Abstract
We use bifurcation method of dynamical systems to study exact traveling wave solutions of a nonlinear evolution equation. We obtain exact explicit expressions of bell-shaped solitary wave solutions involving more free parameters, and some existing results are corrected and improved. Also, we get some new exact periodic wave solutions in parameter forms of the Jacobian elliptic function. Further, we find that the bell-shaped waves are limits of the periodic waves in some sense. The results imply that we can deduce bell-shaped waves from periodic waves for some nonlinear evolution equations.
1. Introduction
The Benjamin-Bona-Mahony (BBM) equation [1] was proposed as the model for propagation of long waves where nonlinear dispersion is incorporated. The Kadomtsev-Petviashvili (KP) equation [2] was given as the generalization of the KdV equation. In addition, both BBM and KdV equations can be used to describe long wavelength in liquids, fluids, and so forth. Combining the two equations, the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation was presented in [3] for further study. Some methods are developed and applied to find exact solutions of nonlinear evolution equations because exact solutions play an important role in the comprehension of nonlinear phenomena. For instance, extended tanh method, extended mapping method with symbol computation, and bifurcation method of dynamical systems are employed to study (3) [4–6], and some solitary wave solutions and triangle periodic wave solutions were obtained.
However, there is no method which can be applied to all nonlinear evolution equations. The research on the solutions of the KP-BBM equation now appears insufficient. Further studies are necessary for the traveling wave solutions of the KP-BBM equation. The purpose of this paper is to apply the bifurcation method [7–10] of dynamical systems to continue to seek traveling waves of (3). Firstly, we obtain bell-shaped solitary wave solutions involving more free parameters, and some results in [6] are corrected and improved. Then, we get some new periodic wave solutions in parameter forms of Jacobian elliptic function, and numerical simulation verifies the validity of these periodic solutions. The periodic wave solutions obtained in this paper are different from those in [5]. Furthermore, we find an interesting relationship between the bell-shaped waves and periodic waves; that is, the bell-shaped waves are limits of the periodic waves in forms of Jacobian elliptic function as modulus approaches 1.
This paper is organized as follows. First, we draw the bifurcation phase portraits of planar system according to the KP-BBM equation in Section 2. Second, bell-shaped solitary wave solutions to the equation under consideration are presented in Section 3. Third, periodic solitary waves are given in the forms of Jacobian elliptic function and numerical simulation is done. Finally, the relationship between the bell-shaped solitary waves and periodic waves is proved in Section 4.
2. Bifurcation Phase Portraits of System (6)
Suppose (3) possesses traveling wave solutions in the form , , where is the wave speed and is a real constant. Substituting , into (3) admits to the following ODE: where the derivative is for variable . Integrating (4) twice with respect to and letting the first integral constant take value zero, it follows that where is the second integral constant.
Equation (5) is equivalent to the following two-dimensional system: It is obvious that system (6) has the first integral where is the constant of integration.
Define . When , there are two equilibrium points and of (6) on -axis, where , . The Hamiltonian of and is denoted by and .
In the case of and , the bifurcation phase portraits of system (6) see Figures 1 and 2 in [6], respectively, in which there are some homoclinic and periodic orbits of system (6). For our purpose, we redraw the homolinic and periodic orbits in this paper(see Figures 1–3).
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3. Exact Explicit Expressions of Solitary Wave Solutions
In this section, we discuss bell-shaped wave solutions under and , respectively.
3.1. The Case
System (6) can be rewritten as The first integral of (8) is
When , there are two homoclinic orbits and (see Figures 1(a) and 1(b)). In plane, and can be described by where . That is, Substituting (11) into and integrating along homoclinc orbits and , respectively, we get Completing the above integration, it follows that
Remark 1. is a bright soliton solution when and a dark soliton solution when . If the real number in (13) takes value 1, then solution (13) is the same to solution (1.2) in [6]. Solution (1.1) in [6] is not a real solution of the KP-BBM equation; it is obvious that solution (1.1) tends to infinite as , and it does not satisfy the KP-BBM equation (3).
When , there are two homoclinic orbits and (see Figures 2(a) and 2(b)). Similarly solitary wave solutions according to and are obtained as follows:
Remark 2. If the real number in (14) takes value 1, then solution (14) is the same to solution (1.4) in [6]. Solution (1.5) in [6] is not a real solution of the KP-BBM equation; it is easy to verify that it does not satisfy the KP-BBM equation (3).
3.2. The Case
There are two homoclinic orbits and when (see Figures 3(a) and 3(b)). and can be described by
When , the corresponding homoclinic orbit has a double zero point and a zero point on -axis (see Figure 3(a)), so (15) can be rewritten as that is, Substituting (17) into and integrating along homoclinic orbits , we get where and if and ; then, completing (18) we get the following solution: In (18), and if and ; then, completing (18) we get the following solution:
Remark 3. If the real number in (19) and (20) takes value 1, then solutions (19) and (20) are the same to solutions (1.6) and (1.8) in [6]. Solutions (1.7) and (1.9) in [6] are not real solutions of the KP-BBM equation, and it is easy to verify that they do not satisfy the KP-BBM equation (3).
When , the corresponding homoclinic orbit has a double zero point and a zero point on -axis (see Figure 3(b)), so (15) can be rewritten as that is, Substituting (22) into and integrating along homoclinic orbits , we get where and if and ; then, completing (23) we get the solution . In (23), and if and ; then, completing (23) we get the solution .
4. Periodic Wave Solutions
So as to explain our work conveniently, in this section the Jacobian elliptic function with modulus will be expressed by . We discuss the periodic wave solutions under conditions and , respectively.
4.1. The Case
When , system (6) has periodic orbits and (see Figures 1(a) and 1(b)). Their expressions are (9) on plane, where (or ). Let then, we have the following results.
Claim 1. In the case of , and (or ); then, the function must have three different real zero points.
Proof. Since is a cubic polynomial about , we can use the Shengjin Theorem [11] to distinguish its solutions. We only discuss the case , and the case is the same. Under the above conditions, So . In the coefficients , , By Shengjing Theorem [11], it follows that the function has three different real zero points. Let , , and ; then, .
Let be three different real zero points of . Then Claim 1 means that (9) has three intersection points , , and on -axis. Therefore, (9) can be rewritten as where when and when .
When , the orbit is according to a periodic solution of (6) and its expression is given by Substituting (27) into and integrating along orbit , we get By formula (236) in [12], we have where , , and . Solving (29), we get That is, where the modulus of is .
Similarly, when , the expression is Substituting (32) into and integrating along orbit , we have The according periodic solution of can be obtained as where the modulus of is .
When , system (6) has periodic orbits and (see Figure 2). Their expressions are (9), where (or ). Similarly, we can get the according periodic solutions of and as and .
To verify validity of the periodic wave solutions, we take , , and to make the conditions in Claim 1 satisfied. By simple calculation, we get that , , and . The specific periodic wave solution is where the modulus of sn is .
4.2. The Case
System (6) has periodic orbits and (see Figure 3). Their expressions are (7) on the plane, where (or ). Let then, we have the following results about .
Claim 2. If and (or ), then the function must have three different real zero points.
Proof. We only prove the case , and the case is the same. Under the above conditions, So . For , we have , , , and . Again, , which is monotonous in the intervals , , and . By zero point theorem of continuous function, there must be one real zero point of that lies in each of the three intervals, proving the claim.
Let be three different real zero points of . Then Claim 2 means that (7) has three intersection points on -axis denoted by , , and . Then (7) can be rewritten as where .
When , the expression of periodic orbit is
When , the expression of periodic orbit is
Substituting (39) and (40) into and integrating along orbits and , respectively, it is the same to the proceeding for solving and and we can get the according periodic solutions of and as follows: where the moduli for are and in (41).
For example, we take , , , and such that the conditions in Claim 2 are satisfied. By simple calculation, we get that , , and . The according periodic wave solution is where the modulus of sn is .
5. Relationship between Bell-Shaped Waves and Periodic Waves
In Sections 3 and 4, the bell-shaped solitary wave and periodic wave solutions are obtained. Via further study, we find that there exists an interesting relationship between these two kinds of solutions; that is, the bell-shaped solutions are limits of the periodic waves in some sense. The results are detailed as follows.
Proposition 4. Let be solutions of (3), let , , , , , and be parameters in (5), and let be modulus of the Jacobian elliptic function sn; then, one has the following.
Case 1. When and , for modulus , the periodic waves and degenerate bell-shaped wave .
Case 2. When and , for modulus , the periodic waves and degenerate bell-shaped wave .
Case 3. When and , for modulus , the periodic wave degenerates bell-shaped wave .
Case 4. When and , for modulus , the periodic wave degenerates bell-shaped wave .
Here, we only prove Cases 1 and 3 for simplicity. The remaining cases are the same. In the following proofs, we use the property of elliptic function that when the modulus [5, 13].
Proof of Case 1. When , it means and ; then, we calculate
Substituting into admits to as follows:
When , it means ; then, we calculate and , and substituting into we get .
Proof of Case 3. When , it means and ; then, we calculate and , and substituting into admits to as follows:
The results provide a manner that we can get bell-shaped waves from periodic waves for some nonlinear development equations.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11201070) and Guangdong Province (no. 2013KJCX0189 and no. Yq2013161). The author would like to thank the editors for their hard working and the anonymous reviewers for helpful comments and suggestions.