New peaked solitary wave solutions of the generalized Camassa-Holm equation. (English) Zbl 1068.35123

Tian, Lixin; Song, Xiuying

Chaos Solitons Fractals 19, No. 3, 621-637 (2004).

Summary: We consider generalized Camassa-Holm equations and the generalized weakly dissipative Camassa-Holm equations and derive some new exact peaked solitary wave solutions. For \(m=3\), where \(m\) is a representative of the strength of the nonlinearity, we give two types new exact traveling wave solutions of the generalized weakly dissipative Camassa-Holm equations.

MSC:

35Q51	Soliton equations
35Q53	KdV equations (Korteweg-de Vries equations)

References:

[1]	Camassa, R.; Holm, D., An integrable shallow water equation with peaked soliton, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[2]	Fisher, M.; Sshiff, J., The Camassa-Holm equation: conserved quantities and the initial value problem, Phys. Lett. A, 259, 3, 371-376 (1999) · Zbl 0936.35166
[3]	Clarkson, D. A.; Mansfiel, E. L.; Priestley, T. J., Symmetries of a class of nonlinear third-order partial differential equations, Math. Comput. Modell., 25, 819, 195-212 (1997) · Zbl 0879.35005
[4]	Kraenkel, R. A.; Senthilvelsn, M.; Zenchuk, A. I., On the integrable perturbations of the Camassa-Holm equation, J. Math. Phys., 41, 5, 3160-3169 (2000) · Zbl 1052.37058
[5]	Cooper, F.; Shepard, H., Solitons in the Camassa-Holm shallow water equation, Phy. Lett. A, 194, 246-250 (1994) · Zbl 0961.76512
[6]	Tian, L.; Xu, G.; Liu, Z., The concave or convex peaked and smooth solutions of Camassa-Holm equation, Appl. Math. Mech., 23, 5, 557-567 (2002) · Zbl 1020.35075
[7]	Chen, S.; Foias, C.; Holm, D. D.; Olson, E. J.; Titi, E. S.; Wynne, S., The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81, 5338-5341 (1998) · Zbl 1042.76525
[8]	Dullin, H. R.; Gottwald, G.; Holm, D. D., An integrable shallow water equation with linear and nonlinear dispersion, Phys. Rev. Lett., 87, 19, 194501-194504 (2001)
[9]	Foias, C.; Holm, D. D.; Titi, E., The Navier-Stokes-alpha model of fluid turbulence, Physica D, 152, 505-519 (2001) · Zbl 1037.76022
[10]	Liu, Z.; Qian, T., Peakons and their bifurcation in a generalized Camassa-Holm equation, Int. J. Bifur. Chaos, 11, 781-792 (2001) · Zbl 1090.37554
[12]	Rosenau, P.; Hyman, J. M., Compactons: solitons with finite wavelength, Phys. Rev. Lett., 70, 5, 564-566 (1993) · Zbl 0952.35502
[13]	Wazwaz, A. M., Compactons dispersive structures for varians of the \(k(n,n)\) and the KP equations, Chaos, Solitons & Fractals, 13, 1053-1062 (2002) · Zbl 0997.35083
[14]	Yan, Z., New families of solitons with compact support for Boussinesq-like \(B(m,n)\) equations with fully nonlinear dispersion, Chaos, Solitons & Fractals, 14, 1151-1158 (2002) · Zbl 1038.35082
[15]	Yan, Z.; Bluman, G., New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Comput. Phys. Commun., 149, 11-18 (2002) · Zbl 1196.68338
[16]	Zhang, Y.; Zheng, H., Two types traveling wave solutions in Burgers-KdV equation, Appl. Math. Mech., 21, 10, 1009-1011 (2000)
[17]	Cheng, L.; Liang, C., Soliton theory and application (1997), Xidian University Press