New Exact Explicit Nonlinear Wave Solutions for the Broer-Kaup Equation

We study the nonlinear wave solutions for the Broer-Kaup equation. Many exact explicit expressions of the nonlinear wave solutions for the equation are obtained by exploiting the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which are new. Some previous results are extended.

The remainder of this paper is organized as follows.In Section 2, we show the bifurcation of phase portraits corresponding to (1).We state our main results and the theoretical derivation for the main results in Section 3. A short conclusion will be given in Section 4.

Bifurcation of Phase Portraits
In this section, we give the process of obtaining the bifurcation of phase portraits corresponding to (1).
For given constant , substituting (, ) = (), (, ) = () with  =  −  into (1), it follows Integrating (2) once leads to where both  and  are integral constants, respectively.From the first equation of system (3), we obtain Substituting (4) into the second equation of system (3) leads to Journal of Applied Mathematics Figure 1: The phase portraits of system (7).
By setting  =  + , (5) becomes Letting  =   , we obtain a planar system d d = , with first integral Now, we study the bifurcation of phase portraits of system (7).Set Obviously,  0 () has three zero points, which can be expressed as where Additionally, it is easy to obtain the two extreme points of () as follows: Let which is the absolute value of extreme values of  0 ().
Let (  , 0) be one of the singular points of system (7).Then the characteristic values of the linearized system of system (7) at the singular point (  , 0) are From the qualitative theory of dynamical systems, we therefore know that Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (7) in Figure 1.

Main Results and the Theoretic Derivations of the Main Results
In this section, we state our results about solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions for the first component of system (7).
To relate conveniently, we omit  =  +  and the expression of the second component of system (7), that is,  =  +  − (1/2) 2 , in the following theorem.
Theorem 1.For given constants  and  (1 <  < 3), which will be given later, the Broer-Kaup equation (1) has the following exact explicit nonlinear wave solutions.
(1) When  = , one obtains two kink-shaped solutions two singular solutions and four periodic singular solutions (2) When  <  <  0 + , one gets two solitary wave solutions two singular solutions and two periodic singular solutions where  2 ,  6 , and  7 will be given in the proof of the theorem.
(1) When  = , we consider the following two kinds of orbits.
(ii) Second, from the phase portrait in Figure 1, we note that there are two special orbits Γ ± 2 , which have the same Hamiltonian as that of the center point (0, 0).In (, )-plane, from (8), the expressions of these two orbits are given as Substituting (24) into the first equation of system (7) and integrating along the two orbits Γ ± 2 , it follows that From (25), we have At the same time, we note that if  = () is a solution of system, then  = (+) is also a solution of system.Specially, we take  = /2; we obtain another two solutions Noting that  = () and  =  − , we get four periodic singular solutions  3 ± (, ) and  4 ± (, ) as ( 16).
(ii) Second, from the phase portrait in Figure 1, we note that there are another two special orbits Γ ± 4 , which have the same Hamiltonian as that of the center point ( 2 , 0).In (, )plane, from (8), the expressions of these two orbits are given as where Substituting (33) into the first equation of system (7) and integrating along these two special orbits Γ ± 4 , it follows that From (35), we have Noting that  = () and  =  − , we get two periodic singular solutions  7 ± (, ) as (19).
(3) When  =  0 + , from the phase portrait in Figure 1, we note that there are two orbits Γ ± 5 , which have the same Hamiltonian as the degenerate saddle point ( 8 , 0).In (, )plane, from (8), the expressions of these two orbits are given as where Substituting (37) into the first equation of system (7) and integrating along these two orbits Γ ± 5 , it follows that Noting that  = () and  =  − , we get four singular solutions  8 ± (, ) and  9 ± (, ) as (20).
Remark 2. One may find that we only consider some special orbits in Figure 1 when  ≥ .In fact, we may obtain exactly the same results when  < .Remark 3. We employ the software Mathematica to check the correctness of the above nonlinear wave solutions.To illustrate, we show the commands of verifying  (41)

Conclusions
In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the nonlinear wave solutions for the Broer-Kaup equation ( 1) and obtain exact explicit expressions of the various kinds of nonlinear wave solutions, which include solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions.To the best of our knowledge, most of the nonlinear wave solutions are newly obtained.