All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

: In this article, we prove that the (cid:104) p , q (cid:105) condition holds, ﬁrst by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W . Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions u r ( x + ν t ) and simply periodic solutions u s ,1 ( x + ν t ) of Kawahara equation are solitary wave solutions, while simply periodic solutions u s ,2 ( x + ν t ) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.


Introduction and Main Results
In this article, we study all traveling wave exact solutions of the Kawahara equation [1]: where a, c, d are constants, x and t represent spatial and time variables, uu x is the nonlinear disturbance term, and u xxx and u xxxxx are the dispersion terms of order three and five, respectively. The solutions of Equation (1) could be used to analyze and interpret a lot of nonlinear dispersive phenomena that can arise in optical fiber, ocean, plasma physics, etc. [2,3]. Given the importance of this equation, it remains the subject of study for many researchers. Many different techniques were devoted to studying various solutions of Equation (1) and its family [4][5][6][7][8][9][10][11][12][13]. For example, Kudryashov [14], obtained exact meromorphic solutions of the Kawahara equation using the Laurent series. Wazwaz [15], found some different solutions (compacton and solitons) to Equation (1) in terms of trigonometric functions. Khan [16], analyzed the Kawahara equation using the variational approach and derived new conditions for obtaining solitary wave solutions. Using the traveling wave ansatz, Baiswas [17], studied the generalized Kawahara equation and derived a solitary wave solution for the family of the Kawahara equation. A lot of effective methods are applied to study the exact solutions, which makes the research more abundant [18][19][20][21][22][23]. Wang [24] used ansatz method to derive the exact solitary wave solution for the generalized Korteweg-de Vries-Kawahara (GKdV-K) equation. Aiman Zara [25] studied numerical approximation of the modified Kawahara equation using the Kernel smoothing method. El-Tantawy [26] derived a set of novel exact and approximate analytic solutions to the family of the forced damped Kawahara equation (KE) using the ansatz method.
In recent years, Yuan et al. [27,28] introduced the p, q condition to study a class of constant-coefficient complex algebraic differential equations with dominant terms, and obtained all possible meromorphic solutions which belong to the class W. That is to say, the general representation of rational function solutions, finitely growing simply periodic solutions and elliptic solutions, and both rational function solutions and simply periodic solutions can be obtained by degenerating elliptic general solutions. Futhermore, a complex method for obtaining exact solutions in the mathematical physics is presented. Using the complex method [27,28], we prove that all meromorphic solutions of complex Kawahara equation belong to the class W, and then get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Equation (1). This method is simpler than other methods in the process of finding solutions, and can also be applied to the solution of other nonlinear differential equations.
Substitute the traveling wave transform into Equation (1); integrate it to get where C 0 is an integral constant. Multiply Equation (3) by w , and integrate it; thus, we get the complex ordinary differential Kawahara equation where A = a 6d , B = − c 2d , C = − ν 2d , D = − C 0 d , E = − C 1 d , C 0 and C 1 are integral constants. Now, we give the main results in our paper. Theorem 1. Let A = 0, then all meromorphic solutions w(z) of Equation (4) belong to the class W. Here, class W consists of elliptic functions and their degenerations with the form R(z) or R(e αz ), α ∈ C, where R is a rational function.
In addition, the solutions of Equation (4) have the forms: (1) All elliptic function solutions D and N are constants.
All traveling wave exact solutions of Equation (1) are obtained by substituting (2) into meromorphic solutions w of Equation (4). So, we get the following Theorem. Theorem 2. Let ad = 0, all traveling wave exact solutions u(x, t) of Equation (1) have the following forms: (1) All elliptic function solutions (2) All simply periodic solutions and where γ is a root of 31z 2 − 31z + 10 = 0.
(3) All rational function solutions where c = 0, x 0 and t 0 are arbitrary real constants.
The rest of the article is organized in the following order: In Section 2, we will present the relevant lemmas and methodology. Section 3 gives the detailed proof process of Theorem 1 and concise proof method of Theorem 2. Section 4 illustrates our main results using computer simulations. In the last section, some conclusions are given.

Preliminary Lemmas and the Complex Method
We need some definitions and lemmas in order to prove Theorem 1 and present the complex method.

Definition 1 ([28]).
A differential polynomial is defined by where Λ is a finite index set, and b r are constants. deg P[w] := max r∈Λ {d(r)} is called the degree of P[w]. Consider the differential equation where a = 0 is a constant. The dominant part of E(z, w) composes all dominant terms which can determine the multiplicity q of w in E(z, w), and is denoted byÊ(z, w). D(q) and D r (q) represent the multiplicity of pole of each term inÊ(z, w) and the multiplicity of pole of each monomial in E(z, w) −Ê(z, w), respectively. Obviously Definition 2 ([28]). The derivative ofÊ(z, w) with respect to w can be calculated by the following formula, for any χ,Ê Definition 3 ([28]). Substituting Laurent series into Equation (15), where c −q = 0, q > 0. Then, we can get p different principle −1 ∑ l=−q c l z l with pole of multiplicity q at z = 0, Equation (15) is said to satisfy weak p, q condition. If Equation (15) has p different meromorphic solutions with pole of multiplicity q at z = 0, Equation (15) satisfies p, q condition.

Definition 5 ([29]). The Fuchs index of Equation
where c −ij can be determined by (18) , c −ij and k 0 are constants.
Any rational function solution w := R(z) is expressed as which has m(≤ p) different poles of multiplicity q. Any simply periodic solution w := R(ξ) is a rational function of ξ = e αz (α ∈ C) and can be given in the form which has m(≤ p) different poles of multiplicity q.
Lemma 2 ( [33,34]). Weierstrass elliptic functions ℘(z) have the addition formula and two successive degeneracies, as shown below: (II) If ∆(g 2 , g 3 ) = 0, Weierstrass elliptic functions degenerate to simply periodic functions, which can be expressed as (III) If g 2 = g 3 = 0, Weierstrass elliptic functions degenerate to rational functions of z, which can be expressed as Next, we give the complex method.

5.
Inserting the inverse transform T −1 into w(z − z 0 ), we obtain all exact solutions u(x, t) of the given partial differential equation.
From, (23), we get the indeterminant form of rational function solutions of Equation (4) at pole z = 0 Inserting (32) into Equation (4), and then setting the coefficients to zero, we get the system of Equations (1) which are given in Appendix A. Computing the system of Equations (1), we get So we find that the rational function solutions of Equation (4) are Hence, all rational function solutions of Equation (4) are where B = 0, D = − C 2 3A , E = C 3 27A 2 and z 0 ∈ C. By (22) and (26), we get the indeterminant form of simply periodic solutions of Equation (4) at pole z = 0 where ℘(z, 3δ 2 , −δ 3 ) = 2δ − 3 2 δ coth 2 3δ 2 z.
Substituting (36) into (35), we get Comparing coefficients of (37) and (28), we obtain the system of Equations (2) which are showed in Appendix A. Solving the system of Equations (2), we have and (39).
From (22), we infer the indeterminant forms of elliptic function solutions of Equation (4) are with pole at z = 0. Expanding ℘(z, g 2 , g 3 ) at z = 0, we have Put (45) into (44) and apply (20), we get Comparing coefficients of w(z) and (28), we obtain the system of Equations (3). which are shown in the Appendix. Computing the system of Equations (3), we derive and g 2 satisfies Substituting (44) and (47) into Equation (4), from the correlation of coefficients, we can get Thus, all elliptic function solutions of Equation (4) are where z 0 ∈ C, g 3 = 31B 3 593190 − 7B 390 g 2 , and g 2 satisfies By using the addition formula, we can get another representation of it as where D and N are constants. So far, we have completed the proof of Theorem 1. Substituting (2) into all meromorphic solutions w(z) of Equation (4), we obtained Theorem 2. According to the theorem in ref. [35], we can get the following corollary.
Corollary. All rational solutions u r (x + νt) and simply periodic solutions u s,1 (x + νt) of Equation (1) are real valued, while simply periodic solutions u s,2 (x + νt) are not real valued.

Computer Simulations
This subsection will show our results through computer simulations of u r (x + νt) and u s,1 (x + νt), as demonstrated in the following figures.

1.
By applying the complex method, we are able to achieve the rational solution u r (x + νt) of Equation (4). Figure 1 describes the 3D graphs of solution u r (x + νt) for a = 1, ν = 1, and d = 1 1680 within the interval −5 ≤ x, t ≤ 5. Figure 2 shows the 2D graphs of solution u r (x + νt) for a = 1, ν = 1, and d = 1 1680 within the interval −10 ≤ x ≤ 10 when t = 0. It could be observed that they have one generation pole, which is shown by Figures 1 and 2.

Conclusions
In this paper, we are the first to utilize the complex method to prove that all meromorphic solutions of complex Kawahara equation belong to the class W, and then we get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Equation (1). In addition, we find that all rational solutions u r (x + νt) and simply periodic solutions u s,1 (x + νt) of Equation (1) are solitary wave solutions, which could be used to analyze and interpret a lot of nonlinear dispersive phenomena. This research enriches the methods of solving differential equations. Our results also reveal that the complex method of looking for traveling wave exact solutions is general and feasible, and can be applied to other nonlinear partial differential equations.