# 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.

show all references

##### 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.
 [1] Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088 [2] Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280 [3] Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459 [4] Jaeho Choi, Nitin Krishna, Nicole Magill, Alejandro Sarria. On the $L^p$ regularity of solutions to the generalized Hunter-Saxton system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6349-6365. doi: 10.3934/dcdsb.2019142 [5] Meiling Yang, Yongsheng Li, Zhijun Qiao. Persistence properties and wave-breaking criteria for a generalized two-component rotational b-family system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2475-2493. doi: 10.3934/dcds.2020122 [6] Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 [7] Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641 [8] Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239 [9] Vural Bayrak, Emil Novruzov, Ibrahim Ozkol. Local-in-space blow-up criteria for two-component nonlinear dispersive wave system. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6023-6037. doi: 10.3934/dcds.2019263 [10] Yongsheng Mi, Chunlai Mu, Pan Zheng. On the Cauchy problem of the modified Hunter-Saxton equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2047-2072. doi: 10.3934/dcdss.2016084 [11] Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613 [12] Xiuting Li, Lei Zhang. The Cauchy problem and blow-up phenomena for a new integrable two-component peakon system with cubic nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3301-3325. doi: 10.3934/dcds.2017140 [13] Marcus Wunsch. On the Hunter--Saxton system. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 647-656. doi: 10.3934/dcdsb.2009.12.647 [14] Mike Hay, Andrew N. W. Hone, Vladimir S. Novikov, Jing Ping Wang. Remarks on certain two-component systems with peakon solutions. Journal of Geometric Mechanics, 2019, 11 (4) : 561-573. doi: 10.3934/jgm.2019028 [15] Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643 [16] Qiaoyi Hu, Zhixin Wu, Yumei Sun. Liouville theorems for periodic two-component shallow water systems. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3085-3097. doi: 10.3934/dcds.2018134 [17] Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure and Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487 [18] Katrin Grunert. Blow-up for the two-component Camassa--Holm system. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2041-2051. doi: 10.3934/dcds.2015.35.2041 [19] Yong Chen, Hongjun Gao, Yue Liu. On the Cauchy problem for the two-component Dullin-Gottwald-Holm system. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3407-3441. doi: 10.3934/dcds.2013.33.3407 [20] Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101

2020 Impact Factor: 1.327