Bifurcations and exact bounded travelling wave solutions for a partial differential equation. (English) Zbl 1181.35214

Summary: A partial differential equation is investigated by using the bifurcation theory and the method of phase portraits analysis, the existence of loop soliton, peakon, generalized compacton, smooth and non-smooth periodic waves, breaking kink and anti-kink waves is proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some conditions, exact parametric representations of these waves in explicit and implicit forms are obtained.


35Q51 Soliton equations
35C07 Traveling wave solutions
35C08 Soliton solutions
35B10 Periodic solutions to PDEs
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI


[1] Camassa, R.; Holm, D.D., An integrable shallow water equation with peaked solitons, Phys. rev. lett., 71, 1661-1664, (1993) · Zbl 0972.35521
[2] Boyd, J.P., Peakons and coshoidal waves: traveling wave solutions of the camassa – holm equation, Appl. math. comput., 81, 173-187, (1997) · Zbl 0871.35089
[3] Cooper, F.; Shepard, H., Solitons in the camassa – holm shallow water equation, Phys. lett. A, 194, 246-250, (1994) · Zbl 0961.76512
[4] Constantin, A., Soliton interactions for the camassa – holm equation, Exp. math., 15, 251-264, (1997) · Zbl 0879.35121
[5] Constantin, A.; Strauss, W.A., Stability of peakons, Commun. pure appl. math., 53, 603-610, (2000) · Zbl 1049.35149
[6] Liu, Z.R.; Qian, T.F., Peakons of the camassa – holm equation, Appl. math. model., 26, 473-480, (2002) · Zbl 1018.35061
[7] Lenells, J., The scattering approach for the camassa – holm equation, J. nonlinear math. phys., 9, 389-393, (2002) · Zbl 1014.35082
[8] Parker, A.; Matsuno, Y., The peakon limits of soliton solutions of the camassa – holm equation, J. phys. soc. Japan, 75, 124001-124009, (2006)
[9] Li, Y.; Zhang, J.E., The multiple-soliton solution of the camassa – holm equation, Proc. R. soc. London, ser. A, 460, 2617-2627, (2004) · Zbl 1068.35109
[10] Zheng, Y.; Lai, S.Y., Peakons, solitary patterns and periodic solutions for generalized camassa – holm equations, Phys. lett. A, 372, 4141-4143, (2008) · Zbl 1221.82106
[11] Degasperis, A.; Procesi, M., (), 23-37
[12] Degasperis, A.; Hone, A.N.W.; Holm, D.D., A new integrable equation with peakon solutions, Theor. math. phys., 133, 1463-1474, (2002)
[13] Manna, M.A.; Merle, V., Asymptotic dynamics of short waves in nonlinear dispersive models, Phys. rev. E, 57, 6206-6209, (1998)
[14] Manna, M.A., Nonlinear asymptotic short-wave models in fluid dynamics, J. phys. A, 34, 4475-4491, (2001) · Zbl 1006.76014
[15] Matsuno, Y., Cusp and loop soliton solutions of short-wave models for the camassa – holm and degasperis – procesi equations, Phys. lett. A, 359, 451-457, (2006) · Zbl 1193.35195
[16] Alber, M.S.; Camassa, R.; Fedorov, Yu.; Holm, D.D.; Marsden, J.E., The geometry of peaked solitons and billiard solutions of a class of integrable PDE’s, Lett. math. phys., 32, 137-151, (1994) · Zbl 0808.35124
[17] Alber, M.S.; Camassa, R.; Fedorov, Yu.; Holm, D.D.; Marsden, J.E., The complex geometry of weak piecewise smooth solutions of integrable nonlinear pde’s of shallow water and Dym type, Commun. math. phys., 221, 197-227, (2001) · Zbl 1001.37062
[18] Hunter, J.K.; Saxton, R., Dynamics of director fields, SIAM J. appl. math., 51, 1498-1521, (1991) · Zbl 0761.35063
[19] Vakhnenko, V.A., Solitons in a nonlinear model medium, J. phys. A, 25, 4181-4187, (1992) · Zbl 0754.35132
[20] Li, J.B.; Liu, Z.R., Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Appl. math. model., 25, 41-56, (2000) · Zbl 0985.37072
[21] Long, Y.; Rui, W.G.; He, B., Travelling wave solutions for a higher order wave equations of KdV type(I), Chaos solitons fractals, 23, 469-475, (2005) · Zbl 1069.35075
[22] Li, J.B.; Rui, W.G.; Long, Y.; He, B., Travelling wave solutions for a higher order wave equations of KdV type (III), Math. biosci. eng., 3, 125-135, (2006) · Zbl 1136.35449
[23] He, B.; Li, J.B.; long, Y.; Rui, W.G., Bifurcations of travelling wave solutions for a variant of camassa – holm equation, Nonlinear anal. real., 9, 222-232, (2008) · Zbl 1185.35217
[24] Li, J.B.; Dai, H.H., On the study of singular nonlinear traveling wave equation: dynamical system approach, (2007), Science Press Bejing
[25] Li, J.B.; Liu, Z.R., Travelling wave solutions for a class of nonlinear dispersive equation, Chin. ann. math., 3, 397-418, (2002) · Zbl 1011.35014
[26] Long, Y.; He, B.; Rui, W.G.; Chen, C., Compacton-like and kink-like waves for a higher-order wave equation of Korteweg-de Vries type, Int. J. comput. math., 83, 959-971, (2006) · Zbl 1134.35096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.