (2.6), we have

𝑢_𝑛±1(𝑡)=𝑢(𝜉_𝑛±1)=𝑢(𝜉_𝑛±𝑑). (3.2)

The expressions of 𝑢_𝑛+1(𝑡) and 𝑢_𝑛−1(𝑡) in sinh(𝜉_𝑛) and cosh(𝜉_𝑛) 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 sinh^𝑙(𝜉_𝑛)cosh^𝑘(𝜉_𝑛)(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)−𝑏₁𝑎𝑐=0,₁[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)(2𝛾𝑎₀+𝛽)−𝑟𝑏₁𝑎𝑐cosh(𝑑)=0,₁[2sinh(𝑑)cosh(𝑑)(2𝛼𝑟+𝛽𝑏₁+2𝛽𝑎₀𝑟+𝛾𝑎₀𝑏₁+𝛾𝑎₀²𝑟)+2sinh(𝑑)(−𝛾𝑎₁²𝑟+2𝛾𝑎₀𝑏₁+𝛾𝑏₁²𝑟+𝛾𝑎₀²𝑟³+𝛽𝑏₁𝑟²+𝛼𝑟³+𝛽𝑎₀𝑟³+𝛽𝑏₁+2𝛾𝑎₀𝑟²𝑏₁)+2𝑐𝑟cosh(𝑑)+𝑐𝑟cosh²(𝑑)+𝑐𝑟³−𝑐𝑟]=0,2𝑏₁sinh(𝑑)(−𝛾𝑏₁²−𝛾𝑎₀²𝑟²+𝛾𝑎₁²−𝛼𝑟²−𝛽𝑎₀𝑟²−2𝛾𝑎₀𝑟𝑏₁−𝛽𝑏₁𝑟)+2𝑎₁²sinh(𝑑)cosh(𝑑)(2𝛾𝑎₀𝑟+2𝛾𝑏₁+𝛽𝑟)−𝑏₁𝑐cosh²(𝑑)+𝑏₁𝑐−𝑏₁𝑐𝑟²𝑎=0,₁[2sinh(𝑑)cosh(𝑑)(𝛽𝑎₀𝑟²−𝛾𝑎₁²+𝛽𝑏₁𝑟+𝛾𝑎₀²𝑟²+2𝛾𝑎₀𝑟𝑏₁+𝛼𝑟²+𝛾𝑏₁²)+2𝑏₁sinh(𝑑)(2𝛾𝑏₁+𝛽𝑟)+4𝛾𝑎₀𝑟𝑎₁𝑏₁𝑎₁𝑐cosh²(𝑑)+𝑐𝑟²−𝑐]=0. (3.3)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)

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,𝑎₀𝛽=−∓𝑟2𝛾(𝛽²−4𝛼𝛾)(𝑟²−1)tanh(𝑑/2)2𝛾(𝑟²𝑏−1)₁=±[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) (3.4)

Case 2. 𝑎₀𝛽=−2𝛾,𝑎₁=±𝛽²−4𝛼𝛾tanh(𝑑/2),𝑏2𝛾₁=±(𝛽²−4𝛼𝛾)(𝑟²−1)tanh(𝑑/2)2𝛾,𝑐=(𝛽²−4𝛼𝛾)tanh(𝑑/2)𝛾; (3.5)

Case 3. 𝑏₁=0,𝑟=0,𝑎₀𝛽=−,𝑎2𝛾₁=±𝛽²−4𝛼𝛾tanh(𝑑)2𝛾,𝑐=(𝛽²−4𝛼𝛾)tanh(𝑑).2𝛾 (3.6)

Thus the solitary wave solutions of (1.1) are

𝑢_𝑛𝛽(𝑡)=∓2𝛾(𝛽²−4𝛼𝛾)(𝑟²−1)tanh(𝑑/2)2𝛾(𝑟²−1)(𝑟−2𝑟²−1−cosh(𝑑)cosh(𝜉_𝑛)+𝑟), (3.7) where 𝜉_𝑛=𝑛𝑑+([2𝑟²−1−cosh(𝑑)](𝛽²−4𝛼𝛾)tanh(𝑑/2)/2𝛾(𝑟²−1))𝑡+𝜉₀;

𝑢_𝑛𝛽(𝑡)=−±2𝛾𝛽²−4𝛼𝛾tanh(𝑑/2)2𝛾sinh(𝜉_𝑛)+√𝑟²−1cosh(𝜉_𝑛,)+𝑟 (3.8) where 𝜉_𝑛=𝑛𝑑+((𝛽²−4𝛼𝛾)tanh(𝑑/2)/𝛾)𝑡+𝜉₀;

𝑢_𝑛𝛽(𝑡)=−±2𝛾𝛽²−4𝛼𝛾tanh(𝑑)2𝛾tanh(𝑛𝑑+(𝛽²−4𝛼𝛾)tanh(𝑑)2𝛾𝑡+𝜉₀). (3.9)

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

𝑢_𝑛(𝑡)=𝑠₀+𝑠₁sin(𝜉_𝑛)cos(𝜉_𝑛+𝑠)+𝑟cos(𝜉²_𝑛,)+𝑟 (3.10) where 𝑠_𝑖(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:

𝑢_𝑛𝛽(𝑡)=−±2𝛾(𝛽²−4𝛼𝛾)(𝑟²−1)tan(𝑑/2)2𝛾(𝑟²−1)(𝑟+1+cosh(𝑑)−2𝑟²cos(𝜉_𝑛)+𝑟), (3.11) where 𝜉_𝑛=𝑛𝑑+([2𝑟²−1−cos(𝑑)](𝛽²−4𝛼𝛾)tan(𝑑/2)/2𝛾(𝑟²−1))𝑡+𝜉₀;

𝑢_𝑛𝛽(𝑡)=−±2𝛾𝛽²−4𝛼𝛾tan(𝑑/2)2𝛾sin(𝜉_𝑛)+√𝑟²−1cos(𝜉_𝑛,)+𝑟 (3.12) where 𝜉_𝑛=𝑛𝑑+((𝛽²−4𝛼𝛾)tan(𝑑/2)/𝛾)𝑡+𝜉₀;

𝑢_𝑛𝛽(𝑡)=−±2𝛾𝛽²−4𝛼𝛾tan(𝑑)2𝛾tan(𝑛𝑑+(𝛽²−4𝛼𝛾)tan(𝑑)2𝛾𝑡+𝜉₀). (3.13)

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 [3] 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).