| Citation: |
Yanggeng Fu, Jibin Li. BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN THREE MODIFIED CAMASSA-HOLM EQUATIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1051-1061. doi: 10.11948/20200347
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BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN THREE MODIFIED CAMASSA-HOLM EQUATIONS
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Abstract
This paper studies traveling wave solutions of three modified Camassa-Holm equations posed by Anco and Recio in 2019. The corresponding traveling system is a singular system of second class. The bifurcations of traveling wave solutions in the parameter space are investigated from a dynamical systems theoretical point of view. The existence of solitary wave solution, periodic wave solution and so-called $M$-shape-solution are proved.
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References
[1] S. Anco, and D. Kraus, Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation, Discrete Continuous Dyn. Syst. A, 2018, 38, 4449-4465. doi: 10.3934/dcds.2018194 [2] S. Anco, and E. Recio, Accelerating dynamical peakons and their behaviour, Discrete Continuous Dyn. Syst. A, 2018, 39, 6131-6148. [3] S. Anco, and E. Recio, Conserved norms and related conservation laws for multi-peakon equations, J. Phys. A: Math. Theor., 2018, 51, 065203. doi: 10.1088/1751-8121/aaa0e0 [4] S. Anco, and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A: Math. Theor., 2019, 52, 125203. doi: 10.1088/1751-8121/ab03dd [5] Y. Fu, J. Li, Exact stationary-wave solutions in the standard model of the Kerr-Nonlinear optical fiber with the Bragg grating, J. Appl. Anal. Comput., 2017, 7(3), 1177-1184. [6] J. Li, Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. [7] J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, International Journal of Bifurcation and Chaos, 2007, 17, 4049-4065. doi: 10.1142/S0218127407019858 [8] J. Li and Z. Qiao, Explicit soliton solutions of the Kaup-Kupershmidt equation through the dynamical system approach, J. Appl. Anal. Comput., 2011, 1(2), 243-250. [9] J. Li and Y. Zhang, Exact M/W-shape wave solutions determined by a singular traveling wave equation, Nonlinear Analysis: RWA, 2009, 10, 1797-1802. doi: 10.1016/j.nonrwa.2008.02.016 [10] Z. Qiao, A new integrable equation with cuspons and W/M-shape peaks solitons, J. Math. Physics, 2006, 47, 112701-9. doi: 10.1063/1.2365758 -
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Yanggeng Fu, Jibin Li. BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS IN THREE MODIFIED CAMASSA-HOLM EQUATIONS[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 1051-1061. doi: 10.11948/20200347
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Figure 1. The bifurcations of phase portraits of system (1.10) for a fixed
$ c>0, p\geq 1 $ . -
Figure 2. The bifurcations of phase portraits of system (1.11) for a fixed
$ c>0, p\geq 2 $ . -
Figure 3. The graph of
$ (\phi^2-y^2)^p-c = 0 $ and the vector fields defined by (2.1) and (1.10) -
Figure 4.
$ M- $ shape wave solution of system (2.1) in the$ (\xi, \phi)- $ plane when$ 0<g<g^*. $ -
Figure 5. The
$ M $ -shape wave consisting of three breaking waves of equation (1.4)