Exact traveling wave solutions for system of nonlinear evolution equations

In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

The aim of this work is to demonstrate the efficiency of the generalized Kudryashov method for finding exact traveling wave solutions transmutable to the solitary wave solutions for system of nonlinear evolution equations. For this purpose, we consider the one dimensional variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations.

Algorithm of the generalized Kudryashov method
Let us consider the nonlinear evolution equation in two independent variables x and t: where u = u(x, t) is an unknown function, x is the spatial variable and t is the time variable, P is a polynomial in u and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved.
The main steps of generalized Kudryashov method are as follows (Demiray et al. 2014a, b;Baskonus and Bulut 2015): Step 1: The traveling wave variable ξ = x − ω t transforms Eq. (1) into an ordinary differential equation of the form: where the prime indicates differentiation with respect to ξ, and ω ∊ R\{0} is the velocity of the relative wave mode.
Step 2: Suppose that the solution of Eq. (2) has the following form: where a i (i = 0, 1, 2, . . . , N ) and b j (j = 0, 1, 2, . . . , M) are constants to be determined afterward such that a N � = 0 and b M � = 0, and Q = Q(ξ ) satisfies the following ordinary differential equation: The solution of Eq. (4) is as follows: where A is a constant of integration. (1) Step 3: The positive integers N and M appearing in Eq.
(3) can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms come out in Eq. (1) or Eq. (2). Moreover precisely, we define the degree of u(ξ ) as D(u(ξ )) = N − M which gives rise to the degree of other expression as follows: where p, q, s are integer numbers. Therefore, we can find the value of N and M in Eq. (3).
(2), we obtain a polynomial in Q i−j , (i, j = 0, 1, 2, . . .). In this polynomial equating the coefficients of all terms of the same powers of Q to zero, we obtain a system of algebraic equations which can be solved by using Maple or Mathematica to get the unknown parameters a i (i = 0, 1, 2, . . . , N ), b j (j = 0, 1, 2, . . . , M), and ω. Consequently, we obtain the exact solutions of Eq. (1).

Applications
In this section, we will apply the generalized Kudryashov method to construct the exact traveling wave solutions transmutable to the solitary wave solutions for the following two nonlinear evolution equations: The traveling wave transformation is defined by, Using traveling wave Eqs. (7), (6) transform into the following ordinary differential equations: Integrating Eqs. (8) and (9) with respect to ξ, choosing the constant of integration as zero (under the boundary conditions described in "Algorithm of the generalized Kudryashov method" section (Step 1) and using similar boundary conditions for H(ξ)), we obtain the following ordinary differential equations respectively: From Eq. (10), we get Substituting Eq. (12) into Eq. (11), we obtain Now balancing the highest order derivative u ′′ and nonlinear term u 3 , we get 3N − 3M = N − M + 2 or equivalent to N = M + 1.
Setting M = 1, we obtain N = 2. Therefore, Eq. (3) reduces to Substituting Eq. (14) along with Eq. (4) into Eq. (13), we get a polynomial of Q k , (k = 0, 1, 2, . . .). Equating the coefficients of this polynomial of the same powers of Q to zero, we obtain a system of algebraic equations. This system of equations yields the values for ω, a 0 , a 1 , a 2 , b 0 and b 1 .

Set 1 corresponds to the following solutions for the variant Boussinesq equations:
Set 2 corresponds to the following solutions for the variant Boussinesq equations: Set 3 corresponds to the following solutions for the variant Boussinesq equations: Set 4 corresponds to the following solutions for the variant Boussinesq equations: Set 5 corresponds to the following solutions for the variant Boussinesq equations: Set 6 corresponds to the following solutions for the variant Boussinesq equations: Remark The bottom depth H(x,t) must be a non-negative and real physical quantity. Solutions (17)-(22) of the variant Boussinesq equations are significant both mathematically and physically for their positive sign for H (x, t). Besides solutions (15) and (23) are valid mathematically and physically for their positive and negative signs for u(x, t) but their corresponding solutions (16) and (24) are valid only mathematically. Solutions (25) and (26) are complex solutions, therefore although they are logically true but they have no physical significance (Figs. 1, 2).
We can obtain some traveling wave solutions since A is an arbitrary constant of integration, for example.
If we put A = 1 into Eqs. (19) and (20) and considering u(x, t) > 0 as well as a wave moving to the right, i.e., in the positive direction of x-axis, we obtain Example 2: The (2 + 1)-dimensional breaking soliton equations: Now, we will investigate explicit exact traveling wave solutions of the following (2 + 1)-dimensional breaking soliton equations  where α is a nonzero constant. Equations (29) and (30) describe the (2 + 1)-dimensional interaction of a Riemann wave propagation along the y-axis with a long wave propagated along the x-axis.
Applying the traveling wave variable ξ = x + y − ω t and proceeding as before, we obtain Integrating Eq. (32), we obtain choosing constant of integration as zero under the boundary conditions elucidated in "Algorithm of the generalized Kudryashov method" section (Step 1) and similar boundary conditions for v(ξ).
Substituting Eq. (33) into Eq. (31) and integrating, we get choosing constant of integration to zero under the boundary conditions mentioned in "Algorithm of the generalized Kudryashov method" section (Step 1). Considering the homogeneous balance between u ′′ and u 2 in Eq. (34), we obtain N = M + 2.
Setting M = 1, we obtain N = 3. Therefore, Eq. (3) takes the form Substituting Eq. (35) along with Eq. (4) into Eq. (34), we get a polynomial of Q k , (k = 0, 1, 2, . . .). Equating the coefficients of the polynomial of the same powers of Q to zero, we obtain a system of algebraic equations. This system of equations yields the values for ω, a 0 , a 1 , a 2 , b 0 and b 1 .
(29) u t + α u xxy + 4α(uv) x = 0, The remaining solutions of Khan and Akbar (2013) given in "Appendix 1" are obtained changing ξ by −ξ in our Eq. (19). Note that all the solutions obtained here are also valid when one replaces the traveling wave variable ξ by −ξ.
2.  investigated exact traveling wave solutions to the (2 + 1)-dimensional breaking soliton equation by means of the functional variable method and found only one solution (see "Appendix 2"). On the other hand, by using the generalized Kudryashov method we found four solutions from which one of our solutions coincides with the solution of Zayed et al. If we set c = α into the solution (46) (see "Appendix 2") obtained by , then our solution (41) coincides with that solution.
From the above discussion, we conclude that the generalized Kudryashov method is a more reliable technique, in principle, than the modified simple equation method and the functional variable method.

Conclusions
In this article, we have successfully presented a mathematical tool named the generalized Kudryashov method for finding exact traveling wave solutions to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. The obtained results will serve as a very important milestone in the study of plasma physics and water waves phenomena. We also have demonstrated that the generalized Kudryashov method is an effective tool for obtaining exact analytical solutions for large classes of system of nonlinear evolution equations.