# American Institute of Mathematical Sciences

November  2018, 38(11): 5505-5521. doi: 10.3934/dcds.2018242

## Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity

 Department of mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Received  September 2017 Revised  November 2017 Published  August 2018

In this paper, we consider the orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, which admits the single peakons and multi-peakons. We firstly show the existence of the single peakon and prove two useful conservation laws. Then by constructing certain Lyapunov functionals, we give the proof of stability result of peakons in the energy space $H^1(\mathbb{R})$-norm.

Citation: Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242
##### References:

show all references

##### References:
 [1] Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090 [2] Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 [3] David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 [4] Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194 [5] Ying Fu. A note on the Cauchy problem of a modified Camassa-Holm equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2011-2039. doi: 10.3934/dcds.2015.35.2011 [6] Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065 [7] Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067 [8] Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305 [9] Joachim Escher, Tony Lyons. Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach. Journal of Geometric Mechanics, 2015, 7 (3) : 281-293. doi: 10.3934/jgm.2015.7.281 [10] Min Zhu, Shuanghu Zhang. Blow-up of solutions to the periodic modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7235-7256. doi: 10.3934/dcds.2016115 [11] Min Zhu, Ying Wang. Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 645-661. doi: 10.3934/dcds.2017027 [12] Xingxing Liu, Zhijun Qiao, Zhaoyang Yin. On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1283-1304. doi: 10.3934/cpaa.2014.13.1283 [13] Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243 [14] Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047 [15] Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883 [16] Milena Stanislavova, Atanas Stefanov. Attractors for the viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 159-186. doi: 10.3934/dcds.2007.18.159 [17] Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871 [18] Guangying Lv, Mingxin Wang. Some remarks for a modified periodic Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1161-1180. doi: 10.3934/dcds.2011.30.1161 [19] Kai Yan, Zhaoyang Yin. On the initial value problem for higher dimensional Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1327-1358. doi: 10.3934/dcds.2015.35.1327 [20] Kai Yan. On the blow up solutions to a two-component cubic Camassa-Holm system with peakons. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4565-4576. doi: 10.3934/dcds.2020191

2019 Impact Factor: 1.338