Exact and Explicit Solutions of Whitham-Broer-Kaup Equations in Shallow Water

In this paper, a simple direct method is presented to find equivalence transformation of a nonlinear WhithamBroer-Kaup equations. Applying this equivalence transformation, we can obtain the symmetry group theorem of the Whitham-Broer-Kaup equations and then derive series of new exact and explicit solutions of the Whitham-Broer-Kaup equations according to solutions of the previous references.


Introduction
Since the discovery of the soliton in 1965 by Zabusky and Kruskal, a large class of nonlinear evolution equations (NLEEs) have been derived and widely applied in various branches of physics and applied mathematics like condensed matter, nonlinear optics, fluid mechanics, plasma physics, theory of turbulence, ocean dynamics, biophysics and star formation. On the other hand, to better understand the nonlinear mechanisms in different physical contexts, many authors have diligently applied themselves to finding the exact analytical solutions for these NLEEs including the soliton solutions, periodic solutions and rational solutions.
In the present paper, we would like to consider the coupled Whitham-Broer-Kaup (WBK) equations which have been studied by Whitham [16], Broer [2] and Kaup [10]. The WBK equations are as follows, Kaup [10]  However, there are still many researchers are discussing the exact and explicit solutions for this WBK equation [4,5,7,9,14,19,21] by many new methods, and all kinds of solutions, such as traveling wave solutions, tanh-function solutions, elliptic function solutions, Jacobi elliptic function solutions, rational function solutions, exponential function solutions of the WBK equation are obtained in this references.
Naturally, an interesting questions is: Can we find a simple method to obtain a transformation between the present solutions and the new ones? If we find this transformation, we can obtain more and more all kinds of new solutions by the present references. As we know, one of the way to realize this aim is backland method of transformation and method of Lie-point symmetry group [13].
On the other hand, Clarkson and Kruskal [6] introduced a simple direct method to derive symmetry reductions of a nonlinear system without using any group theory. For many types of nonlinear systems, the method can be used to find all the possible similarity reductions. Recently, Lou and Ma [11,12] generalized a new simple direct method basing on CK's method, which is much simpler than the traditional Lie-point symmetry group.
Thus, in this paper, motivated by these ideas, the authors would like to obtain some new exact and explicit solutions by the simple direct method. And the organization of the paper is as follow. In the next section, we obtain the equivalence transformation of the WBK equation by the direct method. In section 3, we obtain series of new solutions of the WBK equation basing on the present references.

Symmetry Group Theorem of the WBK Equations
Following the idea of the direct method, we first suppose the WBK equations has the solutions in the following form Substituting Eq.(2) into Eq.(1) and using Eq.(3) one can get: where the function ( , , , , , , ) So it is necessary to take the coefficient of U ξξξξ and V ξξ in Eq.(4), U ξξξξξξ and V ξξξξ in Eq.(5) being zero. which leads to, 3 2 ( , ) 0 Then we can get 0 At this time, if we substitute Eq. (7) into Eq. (4) and (5), then these two equations are reduced to the following where the function 1 ( , , , , , ) In the same way, it is necessary to take the coefficients of , U UU ξξ ξ and 2 U in Eq.(8), U ξξξ and V ξξ in Eq.(9) being zero. That is, Thus, it is easy to obtain 2 '( ) Therefore, if we substitute Eq.(12) into Eq. (8) and (9), then these two equations can be reduced to the following [ Also, it is necessary to take all the coefficients of the Eq.(13) and Eq. (14),which leads to 0, '( ) 0, 0.
Thus, we can easily get the following symmetry group theorem.
is also a solutions of the WBK equations. According to the formula (18) and (19), we obtain the relationship between the new explicit solutions and the old ones of the WBK equations, which is the so-called equivalence transformation of the Eq.(1). Also it is obvious to see that (18) and (19) is an auto-backlund transformation of the WBK equations (1).
So we obtain following hyperbolic function solutions of the WBK equations:

Conclusions
From the theorem, it follows that the symmetry group is the product of the usual Lie point symmetry group (see [13]). Because if we take the constants in Eq. (18) where ( ) U σ and ( ) V σ is the symmetry of the WBK equations.
In fact, finishing above discussion, we can see that the equivalence transformation obtained by the direct method is more extensive and simpler than that obtained by the Lie-point group, and we can obtain even more new exact and explicit solutions if we take above solutions (20)-(21) as seed solutions.
Moreover, by applying this direct method, we can find many new solutions of other nonlinear evolution equations with variant coefficients, and this left for the future work.