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

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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

Abstract
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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.

Keywords: peakon, solitary wave solution, Generalized two-component Hunter-Saxton system, periodic cusp wave solution..

Mathematics Subject Classification: Primary: 34A05, 34C25-28, 34M55, 35Q51, 35Q53; Secondary: 58F05, 58F14, 58F3.

Citation: Jibin Li. Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system. Discrete & 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, (1971). Google Scholar
[2]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solution,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[3]	R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar
[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. doi: 10.1007/s11005-005-0041-7. Google Scholar
[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. doi: 10.1080/03605302.2011.556695. Google Scholar
[6]	A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar
[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. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar*
[8]	J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact solutions,, Science Press, (2013). Google Scholar
[9]	J. Li and G. Chen, On a class of singular nonlinear traveling wave equations,, Int. J. Bifurcation and Chaos, 17 (2007), 4049. doi: 10.1142/S0218127407019858. Google Scholar
[10]	J. Li and H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach,, Science Press, (2007). Google Scholar
[11]	J. Li and Z. Qiao, Peakon, pseudo-peakon, and cuspon solutions for two generalized Cammasa-Holm equations,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4835395. Google Scholar
[12]	J. Li and Z. Qiao, Bifurcations of traveling wave solutions for an integrable equation,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3385777. Google Scholar
[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). doi: 10.1142/S0218127412503051. Google Scholar
[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. doi: 10.1142/S0218127406016033. Google Scholar
[15]	B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system,, Nonlinear Analysis, 89 (2013), 242. doi: 10.1016/j.na.2013.05.004. Google Scholar
[16]	V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar
[17]	Z. Qiao and J. Li, Negative order KdV equation with both solitons and kink wave solutions,, Europhys. Lett., 94 (2011). Google Scholar
[18]	Z. Qiao and G. Zhang, On peaked and smooth solitons for the Camassa-Holm equation,, Europhys. Lett., 73 (2006), 657. doi: 10.1209/epl/i2005-10453-y. Google Scholar
[19]	M. Wunsch, On the Hunter-Saxton system,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 647. doi: 10.3934/dcdsb.2009.12.647. Google Scholar
[20]	M. Wunsch, The generalized Hunter-Saxton system,, SIAM J. Math. Anal., 42 (2010), 1286. doi: 10.1137/090768576. Google Scholar
[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. doi: 10.1016/j.na.2011.04.041. Google Scholar

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References:

[1]	P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists,, Springer, (1971). Google Scholar
[2]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solution,, Phys. Rev. Lett., 71 (1993), 1661. doi: 10.1103/PhysRevLett.71.1661. Google Scholar
[3]	R. Camassa, D. D. Holm and J. M. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar
[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. doi: 10.1007/s11005-005-0041-7. Google Scholar
[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. doi: 10.1080/03605302.2011.556695. Google Scholar
[6]	A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions,, Theoret. Math. Phys., 133 (2002), 1463. doi: 10.1023/A:1021186408422. Google Scholar
[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. doi: 10.1016/S0375-9601(03)00114-2. Google Scholar*
[8]	J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact solutions,, Science Press, (2013). Google Scholar
[9]	J. Li and G. Chen, On a class of singular nonlinear traveling wave equations,, Int. J. Bifurcation and Chaos, 17 (2007), 4049. doi: 10.1142/S0218127407019858. Google Scholar
[10]	J. Li and H. Dai, On the Study of Singular Nonlinear Travelling Wave Equations: Dynamical Approach,, Science Press, (2007). Google Scholar
[11]	J. Li and Z. Qiao, Peakon, pseudo-peakon, and cuspon solutions for two generalized Cammasa-Holm equations,, J. Math. Phys., 54 (2013). doi: 10.1063/1.4835395. Google Scholar
[12]	J. Li and Z. Qiao, Bifurcations of traveling wave solutions for an integrable equation,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3385777. Google Scholar
[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). doi: 10.1142/S0218127412503051. Google Scholar
[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. doi: 10.1142/S0218127406016033. Google Scholar
[15]	B. Moon, Solitary wave solutions of the generalized two-component Hunter-Saxton system,, Nonlinear Analysis, 89 (2013), 242. doi: 10.1016/j.na.2013.05.004. Google Scholar
[16]	V. Novikov, Generalizations of the Camassa-Holm equation,, J. Phys. A: Math. Theor., 42 (2009). doi: 10.1088/1751-8113/42/34/342002. Google Scholar
[17]	Z. Qiao and J. Li, Negative order KdV equation with both solitons and kink wave solutions,, Europhys. Lett., 94 (2011). Google Scholar
[18]	Z. Qiao and G. Zhang, On peaked and smooth solitons for the Camassa-Holm equation,, Europhys. Lett., 73 (2006), 657. doi: 10.1209/epl/i2005-10453-y. Google Scholar
[19]	M. Wunsch, On the Hunter-Saxton system,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 647. doi: 10.3934/dcdsb.2009.12.647. Google Scholar
[20]	M. Wunsch, The generalized Hunter-Saxton system,, SIAM J. Math. Anal., 42 (2010), 1286. doi: 10.1137/090768576. Google Scholar
[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. doi: 10.1016/j.na.2011.04.041. Google Scholar

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