Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))-expansion method

In this paper, we have described two dreadfully important methods to solve nonlinear partial differential equations which are known as exp-function and the exp(−ϕ(ξ)) -expansion method. Recently, there are several methods to use for finding analytical solutions of the nonlinear partial differential equations. The methods are diverse and useful for solving the nonlinear evolution equations. With the help of these methods, we are investigated the exact travelling wave solutions of the Vakhnenko- Parkes equation. The obtaining soliton solutions of this equation are described many physical phenomena for weakly nonlinear surface and internal waves in a rotating ocean. Further, three-dimensional plots of the solutions such as solitons, singular solitons, bell type solitary wave i.e. non-topological solitons solutions and periodic solutions are also given to visualize the dynamics of the equation.


Introduction
The effort in finding exact solutions to nonlinear equations is witnessed significant curiosity and progress in finding solutions to nonlinear partial differential equations (NPDEs) that resemble physical phenomena. The nonlinear wave phenomena observed in fluid dynamics, plasma and optical fibers are often modeled by the bell (i.e. non-topological solitons) shaped sech solutions and the kink (i.e. topological solitons) shaped tanh solutions. Both mathematicians and physicists have devoted considerable effort of research regarding this matter. A peek at the literature reveals a lot of effective methods that solve this type of NPDEs.
Recently, a remarkable and important discover has been made by Vakhnenko and Parkes (Vakhnenko and Parkes 1998), who have confirmed an integrable equation as follows: The traveling wave solutions of this Vakhnenko-Parkes equation was investigated in (Kangalgil and Ayaz 2008;Parkes 2010b;Gandarias and Bruzon 2009;Yasar 2010;Abazari 2010;Liu andHe 2013, Ostrovsky 1978) and Liu (Liu and He 2013) found traveling wave solutions of this equations by improved (G′/G) -expansion method with auxiliary equation GG″ = AG 2 + BGG′ + C(G′) 2 .
In this paper, we investigate the traveling wave solutions of the Vakhnenko-Parkes equation (1) via two methods namely the Exp-function and the exp(−ϕ(ξ)) -expansion methods.
The rest of the paper is organized as follows: In section 2, we build up an introduction of exp-function and the exp(−ϕ(ξ)) -expansion method. By these methods, we gain the exact solutions of Vakhnenko-Parkes equation in section 3. In section 4, we out line results and discussion of the achieved solutions. Finally, some conclusions are drawn in the section 5.

The methodologies
In this section, we will go over the main points of the expfunction method and the exp(−ϕ(ξ)) -expansion method to raise the rational solitary wave and periodic wave solutions for the Vakhnenko-Parkes equation which have been paid attention by many researchers in mathematical physics.
Consider a nonlinear equation with two independent variable x and t, is given by where U = U(x,t) is an unknown function, P is a polynomial in U = U(x,t) and its partial derivatives, in which the highest order derivatives term and nonlinear terms are involved.
Combining the independent variable x and t into one traveling wave variable ξ = x ± wt, we suppose that The travelling wave variable (3) permits us to convert the Eq. (2) to an ODE for u = u(ξ) is

The exp-function method
We now discuss the exp-function method to solve partial differential equation Eq. (1).
Step-2.1.1. Assume the solution of the Eq. (1) can be expressed in the following form (He and Wu, 2006): where c, d, p and q are positive unknown integers that could be determine subsequently, a n and b m are unknown constants, Eq. (5) can be re-written in the following form: Step-2.1.2: To determine the values of c and p, we balance the highest order linear term with the highest order nonlinear term in Equation Eq. (4). Similarly, to determine the values of d and q, we have to balance the lowest order linear term with the lowest order nonlinear term in Equation Eq. (4). This confirms the determination of the values of c, d, p and q.
Step-2.1.3: Inserting the values of c, d, p and q into Eq. (6) and then substituting Eq. (6) into Eq. (4) and simplifying, we attain; Then collecting all coefficient Cj and setting each of them to zero, yields a system of algebraic equations for a c 's and b p 's. Then unknown a c 's and b p 's can be evaluated by solving the system of algebraic equations with the help of maple-13. Substituting these values into Eq. (6), we gain traveling wave solutions of the Eq. (1).

The exp(−ϕ(ξ)) -expansion method
Step 2.2.1. Assume that the solution of ODE (4) can be expressed by a polynomial in exp(−ϕ(ξ)) as follows: where ϕ′(ξ) satisfies the ODE The well-known solutions of the ODE (9) are as follows: When λ 2 −4μ < 0; then ϕ ξ ð Þ ¼ ln When When l i , w, λ; i = 0, ⋯ ⋯, m and μ are constants to be determined later, l m ≠ 0, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms arising in the ODE(4).
Step 2.2.2. By substituting Eq. (8) into Eq. (4) and using the ODE (9), and then collecting all terms with the same order of exp(−ϕ(ξ)) together, the left hand side of Eq. (4) is converted into new polynomial in exp (−ϕ(ξ)). Setting each coefficient of this polynomial to zero, yields a system of algebraic equations for l i , ⋯ w, λ; i = 0, ⋯ ⋯, m and μ. Solving the system of algebraic equations and substituting l i , ⋯ w; i = 0, ⋯ ⋯, m, and the general solutions of Eq. (9) into Eq. (8). We have more traveling wave solutions of nonlinear evolution equation Eq. (1).

Application
In this section, we exert the exp-function method and the exp(−ϕ(ξ)) -expansion method to construct the rational solitary wave, non-topological soliton, periodic wave solutions for some nonlinear evolution equations in mathematical physics via the Vakhnenko-Parkes equation Eq. (1).
Inserting Eq. (3) into Eq. (1), we amend the Eq. (1) into the ODE: Integrating Eq. (15) with respect to ξ and setting the integration constant equal to zero yields To determine the values of c and p, according to Step 2.1.2, we balance the term of the highest order in uu″ and the highest nonlinear terms u 3 in Eq. (16). With the aid of computational software Maple 13, yields p = c. To find out the values of q and d, we balance the term of lowest order uu″ in Eq. (16) with lowest order nonlinear term u 3 , with the aid of computational software Maple 13, yields to result q = d. The parameters are free, so we can arbitrarily prefer the values of c and d, but the ultimate solution does not depend upon the choices of them.
Since there are some free variables, for simplicity, we Now, substituting Eq. (18) into Eq. (16) and by employing the computer algebra, such as Maple 13, we gain Setting these equations to zero and solving the system of algebraic equations with the aid of commercial software Maple-13, we achieve the following solution.
Setting these values in the Eq. (18) we acquire the solution If we set If we choose Case 2: Suppose p = c = 2 and q = d = 1.
Since there are some free parameters, for simplicity, Executing the same procedure as described in case-1, we gain Setting these values in the Eq. (22) we acquire the solution which is same obtain in the previous case-1.
Executing the same procedure as described in the case-1 and in the case-2, we attain where ξ = x − wt.
This is also similar solutions achieved in the previous cases and so we should not repeat the procedure again and again for different values of the parameters. Actually the solution is a bell shape soliton solution which referred to as non-topological solitons solution. But in generally, we can obtain all of the above solutions and another family of solutions in case 4.
Now, substituting Eq. (27) into Eq. (15) and by employing the computer algebra, such as Maple 13, we gain e ξ A ðC 1 e 8ξ þ C 2 e 7ξ þ C 3 e 6ξ þ C 4 e 5ξ þ C 5 e 4ξ þ C 6 e 3ξ þ C 7 e 2ξ þ C 8 e ξ þ C 9 Þ ¼ 0 Where A = (b 1 e 2ξ + b 0 e ξ + b -1 ) 5 , others are omitted for simplicity and setting these equations to zero and solving the system of algebraic equations with the aid of commercial software Maple-13, we achieve the following solution.
The solution (i) is same obtained in case 1.
Setting these values of (ii) in the Eq. (18) we acquire the solution If we choose Or if choose Remark-1: We have the solution (19) in the form via Exp-function method, u ξ ð Þ ¼

Results and discussion
In this paper we exerted the exp-function methods and the exp(−ϕ(ξ)) -expansion method as useful mathematical tools to construct topological soliton, non-topological soliton, periodic wave solutions for the Vakhnenko-Parkes equation. The methods have successfully handled with the aid of commercial software Maple-13 that greatly reduces the volume of computation and improves the results of the equation. We have achieved a family of solutions via exp-function method. It is worth declaring that some of our obtained solutions via the exp(−ϕ(ξ)) -expansion method is in good agreement with already published results which is presented in the Tables 1 and 2. The others are completely new solutions achieved by exp(−ϕ(ξ)) -expansion method.

Physical interpretation
In this subsection, we describe the physical interpretation of the solutions for the Vakhnenko-Parkes equation. Solitons are solitary waves with stretchy dispersion possessions, which described many physical phenomena in soliton physics. Soliton preserve their shapes and speed after colliding with each other. Soliton solutions also give ascend to particle-like structures, such as magnetic monopoles etc. The solution (19) in Figure 1 of the equation (1) is represented the exact Bell type solitary (non-topological soliton) wave solution for the parameters b 0 = 4, w = 1 with − 3 ≤ x, t ≤ 3 via exp-function   Figure 1. The solution (32) obtained by the exp(−ϕ(ξ)) -expansion method is cuspon whose shape is depicted in the Figure 2 for the parameters λ = 3, μ = c = w = 1 with − 3 ≤ x, t ≤ 3.
The solution (33) of the equation Eq. (11) is presented the periodic travelling wave solution for various values of the physical parameters. The Figure 3 has been shown the shape of the solution (33) for the parameters λ = 1, μ = c = 2, w = 1 with − 3 ≤ x, t ≤ 3.  (1) represent singular soliton solution for the parameters λ = w = 1, μ = c = 0 with − 3 ≤ x, t ≤ 3 whose shape is given by the Figure 4.
Finally, solution (35) and (36) are similar type solutions and they represent the multiple soliton solution. Omitting one figure we depicted the Figure 5 of the Eq.

Conclusion
In this research some new solitary wave solutions of the Vakhnenko-Parkes equation is found using the exp-function method and the exp(−ϕ(ξ)) -expansion method. As a results two family of bell type solitary wave solutions Eq. (19) or Eq. (26) and Eq. (30) using exp-function method and five solutions Eq. (32)-Eq. (36) including cuspon, singular soliton, multiple soliton and periodic solutions are achieved via exp(−ϕ (ξ)) -expansion method of the Vakhnenko-Parkes equation exist for real sense depends on different relevant physical parameters. Numerical results of the solutions for real sense by using Maple software have been shown graphically and discussed. This will have a good sense to encourage the extensive application of the equations.