Abstract
We investigate the traveling solitary wave solutions of the generalized Camassa-Holm equation on the nonzero constant pedestal . Our procedure shows that the generalized Camassa-Holm equation with nonzero constant boundary has cusped and smooth soliton solutions. Mathematical analysis and numerical simulations are provided for these traveling soliton solutions of the generalized Camassa-Holm equation. Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.
1. Introduction
In 1993, Camassa and Holm [1] derived a nonlinear wave equation (Camassa-Holm equation) and obtained the peakon wave solution of the form . Whereafter, (1) has been researched by many authors [2–7]. Because (1) possesses rich dynamics and complex properties, recently, many authors are interested in its generalized forms. In particular, Liu and Qian [8] suggested a generalized Camassa-Holm equation, and obtained the explicit expressions of the peakon solution of (2). Afterwards, Tian and Song [9] gave some physical significance of this equation and obtained some peakon solutions with special wave speeds. Kalisch [10] studied the stability of solitary wave solution of (2). He et al. [11] constructed some exact traveling wave solutions by using the integral bifurcation method. Liu and Liang [12] studied the explicit nonlinear wave solutions and their bifurcations of (2).
When , (2) transforms into the following equation: For (3), there are some related works. Shen and Xu [13] discussed the existence of smooth and nonsmooth traveling waves. Khuri [14] obtained a singular wave solution composed of triangle functions. Wazwaz [15, 16] acquired eleven exact traveling wave solutions composed of triangle functions or hyperbolic functions. Liu and Ouyang [17] obtained a peakon solution composed of hyperbolic functions. Liu and Guo [18] investigated the periodic blow-up solutions and their limit forms. Wang and Tang [19] obtained two exact solutions. Yomba [20, 21] gave two methods, the sub-ODE method and the generalized auxiliary equation method, to obtain the exact solution of (3). Liu and Pan [22] studied the coexistence of multifarious solutions.
In this paper, we use the Qiao and Zhang method [23] to investigate the traveling solitary wave solutions of (3) on the nonzero constant pedestal
Since Qiao and Zhang presented this method, many authors applied it to different nonlinear models and obtained a variety of new type soliton solutions. Zhang and Qiao [24] discussed the traveling wave solutions for the Degasperis-Procesi equation on the nonzero constant pedestal and found new cusped and peaked soliton solutions. Qiao [25] proposed a new completely integrable wave equation: and obtained new cusped, one-peak, W/M-shape-peaks soliton solutions. Later, Chen et al. [26, 27] studied the osmosis equation under the inhomogeneous boundary condition and obtained smooth, peaked, cusped soliton solutions of the osmosis equation by using the phase portrait analytical technique. Wei et al. [28] investigated the generalized KP-MEW(2,2) equation on the nonzero constant pedestal and acquired smooth, peaked, cusped, and loop soliton solutions. More works on single peak soliton are reported [29–32].
2. Some Preliminary Results
Substituting and into (3), we have where “” is the derivative with respect to . Integrating (9) once, we yield where is an integration constant.
Further, we get where is an integration constant.
Let us solve (11) with the following boundary condition: where is a constant. Equation (11) can be cast into the following ordinary differential equation: When , then (13) reduces to where Obviously, .
Remark 1. In the existing research on this method, the cases on and have been discussed, but the case on has not been discussed. So we consider it is very meaningful researching this new case on this method, and we can obtain some new soliton solutions from this case.
Definition 2. A wave function is called smooth soliton solution, if is smooth and .
Definition 3. A wave function is called cuspon solution, if is smooth locally on either side of and (or −∞).
Without loss of generality, we assume .
3. The Parametric Conditions and Phase Portraits of Existence of Soliton Solutions of the Generalized Camassa-Holm Equation (3)
By virtue of the above analysis, we know that soliton solitons for the generalized Camassa-Holm Equation (3) must satisfy the following initial and boundary values problem:
Lemma 4. Suppose that one of the following five conditions holds:(i), ;(ii), ;(iii), ;(iv), ;(v), .
Then (3) has trivial solution .
Proof. (i) If and , then we have . When , (13) leads to . For , (13) can be cast into .
(ii) When and , then we have . If , (14) changes into . If , (14) transforms into .
(iii) For and , then we obtain . If , (14) leads to . If , (14) changes into .
(iv) If and , then we get and (14) can be cast into .
(v) When and , then we have and (14) transforms into .
The fact that implies and .
Obviously, we get that the generalized Camassa-Holm Equation (3) with nonzero boundary condition has soliton solutions when and do not belong to the above five cases. Then we obtain the generalized Camassa-Holm Equation (3) with nonzero boundary condition having soliton solutions, when , ; , ; , ; and , ; , .
For the cases on , , ; , , and , , , Liu and Qian [8] and Tian and Song [9] researched that the generalized Camassa-Holm Equation (3) has smooth soliton and peakon solutions as similar as follows: where , , and are constants, and is an integration constant.
In fact, when , ; , ; , ; and , ; , , the generalized Camassa-Holm Equation (3) has also other forms of the smooth soliton and cuspon. Because (11) is equivalent to the two-dimensional system From (18), we can obtain the phase portraits of existence of soliton solutions of the generalized Camassa-Holm Equation (3) under the inhomogeneous boundary condition, when and belong to the above five cases (see Figure 1).
The phase portraits of (3) are shown in Figure 1 under different parametric conditions.
(1-1) , ; (1-2) , ; (1-3) , , ; (1-4) , , ; (1-5) , ; (1-6) , ; (1-7) , ; (1-8) , ; (1-9) , , ; (1-10) , , .
4. Cusped and Smooth Solitons for the Generalized Camassa-Holm Equation (3)
In this section, by using the phase portrait analytical technique, which has been developed by Li and Dai [33], we get cusped and smooth soliton solutions of the generalized Camassa-Holm Equation (3) under the inhomogeneous boundary condition.
Case 1 (, ). By the standard phase portrait analysis (see Figure 1(1-1)), we have . From (13), we yield Taking the integration of both sides of (19), we can obtain the implicit cuspon solution defined by where is an integration constant,
Remark 5. is the elliptic integral of first kind, and is the elliptic integral of third kind [34].
The profile of cusped soliton solution is shown in Figure 2(2-1).
Case 2 (, ). Equation (14) can be cast into By the standard phase portrait analysis (see Figure 1(1-2)), we have . From (22), we get Let ; then , and Inserting into (24) and using the initial condition , we obtain Thus, which implies . Therefore, we have So we can get the implicit cuspon solution defined by where
Remark 6. The proof of other cuspons is similar to the above proof.
Because , the constant is defined by
The profile of cusped soliton solution is shown in Figure 2(2-2).
Case 3 (, ). In this case, we discuss two conditions: (1) ; (2) .
(1) When , by the standard phase portrait analysis (see Figure 1(1-3)), we have . From (14), we have
As same as the above, we can obtain the implicit smooth soliton solution defined by
where
For , the constant is defined by . For this smooth soliton solution, we get an exact explicit form [35]
The profile of smooth soliton solution is shown in Figure 2(2-3).
(2) When , by the standard phase portrait analysis (see Figure 1(1-4)), we have . Taking the integration of both sides of (31), we can yield the implicit cuspon solution defined by
where
By view of , the constant is defined by . The profile of cusped soliton solution is shown in Figure 2(2-4).
Case 4 (, ). (1) When , by the standard phase portrait analysis (see Figure 1(1-5)), we have . Equation (14) transforms into
From (37), we have
Taking the integration of both sides of (38), we can obtain the implicit smooth soliton solution defined by
where
For , we obtain . The profile of smooth soliton solution is shown in Figure 2(2-5).
(2) When , by virtue of (37), we have
In this case, we discuss two conditions: (i) ; (ii) .
(i) When , by the standard phase portrait analysis (see Figure 1(1-6)), we have . Taking the integration of (41) on the interval , thus, we obtain the implicit smooth soliton solution defined by
where
Because , we obtain . For this smooth soliton solution, we get an exact explicit form
The profile of smooth soliton solution is shown in Figure 2(2-6).
(ii) When , by the standard phase portrait analysis (see Figure 1(1-7)), we have . Integrating (41) on the interval , we obtain the implicit cuspon solution defined by
where
From , we obtain . The profile of cusped soliton solution is shown in Figure 2(2-7).
Case 5 (, ). (1) When , (14) can be cast into
Because , we have . By the standard phase portrait analysis (see Figure 1(1-8)), we get . By view of (47), we obtain
As same as the above, we can get the implicit smooth soliton solution defined by
For , the constant is defined by . For this smooth soliton solution, we can give an exact explicit form
The profile of smooth soliton solution is shown in Figure 2(2-8).
(2) When , we discuss three cases: (i) ; (ii) ; (iii) .
(i) When , by the standard phase portrait analysis (see Figure 1(1-9)), we have or .
For , from (14), we have Taking the integration of (51) on the interval , thus, we obtain the implicit smooth soliton solution defined by where The constant is defined by . The profile of smooth soliton solution is shown in Figure 2(2-9).
For , by view of (14), we obtain Taking the integration of both sides of (54), thus, we can yield the implicit solution defined by where For , is defined by . The profile of cusped soliton solution is shown in Figure 2(2-10).
(ii) When , by the standard phase portrait analysis (see Figure 1(1-10)), we have .
For , from (14), we get we yield where and is an integration constant. Taking the integration of on the interval , thus, we obtain the implicit solution defined by where The constant is defined by . Because , we know that the is strictly increasing on the interval ; has the inverse denoted by . The profile of smooth soliton solution is shown in Figure 2(2-11).
For , by view of (14), we obtain where and is an integration constant. Taking the integration of on the interval , thus, we obtain the implicit solution defined by where When , the constant is defined by . From , we know that the is strictly increasing on the interval ; has the inverse denoted by .
For this smooth soliton solution, we give an exact explicit form The profile of smooth soliton solution is shown in Figure 2(2-12).
The profile of soliton solution of (3) is shown in Figure 2 under special values of and .
(2-1) , ; (2-2) , ; (2-3) , ; (2-4) , ; (2-5) , ; (2-6) , ; (2-7) , ; (2-8) ; (2-9, 10) , ; (2-11, 12) , .
5. Conclusion
In this paper, we research the soliton solutions of the generalized Camassa-Holm Equation (3) under inhomogeneous boundary condition. The parametric conditions and phase portraits of existence of the cuspon and smooth soliton solutions are given. We obtain cuspon and smooth soliton solutions of the generalized Camassa-Holm Equation (3). Some exact explicit solutions are obtained. We show some graphs to explain our these solutions.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11361017 and 11161013), the Natural Science Foundation of Guangxi (2012GXNSFAA053003), and the Innovation Project of GUET Graduate Education (XJYC2012021 and XJYC2012022).