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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.

Wadati [

However, (

However, (

(a) hybrid lattice [

(b) discrete mKdV lattice [

(c) modified Volterra lattice [

The travelling wave solutions of (

In the theory of lattice-soliton, there existed several classical methods to seek for solutions of NDDEs, such as the inverse scattering method [

In this study, the method proposed in [

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

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

To compute the travelling wave solutions of (

We assume that the travelling wave solutions of (

The degree

The functions

However, (

The following identity is easily obtained in terms of (

If the

With the aid of symbolic computation software Maple 8, we substitute

Substitute the values obtained in Step

If we set

More important, if the hyperbolic functions in (

We can obviously write (

In what follows, we will apply this method to solve (

By balancing the highest nonlinear term

Therefore, the following formal solutions for (

Starting from (

The expressions of

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:

Thus the solitary wave solutions of (

Similarly, we can also assume that (

Repeating the above process and properly modifying the formulae (

If we properly set the parameters

In fact, the solution given in [

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.

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).