Periodic blow-up solutions and their limit forms for the generalized Camassa–Holm equation

Zhengrong Liu1, Boling Guo2
1School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China
2Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

Tài liệu tham khảo

Camassa, 1993, An integrable shallow water equation with peaked solitons, Phys Rev Lett, 71, 1661, 10.1103/PhysRevLett.71.1661 Dai, 1998, Model equations for nonlinear dispersive waves in a compressible Mooney–Rivlin rod, Acta Mec, 127, 193, 10.1007/BF01170373 Liu, 2002, Peakons of the Camassa–Holm equation, Appl Math Model, 26, 473, 10.1016/S0307-904X(01)00086-5 Liu, 2004, Peaked wave solutions of Camassa–Holm equation, Chaos Soliton Fract, 19, 77, 10.1016/S0960-0779(03)00082-1 Constantin, 1998, Global weak solutions for a shallow water equation, Indiana U Math J, 47, 1527, 10.1512/iumj.1998.47.1466 Constantin, 2000, Global weak solutions for a shallow water equation, Comm Math Phys, 211, 45, 10.1007/s002200050801 Lenells, 2005, Travelling wave solutions of the Camassa–Holm equation, J Differ Eq, 217, 393, 10.1016/j.jde.2004.09.007 Constantin, 1997, On the cauchy problem for the periodic Camassa–Holm equation, J Differ Eq, 141, 218, 10.1006/jdeq.1997.3333 Li, 2000, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J Differ Eq, 162, 27, 10.1006/jdeq.1999.3683 Kwek, 2000, An initial boundary value problem of Camassa–Holm equation, J Math Phys, 41, 8279, 10.1063/1.1288498 Yin, 2004, Well-posedness and blow-up phenomena for the periodic generalized Camassa–Holm equation, Commun Pur Appl Anal, 3, 501, 10.3934/cpaa.2004.3.501 Yin, 2004, On the blow-up scenario for the generalized Camassa–Holm equation, Commun Part Diff Eq, 29, 867, 10.1081/PDE-120037334 Liu, 2006, Global existence and blow-up solutions for a nonlinear shallow water equation, Math Ann, 335, 717, 10.1007/s00208-006-0768-1 Liu, 2004, New bounded traveling waves of Camassa–Holm equation, Int J Bifurcat Chaos, 14, 3541, 10.1142/S0218127404011521 Dullin, 2001, An integrable shallow water equation with linear and nonlinear dispersion, Phys Rev Lett, 87, 4501, 10.1103/PhysRevLett.87.194501 Guo, 2003, Peaked wave solutions of CH-γ equation, Sci China Ser A: Math, 46, 696, 10.1007/BF02942241 Guo, 2005, Two new types of bounded waves of CH-γ equation, Sci China Ser A: Math, 48, 1618, 10.1360/04ys0205 Tang, 2007, Four types of bounded wave solutions of CH-γ equation, Sci China Ser A: Math, 50, 132, 10.1007/s11425-007-2042-8 Liu, 2001, Peakons and their bifurcation in a generalized Camassa–Holm equation, Int J Bifurcat Chaos, 11, 781, 10.1142/S0218127401002420 Tian, 2004, New peaked solitary wave solutions of the generalized Camassa–Holm equation, Chaos Soliton Fract, 19, 621, 10.1016/S0960-0779(03)00192-9 Khuri, 2005, New ansatz for obtaining wave solutions of the generalized Camassa–Holm equation, Chaos Soliton Fract, 25, 705, 10.1016/j.chaos.2004.11.083 Shen, 2005, Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa–Holm equation, Chaos Soliton Fract, 26, 1149, 10.1016/j.chaos.2005.02.021 Wazwaz, 2006, Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Phys Lett A, 352, 500, 10.1016/j.physleta.2005.12.036 Wazwaz, 2007, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations, Appl Math Comput, 186, 130, 10.1016/j.amc.2006.07.092 Liu, 2007, A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations, Phys Lett A, 366, 377, 10.1016/j.physleta.2007.01.074