Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations

Abstract In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions–hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters. PACS numbers 02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg


Introduction
The study of NLEEs, i.e., partial differential equations with time derivatives has a rich and long history, which has continued to attract attention in the last few decays. There are many examples throughout the world where NLEEs play an important role in controlling the natural systems. Because the majority of the phenomena in real world can be described by using NLEEs. NLEEs are frequently used to explain many problems of meteorology, population dynamics, fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, geochemistry, nanotechnology etc. By the aid of exact solutions, when they exist, the phenomena modeled by these NLEEs can be better understood. Therefore, the study of the traveling wave solutions for NLEEs plays an important role in the study of nonlinear physical phenomena.
The purpose of this paper is to apply the MSE method to construct the exact solutions for nonlinear evolution equations in mathematical physics via the DMBBM equation and cKG equation. The DMBBM equation and cKG equation are NLEEs representing the balance of dispersion and weak nonlinearity in physical systems that generate solitary waves.
The article is prepared as follows: The MSE method, Applications, Graphical representation of some obtained solutions, Comparisons, and conclusions.

The MSE method
Consider a general nonlinear evolution in the form ℜ u; u t ; u x ; u y ; u z ; u xx ; u tt ; u xz ; ::::: where ℜ is a polynomial of u(x, y, z, t) 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 (Jawad et al., 2010;Zayed, 2011, Zayed andIbrahim, 2012): Step 1. Using the traveling wave transformation where ℘ is a polynomial in u(ξ) and its derivatives, while u′ ξ ð Þ ¼ d dξ , u″ ξ ð Þ ¼ d 2 dξ 2 ;and so on.
Step 2. We suppose that Eq.(2.3) has the formal solution where β k are arbitrary constants to be determined, such that β N ≠ 0, and Φ(ξ) is an unknown function to be determined later.
Step 3. We determine the positive integer N in Eq. (2.4) by considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Eq. (2.3).
Step 4. We substitute (2.4) into (2.3), we calculate all the necessary derivatives u′ , u″ , ⋯ and then we account the function Φ(ξ). As a result of this substitution, we get a polynomial of Φ′(ξ)/Φ(ξ) and its derivatives. In this polynomial, we equate with zero all the coefficients of Φ − i (ξ), where i = 0, 1, 2, ⋯. This operation yields a system of equations which can be solved to find β k and Φ(ξ). Consequently, we can get the exact solutions of Eq. (2.1).

Applications
The (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona Mahony equation: In this section, we will apply the modified simple equation method to find the exact solutions and then the solitary wave solutions of (1 + 1)dimensional nonlinear DMBBM equation, where α is a nonzero constant. This equation was first derived to describe an approximation for surface long waves in nonlinear dispersive media. It can also characterize the hydro magnetic waves in cold plasma, acoustic waves in inharmonic crystals and acoustic gravity waves in compressible fluids (Yusufoglu 2008;Zayed and Al-Joudi 2010). The traveling wave transformation is Using traveling wave Eq. (3.2), Eq. (3.1) transforms into the following ODE Integrating with respect to ξ choosing constant of integration as zero, we obtain the following ODE Now balancing the highest order derivative u″ and non-linear term u 3 , we get 3N = N + 2, which gives N = 1 where β 0 and β 1 are constants to be determined such that β 1 ≠ 0, while ψ(ξ) is an unknown function to be determined. It is easy to see that Now substituting the values of u″, u, u 3 in equation (3.3) and then equating the coefficients of Φ 0 , Φ − 1 , Φ − 2 , Φ − 3 to zero, we respectively obtain and β 1 ≠ 0 Case I: when β 0 = 0 solving Eqs. (3.9), and (3.10) we get trivial solution. So this case is rejected.
Substituting the values of Φ and Φ′ into Eq. (3.5), we obtain the following exact solution, Putting the values of β 0 , β 1 , l and simplifying, we obtain

The coupled Klein-Gordon equation
Now we will bring to bear the MSE method to find exact solutions and then the solitary wave solutions to the cKG Equation in the form, The traveling wave Eq. (3.23) reduces Eqs. (3.22) into the following ODEs Balancing the highest order derivative u″ and nonlinear term u 3 from Eq. (3.27), we obtain N = 1 Now for N = 1, Eq. (2.4) becomes where β 0 and β 1 are constants to be determined such that β 1 ≠ 0, while Φ(ξ) is an unknown function to be determined. It is easy to see that

Graphical representation of some obtained solutions
In this section, we put forth to illustrate the threedimensional and two-dimensional structure of the determined solutions of the studied NLEEs, to visualize the inner mechanism of them. Figure 1 and Figure    solutions of NLEEs. On the other hand it is worth mentioning that the exact solutions of the studied NLEEs have been achieved in this article without using any symbolic computations software, since the computations are very simple and easy. Similarly for any nonlinear evolution equation it can be shown that the MSE method is much easier than other methods. Furthermore, auxiliary equations are unnecessary to solve NLEEs by means of MSE methods, i.e., there exists no predefined functions or equations in MSE method.

Conclusions
This study shows that the MSE method is quite efficient and practically well suited for use to find exact traveling wave solutions of the DMBBM equation and cKG equation. We have obtained exact solutions of these equations in terms of the hyperbolic and trigonometric functions. This study also shows that the procedure is simple, direct and constructive. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability. We conclude that the studied method can be used for many other NLEEs in mathematical physics and engineering fields.