Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations

Themodified simple equation (MSE)method is executed to find the travelingwave solutions for the coupledKonno-Oono equations and the variant Boussinesq equations. The efficiency of this method for finding exact solutions and traveling wave solutions has been demonstrated. It has been shown that the proposed method is direct, effective, and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Moreover, this procedure reduces the large volume of calculations.


Introduction
Nowadays NLEEs have been the subject of all-embracing studies in various branches of nonlinear sciences.A special class of analytical solutions named traveling wave solutions for NLEEs has a lot of importance, because most of the phenomena that arise in mathematical physics and engineering fields can be described by NLEEs.NLEEs are frequently used to describe many problems of protein chemistry, chemically reactive materials, in ecology most population models, in physics the heat flow and the wave propagation phenomena, quantum mechanics, fluid mechanics, plasma physics, propagation of shallow water waves, optical fibers, biology, solid state physics, chemical kinematics, geochemistry, meteorology, electricity, and so forth.Therefore, investigation, traveling wave solutions is becoming more and more attractive in nonlinear sciences day by day.However, not all equations posed of these models are solvable.As a result, many new techniques have been successfully developed by diverse groups of mathematicians and physicists, such as the modified simple equation method [1][2][3][4], the extended tanh method [5,6], the Exp-function method [7][8][9][10][11], the Adomian decomposition method [12], the F-expansion method [13], the auxiliary equation method [14], the Jacobi elliptic function method [15], modified Exp-function method [16], the (  /)-expansion method [17][18][19][20][21][22][23][24][25][26], Weierstrass elliptic function method [27], the homotopy perturbation method [28][29][30], the homogeneous balance method [31,32], the Hirota's bilinear transformation method [33,34], the tanhfunction method [35,36] and so on.
The objective of this paper is to apply the MSE method to construct the exact and traveling wave solutions for nonlinear evolution equations in mathematical physics via coupled Konno-Oono equations and variant Boussinesq equations.
The paper is prepared as follows.In Section 2, the MSE method is discussed.In Section 3, we apply this method to the nonlinear evolution equations pointed out above, in Section 4, physical explanations, and in Section 5 conclusions are given.

The MSE Method
In this section, we describe the MSE method for finding traveling wave solutions of nonlinear evolution equations.Suppose that a nonlinear equation, say in two independent variables  and , is given by where () = (, ) is an unknown function, R is a polynomial of (, ) and its partial derivatives in which the highest order derivatives and nonlinear terms are involved.In the following, we give the main steps of this method [1][2][3][4].
Step 2. We suppose that (3) has the formal solution where   are constants to be determined, such that   ̸ = 0, and () is an unknown function to be determined later.
Step 3. The positive integer  can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (1) or (3).Moreover precisely, we define the degree of () as (()) =  which gives rise to the degree of other expression as follows: Therefore, we can find the value of  in (4), using (5).
Step 4. We substitute (4) into (3), and then we account the function ().As a result of this substitution, we get a polynomial of (  ()/()) and its derivatives.In this polynomial, we equate the coefficients of same power of  − () to zero, where  ≥ 0. This procedure yields a system of equations which can be solved to find   , () and   ().
Then the substitution of the values of   , () and   () into (4) completes the determination of exact solutions of (1).

The New Coupled Konno-Oono
Equations.Now we will bring to bear the MSE method to find exact solutions, and then the solitary wave solutions of coupled Konno-Oono equations in the form [37], Now let us suppose that the traveling wave transformation equation be Equation ( 7) reduces ( 6) into the following ODEs: By integrating (9) with respect to , we obtain where  is a constant of integration.Substituting ( 10) into (8), we get Balancing the highest order derivative   and nonlinear term  3 from ( 11), we obtain 3 =  + 2, which gives  = 1.Now for  = 1, using (4) we can write where  0 and  1 are constants to be determined such that  1 ̸ = 0, while () is an unknown function to be determined.It is trouble free to find that Now substituting the values of ,  3 ,   into (11) and then equating the coefficients of  0 ,  −1 ,  −2 ,  −3 to zero, we, respectively, obtain Solving ( 14), we get Solving (17), we get Solving ( 15) and ( 16) we get, Integrating (20) with respect to , we obtain where  = (6 2 0 + 2)/ 2 ,  =  2 /2 1  0 , and ,  are constants of integration.
Substituting the values of  and   into (12), we obtain the following exact solution: Case 1.When  0 = 0, (22) yields trivial solution.So this case is discarded.

Physical Explanation
In this section, we will put forth the physical explanation and the graphical representation of determined traveling wave solutions of nonlinear evolution equations through coupled Konno-Oono equations and the variant Boussinesq equations.

Explanations
(i) The equations ( 24) and ( 25) are complex soliton solutions.The shape of ( 24) is known as singular soliton, and the shape of ( 25) is known as kink soliton.Figures 1 and 2 represent the modulus shape of ( 24) and ( 25) with wave speed  = 1,  = 1 and wave speed  = 1,  = 2, respectively, within the interval −3 ≤ ,  ≤ 3. The disturbance of ( 24) and ( 25) is in the positive -direction for positive values of wave speed .If we take negative values of wave speed ,      then the disturbance of ( 24) and (25) will be in the negative -direction.
The disturbances represented in Figures 9-12 are in the positive -direction.

Graphical
Representation.Some of our obtained traveling wave solutions are represented in the following figures with the aid of commercial software Maple.

Conclusions
In this paper, the MSE method has been employed for analytic treatment of two nonlinear coupled partial differential equations.The MSE method requires wave transformation formulae.Via MSE method traveling wave solutions, kink solutions, bell-shaped solutions of coupled Konno-Oono equations, and the variant Boussinesq equations were derived.The procedure is simple, direct, and constructive.Without the help of a computer algebra system all examples in this paper show the efficiency of MSE method.