# American Institute of Mathematical Sciences

August  2014, 19(6): 1719-1729. doi: 10.3934/dcdsb.2014.19.1719

## Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system

 1 Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004

Received  September 2013 Revised  February 2014 Published  June 2014

In this paper, we apply the method of dynamical systems to a generalized two-component Hunter-Saxton system. Through qualitative analysis, we obtain bifurcations of phase portraits of the traveling system. Under different parameter conditions, exact explicit smooth solitary wave solutions, solitary cusp wave solutions, as well as periodic wave solutions are obtained. To guarantee the existence of these solutions, rigorous parametric conditions are given.
Citation: Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1719-1729. doi: 10.3934/dcdsb.2014.19.1719
##### References:
 [1] P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer, Berlin, 1971. [2] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solution, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [4] M. Chen, S. Liu and Y. Zhang, A 2-component generalization of the Cammassa-Holm equation and its solution, Letters in Math. Phys., 75 (2006), 1-15. doi: 10.1007/s11005-005-0041-7. [5] M. Chen, Y. Liu and Z. Qiao, Sability of solitary wave and globl exisence of a generalized two-component Cammsa-Holm Equation, Comnications of partial differential equation, 36 (2011), 2162-2188. doi: 10.1080/03605302.2011.556695. [6] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [7] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, Phys. Lett. A, 308 (2003), 437-444. doi: 10.1016/S0375-9601(03)00114-2. [8] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact solutions, Science Press, Beijing, 2013. [9] J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifurcation and Chaos, 17 (2007), 4049-4065. doi: 10.1142/S0218127407019858. [10] J. Li and H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach, Science Press, Beijing, 2007. [11] J. Li and Z. Qiao, Peakon, pseudo-peakon, and cuspon solutions for two generalized Cammasa-Holm equations, J. Math. Phys., 54 (2013), 123501, 14pp. doi: 10.1063/1.4835395. [12] J. Li and Z. Qiao, Bifurcations of traveling wave solutions for an integrable equation, J. Math. Phys., 51 (2010), 042703, 23pp. doi: 10.1063/1.3385777. [13] J. Li and Z. Qiao, Bifurcations and exact travelling wave solutions of the generalized two-component Cammsa-Holm Equation, Int. J. Bifurcation and Chaos, 22 (2012), 1250305, 13pp. doi: 10.1142/S0218127412503051. [14] J. Li, J. Wu and H. Zhu, Travelling waves for an integrable higher order KdV type wave equations, Int. J. Bifurcation and Chaos, 16 (2006), 2235-2260. doi: 10.1142/S0218127406016033. [15] B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system, Nonlinear Analysis, 89 (2013), 242-249. doi: 10.1016/j.na.2013.05.004. [16] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. [17] Z. Qiao and J. Li, Negative order KdV equation with both solitons and kink wave solutions, Europhys. Lett., 94 (2011), 50003. [18] Z. Qiao and G. Zhang, On peaked and smooth solitons for the Camassa-Holm equation, Europhys. Lett., 73 (2006), 657-663. doi: 10.1209/epl/i2005-10453-y. [19] M. Wunsch, On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 647-656. doi: 10.3934/dcdsb.2009.12.647. [20] M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304. doi: 10.1137/090768576. [21] M. Wunsch, Weak geodesic flow on a semi-direct product and global solutions to the periodic Hunter-Saxton system, Nonlinear Anal., 74 (2011), 4951-4960. doi: 10.1016/j.na.2011.04.041.

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##### References:
 [1] P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer, Berlin, 1971. [2] R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solution, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. [4] M. Chen, S. Liu and Y. Zhang, A 2-component generalization of the Cammassa-Holm equation and its solution, Letters in Math. Phys., 75 (2006), 1-15. doi: 10.1007/s11005-005-0041-7. [5] M. Chen, Y. Liu and Z. Qiao, Sability of solitary wave and globl exisence of a generalized two-component Cammsa-Holm Equation, Comnications of partial differential equation, 36 (2011), 2162-2188. doi: 10.1080/03605302.2011.556695. [6] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [7] D. D. Holm and M. F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, Phys. Lett. A, 308 (2003), 437-444. doi: 10.1016/S0375-9601(03)00114-2. [8] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact solutions, Science Press, Beijing, 2013. [9] J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, Int. J. Bifurcation and Chaos, 17 (2007), 4049-4065. doi: 10.1142/S0218127407019858. [10] J. Li and H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach, Science Press, Beijing, 2007. [11] J. Li and Z. Qiao, Peakon, pseudo-peakon, and cuspon solutions for two generalized Cammasa-Holm equations, J. Math. Phys., 54 (2013), 123501, 14pp. doi: 10.1063/1.4835395. [12] J. Li and Z. Qiao, Bifurcations of traveling wave solutions for an integrable equation, J. Math. Phys., 51 (2010), 042703, 23pp. doi: 10.1063/1.3385777. [13] J. Li and Z. Qiao, Bifurcations and exact travelling wave solutions of the generalized two-component Cammsa-Holm Equation, Int. J. Bifurcation and Chaos, 22 (2012), 1250305, 13pp. doi: 10.1142/S0218127412503051. [14] J. Li, J. Wu and H. Zhu, Travelling waves for an integrable higher order KdV type wave equations, Int. J. Bifurcation and Chaos, 16 (2006), 2235-2260. doi: 10.1142/S0218127406016033. [15] B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system, Nonlinear Analysis, 89 (2013), 242-249. doi: 10.1016/j.na.2013.05.004. [16] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002, 14pp. doi: 10.1088/1751-8113/42/34/342002. [17] Z. Qiao and J. Li, Negative order KdV equation with both solitons and kink wave solutions, Europhys. Lett., 94 (2011), 50003. [18] Z. Qiao and G. Zhang, On peaked and smooth solitons for the Camassa-Holm equation, Europhys. Lett., 73 (2006), 657-663. doi: 10.1209/epl/i2005-10453-y. [19] M. Wunsch, On the Hunter-Saxton system, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 647-656. doi: 10.3934/dcdsb.2009.12.647. [20] M. Wunsch, The generalized Hunter-Saxton system, SIAM J. Math. Anal., 42 (2010), 1286-1304. doi: 10.1137/090768576. [21] M. Wunsch, Weak geodesic flow on a semi-direct product and global solutions to the periodic Hunter-Saxton system, Nonlinear Anal., 74 (2011), 4951-4960. doi: 10.1016/j.na.2011.04.041.
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