×
Zhou, Yuqian; Chen, Guanrong; Li, Jibin

Bifurcations, exact peakon, periodic peakons and solitary wave solutions of the modified Camassa-Holm equation. (English) Zbl 1487.35347

Int. J. Bifurcation Chaos Appl. Sci. Eng. 32, No. 5, Article ID 2250076, 11 p. (2022).
Summary: For the modified Camassa-Holm equation, by using the methodologies of dynamical systems and singular traveling wave theory developed by J. Li and G. Chen [Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, No. 11, 4049–4065 (2007; Zbl 1158.35080)] to its corresponding traveling wave system, under different parameter conditions, all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon, periodic peakons, as well as compactons) are obtained. More than 23 explicit exact parametric representations of the above-mentioned traveling wave system are presented.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
35C07 Traveling wave solutions
35C08 Soliton solutions
35C09 Trigonometric solutions to PDEs
35B10 Periodic solutions to PDEs
35B32 Bifurcations in context of PDEs

Keywords:

bifurcation; solitary wave; periodic wave; peakon; periodic peakon; compacton; modified Camassa-Holm equation

Citations:

Zbl 1158.35080
Cite Review PDF
Full Text: DOI

References:

[1] Byrd, P. F. & Fridman, M. D. [1971] Handbook of Elliptic Integrals for Engineers and Scientists (Springer, Berlin). · Zbl 0213.16602
[2] Camassa, R. & Holm, D. D. [1993] “ An integrable shallow water equation with peak solutions,” Phys. Rev. Lett.71, 1161-1164.
[3] Camassa, R., Holm, D. D. & Hyman, J. M. [1994] “ A new integrable shallow water equation,” Adv. Appl. Mech.31, 1-33. · Zbl 0808.76011
[4] Camassa, R., Hyman, J. M. & Luce, B. P. [1998] “ Nonlinear waves and solitons in physical sysytems,” Physica D123, 1-20. · Zbl 0939.35159
[5] Chern, S. S. & Tenenblat, K. [1986] “ Pseudo-spherical surfaces and evolution equations,” Stud. Appl. Math.74, 55-83. · Zbl 0605.35080
[6] Gorka, P. & Reyes, E. G. [2011] “ The modified Camassa-Holm equation,” Int. Math. Res. Not.2011, 2617-2649. · Zbl 1223.35118
[7] Li, J. & Chen, G. [2007] “ On a class of singular nonlinear traveling wave equations,” Int. J. Bifurcation and Chaos17, 4049-4065. · Zbl 1158.35080
[8] Li, J. [2013] Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions (Science Press, Beijing).
[9] Li, J., Zhou, W. & Chen, G. [2016] “ Understanding peakons, periodic peakons and compactons via a shallow water wave equation,” Int. J. Bifurcation and Chaos26, 1650207-1-27. · Zbl 1352.34054
[10] Lou, Z., Qiao, Z. & Yin, Z. [2020] “ On the Cauchy problem for a modified Camassa-Holm equation,” Monatshefte für Mathematik193, 857-877. · Zbl 1451.35169
[11] Reyes, E. G. [2002] “ Geometric integrability of the Camassa-Holm equation,” Lett. Math. Phys.59, 117-131. · Zbl 0997.35081
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.