Peakons and periodic cusp waves in a generalized Camassa-Holm equation. (English) Zbl 1021.35086

Summary: We study the peakons and the periodic cusp wave solutions of the equation \[ u_t + 2ku_x-u_{xxt} +auu_x = 2u_xu_{xx} + uu_{xxx}, \] with \(a, k\in\mathbb{R}\), which we will call the generalized Camassa-Holm equation, or simply the GCH equation, for \(a = 3\) it is the CH Camassa-Holm equation. Camassa and Holm showed that the CH equation has a class of new solitary wave solutions called “peakons”.
Using the bifurcation method of the phase plane, we first construct peakons and show that \(a = 3\) is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter \(a\) or the wave speed \(c\) varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.


35Q35 PDEs in connection with fluid mechanics
35B10 Periodic solutions to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
Full Text: DOI


[1] Alber, M. S.; Camassa, R.; Holm, D. D.; Marsden, J. E., The geometry of peaked solitons and billiard solutions of a class of integrable PDEs, Lett Math Phys, 32, 2, 137-151 (1994) · Zbl 0808.35124
[3] Boyd, J. P., Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation, Appl Math Comput, 81, 2-3, 173-187 (1997) · Zbl 0871.35089
[4] Camassa, R.; Holm, D. D., An integrable shallow water equation with peaked solitons, Phys Rev Lett, 71, 11, 1661-1664 (1993) · Zbl 0972.35521
[5] Camassa, R.; Holm, D. D.; Hyman, J. M., A new integrable shallow water equation, Adv Appl Mech, 31, 1-33 (1994) · Zbl 0808.76011
[6] Constantin, A., Quasi-periodicity with respect to time of spatially periodic finite-gap solution of the Camassa-Holm equation, Bull Sci Math, 122, 7, 487-494 (1998) · Zbl 0923.35126
[7] Constantin, A., On the Cauchy problem for the periodic Camassa-Holm equation, J Differential Equations, 141, 2, 218-235 (1997) · Zbl 0889.35022
[8] Constantin, A., On the inverse spectral problem for the Camassa-Holm equation, J Funct Anal, 155, 2, 352-363 (1998) · Zbl 0907.35009
[9] Constantin, A., Soliton interactions for the Camassa-Holm equation, Exposition Math, 15, 3, 251-264 (1997) · Zbl 0879.35121
[10] Constantin, A., The Hamiltonian structure of the Camassa-Holm equation, Exposition Math, 15, 1, 53-85 (1997) · Zbl 0881.35094
[11] Constantin, A., On the spectral problem for the periodic Camassa-Holm equation, J Math Anal Appl, 210, 1, 215-230 (1997) · Zbl 0881.35102
[12] Cooper, F.; Shepard, H., Solitons in the Camassa-Holm shallow water equation, Phys Lett A, 194, 4, 246-250 (1994) · Zbl 0961.76512
[13] Dai, H. H.; Pavlov, M., Maxim transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J Phys Soc Jpn, 67, 11, 3655-3657 (1998) · Zbl 0946.35082
[14] Fisher, M.; Schiff, J., The Camassa-Holm equation: conserved quantities and the initial value problem, Phys Lett A, 259, 5, 371-376 (1999) · Zbl 0936.35166
[15] Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys D, 95, 3-4, 229-243 (1996) · Zbl 0900.35345
[16] Guckenheimer, J.; Holmes, P., Dynamical systems and bifurcations of vector fields (1983), Springer: Springer New York · Zbl 0515.34001
[17] Kouranbaeva, S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J Math Phys, 40, 2, 857-868 (1999) · Zbl 0958.37060
[18] Schiff, J., The Camassa-Holm equation: a loop group approach, Phys D, 121, 1-2, 24-43 (1998) · Zbl 0943.37034
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.