1. Introduction

In recent decades, a new attitude with regard to the exploration of nonlinear evolution equations (NLEEs) has been actively progressing in various branches of Sciences. Nonlinear evolution equations have been the important subject of investigation in various branches of mathematical and physical sciences such as physics, fluid mechanics, chemistry, biology and etc. Obtaining the analytical solutions of NLEEs is an important topic in various areas of Science, since many of mathematical and physical models are explained by NLEEs. Among the possible solutions to NLEEs, specific form of solutions may contingent only on a single combination of variables such as solitons. In mathematics and physics, a soliton is a self-reinforcing solitary wave that sustains its form through it travels with constant speed. Solitons are the special solutions of certain nonlinear partial differential equations, with interesting properties. Because of a balance between nonlinear and linear effects, the shape of soliton wave pulses does not change during propagation in a medium. The soliton phenomenon firstly described by John Scott Russell (1808-1882). The main idea of this phenomenon occurred to him when he observed a solitary wave in the Union Canal in Scotland. Russell reproduced same phenomenon in a wave tank and named it the “Wave of Translation" (also known as solitary wave or soliton) [¹].

The mathematical modeling of DNA vibration dynamics is proposed by Peyrard-Bishop, that takes into consideration the inclusion of nonlinear interaction between adjacent displacements along the Hydrogen bonds [⁸]. To study the main important properties of DNA structure, we have investigated the oscillator-chain of Peyrard-Bishop model (PB) ^[8] which has successfully predicated the appearance of solitonic structures. As it is well known, the balance between weak nonlinearity and dispersion in DNA dynamic model yields the typical derivation of model equations mathematically. This dispersion in many cases enters only in the linear level. Unfortunately, this approach eliminates other possibilities related to the presence of nonlinear dispersion ^[9,10]. Some mathematical structure of DNA has been described in various interesting lines of research, to see more structure and physical properties of DNA, the authors can Ref. to [^2-6] and their references. The mathematical and physical modelling of DNA dynamics has reduced to an important nonlinear structures. The nonlinearity of the DNA dynamic model causes it to form localized waves. The localized waves have some interesting properties, such as the ability of transporting energy without dissipation [^2-7].

Taking into consideration the inharmonic potential, Aguero et al. ^[6] and Najera et al. ^[11] have studied the following modified PB model:

where l21,l2,D and α are constants. Krumhansl et al., ^[12], suggested a theory of soliton excitations as an explanation of the open states of DNA modeling system. In ^[12], they firstly developed the possibility that nonlinear effects might concentrate vibrational energy in DNA into localized soliton like objects. Some other similar worksare studied about solitons of DNA dynamics model, such as Yomosa in ^[13], proposed a soliton theory using a plane base-rotor model. This model was further refined by Takeno and Homma ^[14], who allowed discreteness effects to be taken into account, and by Zhang ^[15], who improved the model for base coupling. Zayed and Arnous also applied Homogeneous Balance Method in ^[35] and generalized Riccati equation mapping method in ^[36,37] to find traveling wave solutions of PB model (1.1).

In the recent decade, many powerful and direct methods have been proposed to find special solutions of nonlinear evolution equations (NLEEs), such as the Bácklund transformation ^[16] and Hirota bilinear method ^[17]. With the help of the computer implementations, some other algebraic method proposed, such as tanh/coth method ^[18], homogeneous balance method ^[19], the Miura transformation ^[20], sine/cosine method ^[21] and Exp-function method ^[22] and some other, see ^[23]. But most of the methods may sometimes fail or can only lead to a kind of special solution and the solution procedures become very complex as the degree of nonlinearity increases.

Since 2008, the (G'/G)-expansion method, firstly introduced by Wang et al. ^[24], has become widely used to search for various exact solutions of NLEEs ^[24-31]. The worth of the (G'/G)-expansion method is that it reduced the nonlinear PDEs to ODEs by some essentially linear methods, i.e. the method is based on the explicit linearization of NLEEs for traveling waves which leads to a second-order differential equation with constant coefficients. The F-expansion method is also an effective and direct algebraic method for constructing the exact solutions of nonlinear evolution equations ^[32]. To do this, it reduces the nonlinear evolution equations to a simple algebraic equation. Many of NLEEs have been solved by F-expansion method, (for a deeper discussion we refer the reader to ^[33,34] and the references given there).

In this paper, we first describe briefly the mathematical concept of both (G'/G)-expansion method and F-expansion method, then we use them to investigate the traveling wave solutions of a mathematical modeling of PB model (1.1).

2. Description of the (G'/G)-expansion method

In this section, we introduce briefly the (G'/G)-expansion method for solving certain nonlinear partial differential equations (PDEs). For a deeper discussion of (G'/G)-expansion method we refer the reader to ^[24-27]. Suppose we have a nonlinear PDE for u(x,t), in the form

where P is a polynomial in its arguments, which includes nonlinear terms and the highest order derivatives. Using the transformation u(x,t)=U(ξ),ξ=x-ωt, Eq. (2) reduces to the ordinary differential equation (ODE)

where U=U(ξ), and prime denotes derivative with respect to ξ. We assume that the solution of Eq. (3) can be expressed by a polynomial in (G'/G) as follows:

where αn, n=0,21,2,…,m, are constants to be determined later and G(ξ) satisfies a second order linear ordinary differential equation (LODE):

where λ and μ are arbitrary constants. Using the general solutions of Eq. (5), we have

and it follows, from (4) and (5), that

and so on, here the prime denotes the derivative with respective to ξ. To determine u explicitly, we take the following four steps:

3. F-expansion method

In order to get more solutions of (1), we use the F-expansion method ^[38] to deal with (13). Supposing that Eq. (13) has the following formal solutions:

where α2≠0,α1, and α0, are constants to be determined further. And 𝜑 satisfy

Substituting (21) and (22) into (13) and collecting all terms with the same order of φ together equating each coefficient of this polynomial to zero, one can get

Case 1:

Case 2:

Thus, substitute the obtained results (23) and (24) in (22), we get

Substituting the general solutions (22) into Eq. (21), we obtain the following two types of traveling wave solutions of Eq. (8):

In view of (22) has a lot of fundamental solutions (twenty seven solutions) ^[33], one can find a number of exact travelling wave solutions for (1), which are listed some special solutions as follows.

Family 1: -4AC>0 and AC≠0,

where E and F are two non-zero real constants and satisfies F2-E2>0.

Family 2 : When -4AC<0 and AC≠0,

where E and F are two non-zero real constants and satisfies F2-E2>0.

Remark 1. Using Case. 2, one can get other exact solutions of Eq. (1). The details are omitted here.

4. Conclusions

This study shows that both (G'/G)-expansion method and F-expansion method are quite efficient and practically well suited for use in finding exact solutions for a mathematical modeling of DNA vibration dynamics. The reliability of the methods and the reduction in the size of computational do- main give these methods a wider applicability. With the aid of Maple 12, we have assured the correctness of the obtained solutions by putting them back into the original equation. We hope that they will be useful for further studies in applied sciences and engineering.