Abstract
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.
1. Introduction
In 1975, Broer [1] obtained a dispersive equation as follows: which describes the evolution of horizontal velocity component of water waves of height propagating in both directions in an infinite narrow channel of finite constant depth. Equation (1) plays an important role in nonlinear physics and gains considerate attention [2–4]. The traveling wave solutions for (1) have been studied by many works, such as [5–10].
In this paper, we employ the bifurcation method and qualitative theory of dynamical systems [11–21] to investigate the nonlinear wave solutions for (1), and we obtain many exact explicit expressions of nonlinear wave solutions for (1). These nonlinear wave solutions contain solitary wave solutions, singular solutions, periodic singular solutions, and kink-shaped solutions, most of which, to our knowledge, are newly obtained.
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.
2. 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
By setting , (5) becomes
Letting , we obtain a planar system with first integral
Now, we study the bifurcation of phase portraits of system (7). Set
Obviously, has three zero points, which can be expressed as where , when .
Additionally, it is easy to obtain the two extreme points of as follows:
Let which is the absolute value of extreme values of .
Let be one of the singular points of system (7). Then the characteristic values of the linearized system of system (7) at the singular point are
From the qualitative theory of dynamical systems, we therefore know that(i)if , then is a saddle point;(ii)if , then is a center point;(iii)if , then is a degenerate saddle point.
Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (7) in Figure 1.
3. 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, , in the following theorem.
Theorem 1. For given constants and , 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 , one gets two solitary wave solutions two singular solutions and two periodic singular solutions where , , and will be given in the proof of the theorem.(3)When , one obtains four singular solutions as follows:
Proof. (1) When , we consider the following two kinds of orbits.
(i) First, we see that there are two heteroclinic orbits connected as two saddle points and from Figure 1. In -plane, from (8), the expressions of the heteroclinic orbits are given as
Substituting (21) into the first equation of system (7) and integrating along the heterclinic orbits, it follows that
From (22), we have
Noting that and , we get two kink-shaped solutions and two singular solutions as (14) and (15).
(ii) Second, from the phase portrait in Figure 1, we note that there are two special orbits , which have the same Hamiltonian as that of the center point . 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 , 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 ; we obtain another two solutions
Noting that and , we get four periodic singular solutions and as (16).
(2) When , we set the largest solution of to be , and then we can obtain another two solutions of as follows:
(i) First, we see that there is a homoclinic orbit , which passes the saddle point . In -plane, from (8), the expressions of the homoclinic orbit are given as where
Substituting (29) into the first equation of system (7) and integrating along the homoclinic orbit, it follows that
From (31), we have
Noting that and , we get two solitary solutions and two singular solutions as (17) and (18).
(ii) Second, from the phase portrait in Figure 1, we note that there are another two special orbits , which have the same Hamiltonian as that of the center point . 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 , it follows that From (35), we have
Noting that and , we get two periodic singular solutions as (19).
To illustrate, we give the graphic of in Figure 2 by taking , , and .
(3) When , from the phase portrait in Figure 1, we note that there are two orbits , which have the same Hamiltonian as the degenerate saddle point . 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 , it follows that
From (39), we have
Noting that and , we get four singular solutions and 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 ,
4. 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This research was supported by Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (no. ZQN-PY119) and the Foundation of Huaqiao University (no. 12BS223).