Abstract
We investigate the -dimensional Broer-Kaup-Kupershmidt equations. Some explicit expressions of solutions for the equations are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions, and solitary wave solutions. Some previous results are extended.
1. Introduction
Consider the -dimensional Broer-Kaup-Kupershmidt (BKK) equations [1–10] These equations have been widely applied in many branches of physics like plasma physics, fluid dynamics, nonlinear optics, and so forth. So a good understanding of more solutions of the -dimensional BKK equations (1.1) might be very helpful, especially for coastal and civil engineers to apply the non-linear water models in a harbor and coastal design.
Recently, the -dimensional BKK equations have been studied by many authors. Yomba [1, 2] used the modified extended Fan subequation method to obtain soliton-like solutions, triangular-like solutions, and single and combined nondegenerate Jacobi elliptic wave function-like solutions of (1.1). Zhang and Xia [3] used the further improved extended Fan sub-equation method to obtain soliton-like solutions, triangular-like solutions, single and combined non-degenerate Jacobi elliptic wave function-like solutions, and Weierstrass elliptic doubly-like periodic solutions of (1.1). Abdou and Soliman [4] obtained some traveling wave solutions of (1.1) by using the modified extended tanh-function method. Song et al. [5] used the new extended Riccati equation rational expansion method to study multiple exact solutions of (1.1). Zhang [6] used the Exp-function method to seek generalized exact solutions with three arbitrary functions of (1.1). El-Wakil and Abdou [7] obtained exact travelling wave solutions by using improved tanh-function method. Lu et al. [8] obtained some exact traveling wave solutions of (1.1) by using the first integral method. Davodi et al. [9] obtained some generalized solitary solutions of (1.1). Bai and Zhao [10] used the Repeated General Algebraic Method to obtain exact solutions of (1.1).
In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11–19] to investigate the -dimensional BKK equations (1.1), and we obtain some explicit expressions of solutions for (1.1). These solutions contain kink-shaped solutions, blow-up solutions, periodic blow-up solutions, and solitary wave solutions, most of which are new by comparing with the solutions of the references [1–10].
The remainder of this paper is organized as follows. In Section 2, we present our main results. Section 3 gives the theoretical derivation for our main results. A short conclusion will be given in Section 4.
2. Main Results
In this section, we state our main results. For ease of exposition, we have omitted the expressions of with , , in the entire process.
Proposition 2.1. For given constants and g0, which will be given later in (3.10), the -dimensional BKK equations have the following exact solutions.
(1) If , we get two kink solutions:
where and are constants, two blow-up solutions
and four periodic blow-up solutions
(2) If , we get a solitary wave solution
and two blow-up solutions
where and .
(3) If , we get three blow-up solutions as follows:
3. The Derivations of Main Results
In this section, we will give the derivations for our main results.
For given constant wave speed , substituting , with into the -dimensional BKK equations (1.1), it follows that Integrating the first equation of (3.1) twice and letting integral constants be zero, we have
Integrating the second equation of (3.1) once, we have where is integral constant.
Substituting (3.2) into (3.3), we get
Letting , we get the following planar system:
Obviously, the above system (3.5) is a Hamiltonian system with Hamiltonian function
Now, we consider the phase portraits of system (3.5). Set has three fixed points , , , and their expressions are given as follows:
It is easy to obtain the two extreme points of as follows:
Let then it is easily seen that is the extreme values of .
Let be one of the singular points of system (3.5). Then the characteristic values of the linearized system of system (3.5) at the singular points are
From the qualitative theory of dynamical systems, we therefore know that,(i)if , is a saddle point;(ii)if , is a center point;(iii)if , is a degenerate saddle point.
Therefore, we obtain the phase portraits of system (3.5) in Figure 1.
Now, we will obtain the explicit expressions of solutions for the -dimensional BKK equations (1.1).
(1) If , we will consider two kinds of orbits.
(i) First, we see that there are two heteroclinic orbits and connected at saddle points and . In -plane, the expressions of the heteroclinic orbits are given as
Substituting (3.12) into and integrating them along the heteroclinic orbits and , it follows that where is constant and
From (3.13), we have where .
From (3.14), we have Noting that and , we get two kink-shaped solutions , and two blow-up solutions as (2.1), and (2.2). (ii) From the phase portrait, we note that there are two special orbits and , which have the same Hamiltonian with that of the center point . In -plane, the expressions of these two orbits are given as where Substituting (3.17) into and integrating them along the two orbits and , it follows that
From (3.19), we have
At the same time, we note that if is a solution of system (3.5), then is also a solution of system (3.5). Specially, when we take , we get other two solutions
Noting that and , we get four periodic blow-up solutions and as (2.3).
(2) If , we set the largest solution of as , then we can get another two solutions of as follows: We see that there is a homoclinic orbit , which passes the saddle point . In -plane, the expressions of the homoclinic orbit are given as where
Substituting (3.23) into and integrating them along the homoclinic orbit, it follows that
From (3.25), we have
where and .
Noting that and , we get a solitary wave solution and two blow-up solutions as (2.4) and (2.5).
(3) If , from the phase portrait, we see that there are two orbits and , which have the same Hamiltonian with the degenerate saddle point . In -plane, the expressions of these two orbits are given as where
Substituting (3.27) into and integrating them along these two orbits and , it follows that From (3.29), we have Noting that and , we get three blow-up solutions , , and as (2.6). Thus, we obtain the results given in Proposition 2.1.
Remark 3.1. One may find that we only consider the case when in Proposition 2.1. In fact, we may get exactly the same solutions in the opposite case.
4. Conclusion
In this paper, we have obtained many new solutions for the -dimensional BKK equations (1.1) by employing the bifurcation method and qualitative theory of dynamical systems. The explicit expressions of the solutions have been given in Proposition 2.1. The method can be applied to many other nonlinear evolution equations, and we believe that many new results wait for further discovery by this method.
Acknowledgment
Research is supported by the Natural Science Foundation of Yunnan Province (no. 2010ZC154).