APPLICATION OF THE RATIONAL (G′/G)-EXPANSION METHOD FOR SOLVING SOME COUPLED AND COMBINED WAVE EQUATIONS

In this paper, we explore the travelling wave solutions for some nonlinear partial differential equations by using the recently established rational (G′/G)-expansion method. We apply this method to the combined KdV-mKdV equation, the reaction-diffusion equation and the coupled HirotaSatsuma KdV equations. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are also derived from the travelling waves. We have also given some figures for the solutions.


Introduction
In the past decades, the travelling wave solutions of nonlinear partial differential equations (NLPDEs) play an effective role in physics, engineering and applied mathematics. The mathematical models of these subjects give important information about the behaviour of the physical event. Therefore, it is very important to obtain the traveling wave solutions of NLPDEs [32]. The NLPDEs have interesting structures that deals with many phenomena in physics, chemistry and engineering, for example; in fluid flow, plasma waves, mechanics, solid state physics, oceanic phenomena, atmospheric phenomena and so on. Many researchers have been proposed various different methods to find solutions for nonlinear partial differential equations and nonlinear fractional differential equations [36][37][38][39][40]. Such as the inverse scattering transform method [1], the Hirota's bilinear method [2], truncated SOLVING SOME COUPLED AND COMBINED WAVE EQUATIONS 117 Painlevé expansion method [3], the tanh-function expansion method [4], the Jacobi elliptic function expansion method [5], the homogeneous balance method [6][7][8], the trial function method [9], the exp-function method [10,34], differential transform method [33], the Bäcklund transform method [11], the generalized Riccati equation method [12][13][14][15], the sub-ODE method [17][18][19][20], the original (G ′ /G)-expansion method [16,29], the double (G ′ /G,1/G)-expansion method [35] etc.. Since there is not a common method that can be used to solve all types of nonlinear evolution equations.
Some researchers established several powerful and direct methods. Wang et al. [16] first introduced the (G ′ /G)-expansion method to find travelling wave solutions of nonlinear evolution equations. Later Islam et al. [21] proposed the rational (G ′ /G)-expansion method which aims to derive closed form travelling wave solutions. In this paper we use the rational (G ′ /G)-expansion method and apply for the combined KdV-mKdV equation, the reaction-diffusion equation, and the coupled Hirota-Satsuma KdV equations. We derived abundant solutions for each equation that is different from the solutions in the literature.

Description of the Method
Suppose that u = u(x, t) is an unknown function depends on the x and t variables and we define the polynomial P in u(x, t) and its various order partial derivatives and nonlinear terms as We use the following steps, to solve Eq.(1) by means of the rational (G ′ /G)expansion method.
Step 1: We assign a new variable U (ξ) in terms of x and t variables and a new transformation: where is ξ 0 a constant and s is the velocity of the wave. The transformation in Eq.(2) transforms Eq.(1) into an ordinary differential equation (ODE) for u = U (ξ).
where U and its derivatives with respect to ξ are the elements of the Q polynomial of U (ξ).
Step 2: Next we integrate Eq.(3) one or twice as possible. Suppose that the solution of Eq.(3) can be written in the following form where a j and b j (j = 0, 1, 2, ..., n) , (a n ̸ = 0, b n ̸ = 0) are arbitrary coefficient to be found later. Next we write, the G = G(ξ) function, which satisfies the following second order ODE; where λ and µ are real constants. We convert Eq.(5) into (G ′ /G) form, From Eq.(5) or Eq.(6) the solution for (G ′ /G) as follows where c 1 and c 2 are constants.
Step 3: To determine the value of n, which is the degree of U (ξ), in Step 2, we apply the homogeneous balance method, that is balancing between the highest order nonlinear terms and the highest order derivatives in Eq. (3). The degree of other terms in Eq.(3) can be written as in the following form [21]  where deg[U (ξ)] is the degree of U (ξ).
Step 4: After determining the value of n, we substitute Eq.(4) along with Eq.(5) into Eq.(3). Equating the coefficients of (G ′ /G) to zero, gives a system of algebraic equations. In order to solve these equations we use the computer software programme such as Maple or Matematica. If there is a possible solution, we obtain values for a i , b i , λ, µ and s (i = 0, 1, 2, ..., n).
Step 5: Finally we substitute the values of a i , b i (i = 0, 1, 2, ..., n), λ, µ, s and the solutions given in Eq. The KdV and mKdV equations are widely studied popular soliton equations. The nonlinear terms appearing in the KdV and mKdV equations often exist in applied science and engineering, such as in plasma physics, ocean dynamics and quantum field theory [22][23][24]. If we combine the quadratic nonlinear term of the KdV equation and the qubic nonlinear term of the mKdV equation, then we get the combined KdV-mKdV equation or the Gardner equation [25] u t + αu u x + βu 2 u x + u xxx = 0 (8) where α and β are nonzero parameters. This equation describes the wave propagation of bounded particle,sound wave and thermal pulse [26][27][28].
where s is the velocity of the wave and the superscript of U shows the derivative of U with respect to ξ. Next, we integrate Eq.(9) and deduce the following equation where C is an integration constant to be found later. We use homogeneous balance method, such as balancing the terms U ′′ and U 3 in Eq.(10) we get n = 1, so we can write Eq.(4) as Next we substitute Eq.(11) into Eq.(10) and organize the equation in terms of the powers of (G ′ /G). Hence equating the coefficients of (G ′ /G) and its powers to zero in the resulting equation, gives a system of algebraic equations for a 0 , b 0 , a 1 , b 1 , s and C. Solving the set of equations by using the computer programme Maple, we get the following set of solutions.

Example 2. The reaction-diffusion equation
We have the reaction-diffusion equation [30] u tt + αu xx + βu + γu 3 = 0 (22) where α , β and γ are nonzero constants. The traveling wave variable Eq.(2) reduces the Eq.(22) into an ODE where s is the velocity of the wave. Next we express the solution of the Eq.(23) in terms of (G ′ /G) as it is written in Eq.(4), where G = G(ξ) satisfies the second order linear ODE in Eq.(23). We use homogeneous balance method, such as balancing the terms U ′′ and U 3 in Eq.(23) we get n = 1, hence from Eq.(4), we have Substituting Eq.(23) into Eq. (22) and write the left hand side in terms of (G ′ /G).
Hence equating the coefficients of the resulting equation to zero, gives a system of algebraic equations for a 0 , b 0 , a 1 , b 1 and s. Solving the set of equations by using the computer programme, we get the following set of solutions: where b 1 , λ and µ are all arbitrary constants. Substituting Eq.(25) into Eq.(24) we get the following solution and (G ′ /G) is given in Eq. (7). Substituting Eq. (7) into Eq.(26), we deduce the following travelling wave solutions.

Example 3. The coupled Hirota-Satsuma KdV equations
The coupled Hirota-Satsuma KdV equations (CHSK) describes an interaction of two long waves with different dispersion relations [31]. We will consider the CHSK equations in the following form Making the transformations u( where s is the velocity of the wave to be determined later. We get the CHSK equations in the following form By balancing the highest order derivatives and nonlinear terms in Eq.(34), we get n = 2 and from Eq.(4) we write the solutions of Eq.(33) as Substituting Eq.(35) into Eq.(34), and we convert Eq.(34) into a polynomial in (G ′ /G). Equating the coefficients of the same power of (G ′ /G) to zero, yields a set of simultaneous algebraic equations. Solving the set of equations for a i , b i , e i , d i (i = 0, 1, 2) and s by using the computer programme, we get the following set of solutions SOLVING SOME COUPLED AND COMBINED WAVE EQUATIONS 127 where a 0 , b 0 , e 2 , λ and µ are constants. Substituting Eq.(36) into Eq.(35), hence we reach the following solutions and (G ′ /G) is given in Eq. (7). Substituting Eq. (7) into Eq.(37), we deduce the following travelling wave solutions.

Conclusion
In this paper, we have obtained various types of travelling wave solutions for the combined KdV-mKdV equation, the reaction-diffusion equation, and the coupled Hirota-Satsuma KdV equations that are solved by using the rational (G ′ /G)expansion method. The main idea of this method is to reduce the partial differential equation to an ODE by using the travelling wave transformation (Eq.(2)), after integrating the ODE in Eq.(3), once or twice, then express the ODE in a compact form. This ODE can be written by a n-th degree polynomial in terms of (G ′ /G), where G = G(ξ) is the general solution of the second order LODE in Eq. (5). In order to find the positive integer, we use the homogeneous balance method, that is balancing between the highest order derivative term and nonlinear term. The coefficients of the polynomials can be obtained by solving a set of algebraic equations. Generally, the resulted algebraic equations can be solved by using Maple software program. It is mostly possible to find a solution of the algebraic equations, but it is generally unable to guarantee the existence of a solution. Despite of this, the rational (G ′ /G)-expansion method is still powerful method for finding travelling wave solutions of nonlinear evolution equations. The rational (G ′ /G)-expansion method is also direct, concise, elementary that the general solution of the second order ODE Eq.(5) is well known and effective that it can be used for many other nonlinear evolution equations, such as the generalized shallow water wave equation, the compound KdV-Burgers equations, the Klein-Gordon equation, the generalized KPP equation, the approximate long water wave equations, the coupled nonlinear Klein-Gordon-Zakharov equations, and so on. Therefore, various explicit solutions of these nonlinear evolution equations can be obtained by this method.
Author Contribution Statements Authors contributed equally and they read and approved the final manuscript.
Declaration of Competing Interests The authors declare that they have no competing interest.