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Bifurcations and Exact Solutions in a Nonlinear Wave Equation

Zongguang Li

School of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China

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and

Rui Liu

http://orcid.org/0000-0002-4547-8695

School of Mathematics, South China University of Technology, Guangzhou 510640, P. R. China

E-mail Address: scliurui@scut.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0218127419500986Cited by:4 (Source: Crossref)

Abstract

The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the $b$ -family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

This research was partially supported by the National Natural Science Foundation of China (Nos. 11771152 and 11571116), National Training Program of Innovation and Entrepreneurship for Undergraduates (No. 201810561172), Pearl River Science and Technology Nova Program of Guangzhou (No. 201610010029).

Keywords:

References

Boyd, J. P. [1997] “ Peakons and coshoidal waves: Traveling wave solutions of the Camassa–Holm equation,” Appl. Math. Comput. 81, 173–187. Crossref, Web of Science, Google Scholar
Camassa, R. & Holm, D. D. [1993] “ An integrable shallow water equation with peaked solitons,” Phys. Rev. Lett. 71, 1661–1664. Crossref, Web of Science, Google Scholar
Chen, C. & Tang, M. Y. [2006] “ A new type of bounded waves for Degasperis–Procesi equations,” Chaos Solit. Fract. 27, 698–704. Crossref, Web of Science, Google Scholar
Chen, Y. R., Ye, W. B. & Liu, R. [2016] “ The explicit periodic wave solutions and their limit forms for a generalized $b$ -equation,” Acta Math. Appl. Sin. 32, 513–528. Crossref, Web of Science, Google Scholar
Constantin, A. & Escher, J. [1998] “ Wave breaking for nonlinear nonlocal shallow water equation,” Exp. Math. 15, 251–264. Google Scholar
Constantin, A. & Strauss, W. A. [2000] “ Stability of peakons,” Commun. Pure Appl. Math. 53, 603–610. Crossref, Web of Science, Google Scholar
Degasperis, A. & Procesi, M. [1999] Asymptotic Integrability, Symmetry and Perturbation Theory, eds. Degasperis, A. & Gaeta, G. (World Scientific, Singapore), pp. 23–37. Google Scholar
Guha, P. [2007] “ Euler–Poincaré formalism of (two component) Degasperis–Procesi and Holm–Staley type systems,” J. Nonlin. Math. Phys. 14, 398–429. Crossref, Web of Science, Google Scholar
Guo, B. L. & Liu, Z. R. [2005] “ Periodic cusp wave solutions and single-solitons for the $b$ -equation,” Chaos Solit. Fract. 23, 1451–1463. Crossref, Web of Science, Google Scholar
He, B., Rui, W. G. & Li, S. [2008a] “ Bounded traveling wave solutions for a modified form of generalized Degasperis–Procesi equation,” Appl. Math. Comput. 206, 113–123. Crossref, Web of Science, Google Scholar
He, B., Rui, W. G. & Chen, C. [2008b] “ Exact traveling wave solutions for a generalized Camassa–Holm equation using the integral bifurcation method,” Appl. Math. Comput. 206, 141–149. Crossref, Web of Science, Google Scholar
Holm, D. D. & Staley, M. F. [2003] “ Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a $1 + 1$ nolinear evolutionary PDE,” Phys. Lett. A 308, 437–444. Crossref, Web of Science, Google Scholar
Johnson, R. S. [2002] “ Camassa–Holm, Korteweg–de Vries and related models for water waves,” J. Fluid Mech. 455, 63–82. Crossref, Web of Science, Google Scholar
Khuri, S. A. [2005] “ New ansatz for obtaining wave solutions of the generalized Camassa–Holm equation,” Chaos Solit. Fract. 25, 705–710. Crossref, Web of Science, Google Scholar
Li, J. B. & Chen, G. R. [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. B. [2013] Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions (Science Press, Beijing). Google Scholar
Li, J. B. & Chen, G. R. [2013] “ Bifurcations of traveling wave solutions in a microstructured solid model,” Int. J. Bifurcation and Chaos 23, 1350009. Link, Web of Science, Google Scholar
Li, J. B., Zhu, W. J. & Chen, G. R. [2016] “ Understanding peakons, periodic peakons and compactons via a shallow water wave equation,” Int. J. Bifurcation and Chaos 26, 1650207. Link, Web of Science, Google Scholar
Liu, Z. R. & Qian, T. F. [2001] “ Peakons and their bifurcation in a generalized Camassa–Holm equation,” Int. J. Bifurcation and Chaos 11, 781–792. Link, Web of Science, Google Scholar
Liu, Z. R. & Ouyang, Z. Y. [2007] “ A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations,” Phys. Lett. A 366, 377–381. Crossref, Web of Science, Google Scholar
Liu, R. [2010] “ Coexistence of multifarious exact nonlinear wave solutions for generalized $b$ -equation,” Int. J. Bifurcation and Chaos 20, 3193–3208. Link, Web of Science, Google Scholar
Liu, Z. R. & Tang, H. [2010] “ Explicit periodic wave solutions and their bifurcations for generalized Camassa–Holm equation,” Int. J. Bifurcation and Chaos 20, 2507–2519. Link, Web of Science, Google Scholar
Liu, Z. R. & Liang, Y. [2011] “ The explicit nonlinear wave solutions and their bifurcations of the generalized Camassa–Holm equation,” Int. J. Bifurcation and Chaos 21, 3119–3136. Link, Web of Science, Google Scholar
Liu, R. & Yan, W. F. [2013] “ Some common expressions and new bifurcation phenomena for nonlinear waves in a generalized MKDV equation,” Int. J. Bifurcation and Chaos 23, 1330007. Link, Web of Science, Google Scholar
Lundmark, H. & Szmigielski, J. [2003] “ Multi-peakon solutions of the Degasperis–Procesi equation,” Inverse Prob. 19, 1241–1245. Crossref, Web of Science, Google Scholar
Lundmark, H. & Szmigielski, J. [2005] “ Degasperis–Procesi peakons and the discrete cubic string,” Int. Math. Res. Papers 2, 53–116. Crossref, 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
Shen, J. W. & Xu, W. [2005] “ Bifurcations of smooth and non-smooth traveling wave solutions in the generalized Camassa–Holm equation,” Chaos Solit. Fract. 26, 1149–1162. Crossref, Web of Science, Google Scholar
Tian, L. X. & Song, X. Y. [2004] “ New peaked solitary wave solutions of the generalized Camassa–Holm equation,” Chaos Solit. Fract. 21, 621–637. Crossref, Web of Science, Google Scholar
Wang, Q. D. & Tang, M. Y. [2008] “ New exact solutions for two nonlinear equations,” Phys. Lett. A 372, 2995–3000. Crossref, Web of Science, Google Scholar
Wazwaz, A. M. [2006] “ Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations,” Phys. Lett. A 352, 500–504. Crossref, Web of Science, Google Scholar
Wazwaz, A. M. [2007] “ New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations,” Appl. Math. Comput. 186, 130–141. Crossref, Web of Science, Google Scholar
Yang, J. P., Li, Z. G. & Liu, Z. R. [2018] “ The existence and bifurcation of peakon and anti-peakon to the $n$ -degree $b$ -equation,” Int. J. Bifurcation and Chaos 28, 1850014. Link, Web of Science, Google Scholar
Yomba, E. [2008a] “ The sub-ODE method for finding exact traveling wave solutions of generalized nonlinear Camassa–Holm, and generalized nonlinear Schrödinger equations,” Phys. Lett. A 372, 215–222. Crossref, Web of Science, Google Scholar
Yomba, E. [2008b] “ A generalized auxiliary equation method and its application to nonlinear Klein–Gordon and generalized nonlinear Camassa–Holm equations,” Phys. Lett. A 372, 1048–1060. Crossref, Web of Science, Google Scholar
Zhang, L. J., Chen, L. Q. & Huo, X. W. [2007] “ Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized Degasperis–Procesi equation,” J. Comput. Appl. Math. 205, 174–185. Crossref, Web of Science, Google Scholar