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Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations
Bifurcations and exact travelling wave solutions of the generalized two-component Hunter-Saxton system
1. | Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004 |
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. |
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