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Investigation of Solitary wave solutions for VakhnenkoParkes equation via expfunction and Exp(−ϕ(ξ))expansion method
SpringerPlus volume 3, Article number: 692 (2014)
Abstract
In this paper, we have described two dreadfully important methods to solve nonlinear partial differential equations which are known as expfunction 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, threedimensional plots of the solutions such as solitons, singular solitons, bell type solitary wave i.e. nontopological solitons solutions and periodic solutions are also given to visualize the dynamics of the equation.
1. 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. nontopological 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.
For instance the inverse scattering transform (Ablowitz and Clarkson 1991; Vakhnenko and Parkes 2002; Vakhnenko and Parkes 2012a; Vakhnenko and Parkes 2002b), the complex hyperbolic function method (Zayed et al. 2006; Chow 1995), the rank analysis method (Feng 2000), the ansatz method (Hu 2001a; Hu 2001b; Majid et al. 2012), the (G′/G) expansion method (Wang et al. 2008; Roshid et al. 2013a; Bekir 2008; Roshid et al. 2013b; Zhang 2008; Alam 2013), the modified simple equation method (Jawad et al. 2010), the expfunctions method (He and Wu 2006), the Hirota method (Hirota 1971), the sinecosine method (Wazwaz 2004), the tanhfunction method (Parkes and Duffy 1996), extended tanhfunction method (Fan 2000; Parkes 2010a; Parkes 2010b), the Jacobi elliptic function expansion method (Liu 2005; Chen and Wang 2005), the Fexpansion method (Wang and Zhou 2003; Wang and Li 2005), the Backlund transformation method (Miura 1978), the Darboux transformation method (Matveev and Salle 1991), the homogeneous balance method (Wang 1995; Zayed et al. 2004; Wang 1996), the Adomian decomposition method (Adomain 1994; Wazwaz 2002), the auxiliary equation method (Sirendaoreji and Sun 2003; Sirendaoreji 2007), the exp(−ϕ(ξ)) expansion method (Khan and Akbar 2013) and so on.
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 VakhnenkoParkes equation was investigated in (Kangalgil and Ayaz 2008; Parkes 2010b; Gandarias and Bruzon 2009; Yasar 2010; Abazari 2010; Liu and He 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 VakhnenkoParkes equation (1) via two methods namely the Expfunction 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 expfunction and the exp(−ϕ(ξ)) expansion method. By these methods, we gain the exact solutions of VakhnenkoParkes 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.
2. 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 VakhnenkoParkes 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
2.1. The expfunction method
We now discuss the expfunction method to solve partial differential equation Eq. (1).
Step2.1.1. 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 rewritten in the following form:
Step2.1.2: 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.
Step2.1.3: 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 maple13. Substituting these values into Eq. (6), we gain traveling wave solutions of the Eq. (1).
2.2. 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 wellknown solutions of the ODE (9) are as follows:
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).
3. Application
In this section, we exert the expfunction method and the exp(−ϕ(ξ)) expansion method to construct the rational solitary wave, nontopological soliton, periodic wave solutions for some nonlinear evolution equations in mathematical physics via the VakhnenkoParkes 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
3.1. Solution of Vakhnenko Parkes equation via the expfunction method
Now, we apply the Expfunction method to create the generalized traveling wave solutions of the Vakhnenko Parkes Eq. (1).
According to Step 2.1.1 in the Expfunction method, the solution of Eq. (16) can be written in the form of Eq. (6). 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.
Case 1: Suppose p = c = 1 and q = d = 1.
Since there are some free variables, for simplicity, we presume b_{1} = 1.
Now, substituting Eq. (18) into Eq. (16) and by employing the computer algebra, such as Maple 13, we gain
Where A = (e^{2ξ} + b_{0}e^{ξ} + b_{‒ 1})^{4},
Setting these equations to zero and solving the system of algebraic equations with the aid of commercial software Maple13, 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, we imagine b_{2} = 1, b_{− 1} = 0.
Executing the same procedure as described in case1, we gain
Setting these values in the Eq. (22) we acquire the solution
which is same obtain in the previous case1.
Case 3: Suppose p = c = 2 and q = d = 2.
Since there are some free parameters, for simplicity, we presume a_{− 2} = a_{− 1} = 0, b_{− 2} = b_{− 1} = 0, b_{1} = 1_{.}
Executing the same procedure as described in the case1 and in the case2, we attain
Hence require solution is
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 nontopological solitons solution. But in generally, we can obtain all of the above solutions and another family of solutions in case 4.
Case 4: Suppose p = c = 1 and q = d = 1.
Now, substituting Eq. (27) into Eq. (15) and by employing the computer algebra, such as Maple 13, we gain
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 Maple13, we achieve the following solution.

(i)
$$\begin{array}{l}{a}_{1}=0,\phantom{\rule{0.24em}{0ex}}{a}_{0}=3{b}_{0},\phantom{\rule{0.24em}{0ex}}{a}_{1}=0,\phantom{\rule{0.24em}{0ex}}{b}_{0}=\mathit{const}\phantom{\rule{0.24em}{0ex}}\mathrm{and}\phantom{\rule{0.24em}{0ex}}{b}_{1}=\\ \phantom{\rule{0.24em}{0ex}}{b}_{0}^{2}/4{b}_{1}\text{.}\end{array}$$

(ii)
$$\begin{array}{l}{a}_{1}={b}_{0}^{2}/4{b}_{1},\phantom{\rule{0.24em}{0ex}}{a}_{0}=2{b}_{0},\phantom{\rule{0.24em}{0ex}}{a}_{1}={b}_{1},\phantom{\rule{0.24em}{0ex}}\mathrm{and}\phantom{\rule{0.24em}{0ex}}{b}_{1}=\\ \phantom{\rule{0.24em}{0ex}}{b}_{0}^{2}/4{b}_{1}\text{.}\end{array}$$
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
Remark1: We have the solution (19) in the form via Expfunction method, $u\left(\xi \right)=\frac{12{b}_{0}}{4{e}^{\xi}+4{b}_{0}+{b}_{0}^{2}{e}^{\xi}}$
It note that if b_{0} > 0 and exp(−x_{0}) = 2/b_{0} then it can be written $u\left(\xi \right)=\frac{3}{2}sec{h}^{2}\left(\frac{1}{2}\left(x\mathit{wt}{x}_{0}\right)\right)$ and if b_{0} <0 and exp(−x_{0}) = 2/b_{0} then it can be written $u\left(\xi \right)=\frac{3}{2}cosec{h}^{2}\left(\frac{1}{2}\left(x\mathit{wt}{x}_{0}\right)\right)$. These two solutions are just solutions u_{11} and u_{12} in Parkes (Parkes 2010b) with k = 1/2.
And for the solution (29) in the form via Expfunction method, $u\left(\xi \right)=1+\frac{3{b}_{0}}{{b}_{1}{e}^{\xi}+{b}_{0}+{b}_{0}^{2}{e}^{\xi}/4{b}_{1}}$
It note that if b_{1}/b_{0} > 0 and exp(−x_{0}) = 2b_{1}/b_{0} then it can be written $u\left(\xi \right)=1+\frac{3}{2}sec{h}^{2}\left(\frac{1}{2}\left(x\mathit{wt}{x}_{0}\right)\right)$ and if b_{1}/b_{0} < 0 and exp(−x_{0}) = 2b_{1}/b_{0} then it can be written $u\left(\xi \right)=1\frac{3}{2}cosec{h}^{2}\left(\frac{1}{2}\left(x\mathit{wt}{x}_{0}\right)\right)$. These two solutions are just solutions u_{21} and u_{22} in Parkes (Parkes 2010b) with k = 1/2.
3.2. Solutions of Vakhnenko Parkes equation via the exp(−ϕ(ξ)) expansion method
Balance the highest order derivate term uu″ with the highest nonlinear terms u^{3} in Eq. (16), we obtain m = 2, so assume the equation Eq. (1) has the solution
Inserting Eq. (31) into Eq. (16) and using the ODE (9), and then collecting all terms with the same order of exp(−ϕ(ξ)) together, Eq. (16) is converted into new polynomial in exp(−ϕ(ξ)). Setting each coefficients of this polynomial is to zero, yields a system of algebraic equations for l_{0}, l_{1}, l_{2} , λ; and μ which are as follows:
Solving the system of algebraic equations and we obtained l_{0} = − 6μ, l_{1} = − 6λ, l_{2} = − 6. Substituting the values of l_{0}, l_{1}, l_{2} in the general solutions of Eq. (9) achieve more traveling wave solutions of nonlinear evolution equation Eq. (1) as follows:
When λ^{2} − 4μ > 0, μ ≠ 0, then
When λ^{2} − 4μ < 0, then
When λ^{2} − 4μ > 0, μ = 0, then
When λ^{2} − 4μ = 0, μ ≠ 0, λ ≠ 0, then
Remark2: All of the solutions presented in this latter have been checked with Maple by putting them back into the original equations.
4. Results and discussion
In this paper we exerted the expfunction methods and the exp(−ϕ(ξ)) expansion method as useful mathematical tools to construct topological soliton, nontopological soliton, periodic wave solutions for the Vakhnenko Parkes equation. The methods have successfully handled with the aid of commercial software Maple13 that greatly reduces the volume of computation and improves the results of the equation. We have achieved a family of solutions via expfunction 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.
4.1. 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 particlelike structures, such as magnetic monopoles etc. The solution (19) in Figure 1 of the equation (1) is represented the exact Bell type solitary (nontopological soliton) wave solution for the parameters b_{0} = 4, w = 1 with − 3 ≤ x, t ≤ 3 via expfunction method. Since second family Eq. (30) has a constant different with first family it figure is also the exact Bell type solitary (nontopological soliton) wave solution. Others solutions via expfunction method are similar to this solution or can be obtained from this solution which profiles are similar to the 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.
Solutions (34) of the equation Eq. (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. (35) for the parameters λ = w = 1, μ = c = 0 with − 3 ≤ x, t ≤ 3.
4.2. Graphical representations
The graphical illustrations of the solutions are given below in the figures (Figures 1, 2, 3, 4 and 5) with the aid commercial software of Maple13.
5. Conclusion
In this research some new solitary wave solutions of the VakhnenkoParkes equation is found using the expfunction 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 expfunction 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.
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The authors would like to express their sincere thanks to the anonymous referees for their valuable comments and suggestions.
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The authors, viz HOR, MRK, RCB and BKD with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
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Roshid, H., Kabir, M.R., Bhowmik, R.C. et al. Investigation of Solitary wave solutions for VakhnenkoParkes equation via expfunction and Exp(−ϕ(ξ))expansion method. SpringerPlus 3, 692 (2014). https://doi.org/10.1186/219318013692
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Keywords
 Solitary Wave
 Wave Solution
 Soliton Solution
 Travel Wave Solution
 Expansion Method