PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION

Shaolong Xie; Xiaochun Hong; Tao Jiang

doi:10.11948/2017019

2017 Volume 7 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(1): 278-290

Next Article Previous Article

Shaolong Xie, Xiaochun Hong, Tao Jiang. PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 278-290. doi: 10.11948/2017019

Citation:

Shaolong Xie, Xiaochun Hong, Tao Jiang. PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION[J]. Journal of Applied Analysis & Computation, 2017, 7(1): 278-290. doi: 10.11948/2017019

PLANAR BIFURCATION METHOD OF DYNAMICAL SYSTEM FOR INVESTIGATING DIFFERENT KINDS OF BOUNDED TRAVELLING WAVE SOLUTIONS OF A GENERALIZED CAMASSA-HOLM EQUATION

1 Business School, Yuxi Normal University, 653100 Yuxi, China;
2 School of Statistics and Mathematics, Yunnan University of Finance and Economics, 650221 Kunming, China;
3 School of Information, Beijing Wuzi University, 101149 Beijing, China

Fund Project:

Abstract

Abstract

In this study, by using planar bifurcation method of dynamical system, we study a generalized Camassa-Holm (gCH) equation. As results, under different parameter conditions, many bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakons, loops and kink waves are given. The dynamic properties of these exact solutions are investigated.
- GCH equation /
- periodic wave /
- periodic cusp wave /
- peakon /
- loop /
- kink wave
MSC: 35B10;35C07;35C08

FullText(HTML)

References

[1]	A. Bressan and A. Constantin, Global conservative solutions of the CamassaHolm equation, Arch. Rational Mech., 183(2007)(2), 215-239. Google Scholar
[2]	S. S. Behzadi, Numerical solution of fuzzy Camassa-Holm equation by using homotopy analysis methods, Journal of Applied Analysis and Computation, 1(2011)(3), 315-323. Google Scholar
[3]	P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals for engineers and scientists, Berlin:Springer, 1971. Google Scholar
[4]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71(1993)(11), 1661-1664. Google Scholar
[5]	R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31(1994), 1-33. Google Scholar
[6]	G. M. Coclite, F. Gargano and V. Sciacca, Analytic solutions and singularity formation for the Peakon b-family equations, Acta Appl. Math., 122(2012), 419-434. Google Scholar
[7]	A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133(2002)(2), 1463-1474. Google Scholar
[8]	H. Holden and X. Raynaud, Dissipative solutions for the Camassa-Holm equation, Discrete Cont. Dyn. S., 24(2009)(4), 1047-1112. Google Scholar
[9]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455(2002), 63-82. Google Scholar
[10]	R. S. Johnson, On the solutions of the Camassa-Holm equation, Proc. R. Soc. Lond. A, 459(2003)(2035), 1687-1708. Google Scholar
[11]	J. Li and Z. Liu, Smooth and non-smooth traveling waves in a nonlinearly dispersive equation, Applied Mathematical Modelling, 25(2000)(1), 41-56. Google Scholar
[12]	J. Li and Z. Liu, Traveling wave solutions for a class of nonlinear dispersive equations, Chinese Annals of Mathematics, 23(2002)(3), 397-418. Google Scholar
[13]	M. C. Lombardo, M. Sammartino and V. Sciacca, A note on the analytic solutions of the Camassa-Holm equation, Comp. Rend. Math., 341(2005)(11), 659-664. Google Scholar
[14]	T. Matsuo, A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation, J. Comp. Appl. Math., 234(2010)(4), 1258-1266. Google Scholar
[15]	A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35(2002)(22), 4775-4790. Google Scholar
[16]	A. B. De Monvel, A. Its and D. Shepelsky, Painlev'e-type asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 42(2010)(4), 1854-1873. Google Scholar
[17]	A. B. De Monvel, A. Kostenko, D. Shepelsky and G. Teschlh, Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal., 41(2009)(4), 1559-1588. Google Scholar
[18]	V. Novikov, Generalization of the Camassa-Holm equation, J. Phys. A, 42(2009), 342002-1-14. Google Scholar
[19]	T. Rehman, G. Gambino and S. R. Choudhury, Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations, Commun. Nonlinear Sci. Numer. Simul., 19(2014)(6), 1746-1769. Google Scholar
[20]	G. D. Rocca, M. C. Lombardo, M. Sammartino and V. Sciacca, Singularity tracking for Camassa-Holm and Prandtl's equations, Appl. Num. Math., 56(2006)(8), 1108-1122. Google Scholar
[21]	A. M. Wazwaz, Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa-Holm equations by using new hyperbolic schemes, Appl. Math. Comput., 182(2006)(1), 412-424. Google Scholar
[22]	S. Xie and L. Wang, Compacton and generalized kink wave solutions of the CH-DP equation, Appl. Math. Comput., 215(2010)(11), 4028-4039. Google Scholar
[23]	S. Xie, L. Wang and Y. Zhang, Explicit and implicit solutions of a generalized Camassa-Holm-Kadomtsev-Petviashvili equation, Commun. Nonlinear Sci. Numer. Simul., 17(2012)(3), 1130-1141. Google Scholar