About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics

Volume 2014 (2014), Article ID 379109, 10 pages

http://dx.doi.org/10.1155/2014/379109
Research Article

Bifurcation Analysis and Different Kinds of Exact Travelling Wave Solutions of a Generalized Two-Component Hunter-Saxton System

1Department of Physics, Honghe University, Mengzi, Yunnan 661100, China

2College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China

Received 22 March 2014; Accepted 27 May 2014; Published 18 June 2014

Academic Editor: Hagen Neidhardt

Copyright © 2014 Qing Meng and Bin He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1012]. 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 [1317].

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.

fig1
Figure 1: Bifurcation sets and phase portraits of system (10).

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 [1012].

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).

References

  1. J. K. Hunter and R. Saxton, “Dynamics of director fields,” SIAM Journal on Applied Mathematics, vol. 51, no. 6, pp. 1498–1521, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B. Khesin and G. Misiołek, “Euler equations on homogeneous spaces and Virasoro orbits,” Advances in Mathematics, vol. 176, no. 1, pp. 116–144, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. D. D. Holm, “Peakons,” in Encyclopedia of Mathematical Physics, vol. 4, pp. 12–20, Academic Press, Oxford, UK, 2006. View at Publisher · View at Google Scholar
  4. J. K. Hunter and Y. Zheng, “On a completely integrable nonlinear hyperbolic variational equation,” Physica D: Nonlinear Phenomena, vol. 79, no. 2–4, pp. 361–386, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Wunsch, “On the Hunter-Saxton system,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 647–656, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. V. Gurevich and K. P. Zybin, “Nondissipative gravitational turbulence,” Soviet Physics—JETP, vol. 67, pp. 1–12, 1988.
  7. M. V. Pavlov, “The Gurevich-Zybin system,” Journal of Physics A: Mathematical and General, vol. 38, no. 17, pp. 3823–3840, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Wunsch, “The generalized Hunter-Saxton system,” SIAM Journal on Mathematical Analysis, vol. 42, no. 3, pp. 1286–1304, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. D. Zuo, “A two-component μ-Hunter-Saxton equation,” Inverse Problems, vol. 26, no. 8, Article ID 085003, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  10. B. Moon and Y. Liu, “Wave breaking and global existence for the generalized periodic two-component Hunter-Saxton system,” Journal of Differential Equations, vol. 253, no. 1, pp. 319–355, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. Moon, “Solitary wave solutions of the generalized two-component Hunter-Saxton system,” Nonlinear Analysis, vol. 89, pp. 242–249, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Zhang and L. Tian, “Wave-breaking criterion for the generalized weakly dissipative periodic two-component Hunter-Saxton system,” Journal of Applied Mathematics, vol. 2013, Article ID 809824, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. J. Li and H. Dai, On the Study of Singular Nonlinear Traveling Wave Equation: Dynamical System Approach, Science Press, Bejing, China, 2007.
  14. Z. Liu and B. Guo, “Periodic blow-up solutions and their limit forms for the generalized Camassa-Holm equation,” Progress in Natural Science, vol. 18, no. 3, pp. 259–266, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. He, Q. Meng, and Y. Long, “The bifurcation and exact peakons, solitary and periodic wave solutions for the Kudryashov-Sinelshchikov equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 11, pp. 4137–4148, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. B. He, “Bifurcations and exact bounded travelling wave solutions for a partial differential equation,” Nonlinear Analysis: Real World Applications, vol. 11, no. 1, pp. 364–371, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Li and Z. Liu, “Smooth and non-smooth traveling waves in a nonlinearly dispersive equation,” Applied Mathematical Modelling, vol. 25, pp. 41–56, 2000.
  18. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer, Berlin, Germany, 1971.