PaperNo Access

Bifurcations, Exact Peakon, Periodic Peakons and Solitary Wave Solutions of the Modified Camassa–Holm Equation

Yuqian Zhou

School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P. R. China

Search for more papers by this author

Guanrong Chen

Department of Electrical Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, P. R. China

Search for more papers by this author

, and

Jibin Li

https://orcid.org/0000-0003-4839-5956

School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, P. R. China

School of Mathematical Science, Huaqiao University, Quanzhou, Fujian 362021, P. R. China

Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejing 321004, P. R. China

E-mail Address: lijb@zjnu.cn

E-mail Address: lijb@hqu.edu.cn

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218127422500766Cited by:3 (Source: Crossref)

Abstract

For the modified Camassa–Holm equation, by using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] 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.

This research is partially supported by the National Natural Science Foundations of China (11871231, 12071162, 11701191) and the Sichuan Natural Science Foundation (2021ZYD0009).

Keywords:

References

Byrd, P. F. & Fridman, M. D. [1971] Handbook of Elliptic Integrals for Engineers and Scientists (Springer, Berlin). Crossref, Google Scholar
Camassa, R. & Holm, D. D. [1993] “ An integrable shallow water equation with peak solutions,” Phys. Rev. Lett. 71, 1161–1164. Crossref, Web of Science, Google Scholar
Camassa, R., Holm, D. D. & Hyman, J. M. [1994] “ A new integrable shallow water equation,” Adv. Appl. Mech. 31, 1–33. Crossref, Google Scholar
Camassa, R., Hyman, J. M. & Luce, B. P. [1998] “ Nonlinear waves and solitons in physical sysytems,” Physica D 123, 1–20. Crossref, Web of Science, Google Scholar
Chern, S. S. & Tenenblat, K. [1986] “ Pseudo-spherical surfaces and evolution equations,” Stud. Appl. Math. 74, 55–83. Crossref, Web of Science, Google Scholar
Gorka, P. & Reyes, E. G. [2011] “ The modified Camassa–Holm equation,” Int. Math. Res. Not. 2011, 2617–2649. Web of Science, Google Scholar
Li, J. & Chen, G. [2007] “ On a class of singular nonlinear traveling wave equations,” Int. J. Bifurcation and Chaos 17, 4049–4065. Link, Web of Science, Google Scholar
Li, J. [2013] Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions (Science Press, Beijing). Google Scholar
Li, J., Zhou, W. & Chen, G. [2016] “ Understanding peakons, periodic peakons and compactons via a shallow water wave equation,” Int. J. Bifurcation and Chaos 26, 1650207-1–27. Link, Web of Science, Google Scholar
Lou, Z., Qiao, Z. & Yin, Z. [2020] “ On the Cauchy problem for a modified Camassa–Holm equation,” Monatshefte für Mathematik 193, 857–877. Crossref, Web of Science, Google Scholar
Reyes, E. G. [2002] “ Geometric integrability of the Camassa–Holm equation,” Lett. Math. Phys. 59, 117–131. Crossref, Web of Science, Google Scholar