Abstract
This paper focuses on a generalized two-component Hunter-Saxton system. From a dynamic point of view, the existence of different kinds of periodic wave, solitary wave, and blow-up wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, some exact parametric representations of the travelling waves are presented.
1. Introduction
The Hunter-Saxton equation, was first introduced by Hunter and Saxton [1] as a model of the dynamics of a nematic liquid crystal. Geometrically, the HS equation (1) describes geodesic flow associated with the right-invariant metrics on a homogeneous space [2]. It is a particular case of the Euler-Poincaré equation on the diffeomorphisms in one spatial dimension [3]. Its integrability was proved by Hunter and Zheng [4].
The two-component Hunter-Saxton system is as follows: which is a generalization of the Hunter-Saxton equation, and was proposed by Wunsch [5] in a periodic setting. It is a particular case of the Gurevich-Zybin system [6] pertaining to nonlinear one-dimensional dynamics of dark matter as well as nonlinear ion-acoustic waves [7]. The two-component system in a periodic setting has received some attention lately [8, 9].
Very recently, the generalized two-component Hunter-Saxton system was studied [10–12]. Moon and Liu [10] studied the wave-breaking phenomena and global existence for (3). Moon [11] determined the existence of solitary wave solutions for and classified the solitary waves for . The existence of peaked solitary waves for was shown. The wave-breaking criterion and local well-posedness were studied in [12].
In this paper, we investigated the following generalized two-component Hunter-Saxton system: Obviously, system (4) becomes system (2) when . We will prove the existence of different kinds of travelling wave of (4) and give some new exact travelling solutions using the approach of dynamical system [13–17].
Using the following independent variable transformation: where is the wave speed, and substituting (5) into (4), we obtain where “” is the derivative with respect to .
Integrating equations of (6) once with respect to , respectively, and setting the integral constants that are both , we have The second equation of (7) becomes
Substituting (8) into the first equation of (7) yields
Letting , we get the following planar dynamical system:
The rest of this paper is organized as follows. In Section 2, we discuss the bifurcation sets and phase portraits of system (10), where explicit parametric conditions will be derived. In Section 3, we give all possible exact periodic wave, solitary wave, and blow-up wave solutions of system (4). A short conclusion will be given in Section 4.
2. Bifurcation Sets and Phase Portraits of System (10)
Using the transformation , it carries (10) into the Hamiltonian system: with the following first integral:
For a fixed , the level curve defined by (12) determines a set of invariant curves of system (11) which contains different branches of curves. As is varied, it defines different families of orbits of (11) with different dynamical behaviors.
Write . Clearly, when and , system (11) has three equilibrium points at , , and in -axis, where . When and , system (11) has two equilibrium points at and in -axis. When , system (11) has only one equilibrium point at in -axis. There is no any equilibrium point of system (11) in line .
Let be the coefficient matrix of the linearized system of (11) at equilibrium point , then we have Trace and
For an equilibrium point of a planar integrable system, we know that is a saddle point if , a center point if and Trace , and a cusp if and the Poincaré index of is zero.
Since both systems (10) and (11) have the same first integral (12), then the two systems above have the same topological phase portraits. Therefore we can obtain the bifurcation sets and phase portraits of system (10) from that of system (11).
By using the properties of equilibrium points and bifurcation method of dynamical systems, we can show that bifurcation sets and phase portraits of system (10) are drawn in Figure 1.
| (a) |
| (b) |
| (c) |
| (d) |
| (e) |
| (f) |
| (g) |
| (h) |
| (i) |
| (j) |
| (k) |
| (l) |
| (m) |
| (n) |
3. Exact Travelling Wave Solutions of System (4)
Denote that From Figure 1, we give some exact travelling wave solutions of system (4) as follows.
3.1. Different Kinds of Periodic Wave Solutions
From Figures 1(d) and 1(k), we see that there is one periodic orbit of system (10) defined by if and only if one of the following conditions holds:;.The periodic orbit passes points and , where . Its expression is
If introducing a new parametric variable and letting then we have
Substituting (15) into (17) and integrating it along the periodic orbit yield the following: Completing (18) and using (8) and (16), we can get a periodic wave solution of system (4) as follows:
From Figures 1(a), 1(b), 1(c), 1(h), 1(i), and 1(j), we see that there are one periodic orbit and an open curve of system (10) defined by if and only if one of the following conditions holds:;;;;;;;;.The periodic orbit passing points and and the open curve passing point , where , are three real roots of . Their expressions are, respectively,
Substituting (20) into (17) and integrating it along the periodic orbit yield equation Completing (22) and using (8) and (16), we can get a periodic wave solution of system (4) as follows: where , , is the Jacobian elliptic function with the modulus , is the normal elliptic integral of the second kind, and reads amplitude (see [18]).
Substituting (21) into (17) and integrating it along the open curve yield the following: Completing (24) and using (8) and (16), we can get a periodic blow-up wave solution of system (4) as follows: where , , , , and are the Jacobian elliptic functions (see [18]).
From Figures 1(e), 1(f), 1(g), 1(l), 1(m), and 1(n), we see that there are one periodic orbit and an open curve of system (10) defined by if and only if one of the following conditions holds:;;;;;;;;.The periodic orbit passes points and , and the open curve passes point , where are three real roots of . Their expressions are, respectively,
Substituting (26) into (17) and integrating it along the periodic orbit yield the following: Completing (28) and using (8) and (16), we can get a periodic wave solution of system (4) as follows: where , .
Substituting (27) into (17) and integrating it along the open curve yield the following: Completing (30) and using (8) and (16), we can get a periodic blow-up wave solution of system (4) as follows: where , .
3.2. Solitary Wave and Blow-Up Wave Solutions
From Figure 1(a), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connecting with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (32) into (17) and integrating it along the homoclinic orbit yield the following: Completing (34) and using (8) and (16), we can get a solitary wave solution of system (4) as follows: where .
Substituting (33) into (17) and integrating it along the open curve yield the following: Completing (36) and using (8) and (16), we can get a blow-up wave solution of system (4) as follows: where .
From Figure 1(n), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connects with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (38) into (17) and integrating it along the homoclinic orbit yield the following: Completing (40) and using (8) and (16), we can get a solitary wave solution of system (4) the same as (35).
Substituting (39) into (17) and integrating it along the open curve yield the following: Completing (41) and using (8) and (16), we can get a blow-up wave solution of system (4) same as (37).
From Figure 1(b), we see that there is an open curve of system (10) defined by passing cusp when , . Its expression is
Substituting (42) into (17) and integrating it along the open curve yield the following: Completing (43) and using (8) and (16), we can get a blow-up wave solution of system (4) as follows:
From Figure 1(m), we see that there is an open curve of system (10) defined by passes cusp when , . Its expression is
Substituting (45) into (17) and integrating it along the open curve yield the following: Completing (46) and using (8) and (16), we can get a blow-up wave solution of system (4) the same as (44).
From Figure 1(c), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connects with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (47) into (17) and integrating it along the homoclinic orbit yield the following: Completing (49) and using (8) and (16), we can get a solitary wave solution of system (4) as follows: where , .
Substituting (48) into (17) and integrating it along the open curve yield the following: Completing (51) and using (8) and (16), we can get a blow-up wave solution of system (4) as follows: where , .
From Figure 1(e), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connects with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (53) into (17) and integrating it along the homoclinic orbit yield the following: Completing (55) and using (8) and (16), we can get a solitary wave solution of system (4) the same as (50).
Substituting (54) into (17) and integrating it along the open curve yield the following: Completing (56) and using (8) and (16), we can get a blow-up wave solution of system (4) the same as (52).
From Figure 1(f), we see that there is an open curve of system (10) defined by passes cusp when , . Its expression is
Substituting (57) into (17) and integrating it along the open curve yield the following: Completing (58) and using (8) and (16), we can get a blow-up wave solution of system (4) as follows:
From Figure 1(i), we see that there is an open curve of system (10) defined by passes cusp when . Its expression is
Substituting (60) into (17) and integrating it along the open curve yield the following: Completing (61) and using (8) and (16), we can get a blow-up wave solution of system (4) the same as (59).
From Figure 1(j), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connects with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (62) into (17) and integrating it along the homoclinic orbit yield the following: Completing (64) and using (8) and (16), we can get a solitary wave solution of system (4) as follows: where , .
Substituting (63) into (17) and integrating it along the open curve yield the following: Completing (66) and using (8) and (16), we can get a blow-up wave solution of system (4) as follows: where , .
From Figure 1(l), we see that there are one homoclinic orbit and an open curve of system (10) defined by when , . The homoclinic orbit connects with saddle point and passes point , and the open curve passes saddle point , where . Their expressions are, respectively,
Substituting (68) into (17) and integrating it along the homoclinic orbit yield the following: Completing (70) and using (8) and (16), we can get a solitary wave solution of system (4) the same as (65).
Substituting (69) into (17) and integrating it along the open curve yield the following: Completing (71) and using (8) and (16), we can get a blow-up wave solution of system (4) the same as (67).
4. Conclusion
In this paper, we studied the bifurcations of travelling wave solutions of a generalized two-component Hunter-Saxton system and obtained different kinds of periodic wave solutions, which concluded periodic blow-up wave and periodic loop solutions and so forth. Some solitary wave and blow-up wave solutions are also obtained. The results of this paper have enriched the results of [10–12].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the referees very much for their perceptive comments and suggestions. This work is supported by the Natural Science Foundation of Yunnan Province, China (no. 2013FZ117), and the National Natural Science Foundation of China (no. 11161020).