# American Institute of Mathematical Sciences

November  2018, 38(11): 5415-5442. doi: 10.3934/dcds.2018239

## On a new two-component $b$-family peakon system with cubic nonlinearity

 1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 2 Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China 3 School of Mathematical & Statistical Sciences, University of Texas Rio Grande Valley, 1201 W. University, Dr. Edinburg, Texas 78539, USA 4 College of Sciences, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Received  August 2017 Revised  November 2017 Published  August 2018

Fund Project: Authors to whom correspondence should be addressed through the following three E-mails: kaiyan@hust.edu.cn, zhijun.qiao@utrgv.edu, zhangmath@126.com

In this paper, we propose a two-component $b$-family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component $b$-family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically.

Citation: Kai Yan, Zhijun Qiao, Yufeng Zhang. On a new two-component $b$-family peakon system with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5415-5442. doi: 10.3934/dcds.2018239
##### References:

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##### References:
two-peakon solutions $u$ and $v$ given by (2.10) with $A_1 = 1$, $A_2 = 2$, $A_3 = 3$ and $c = 1$
Two-peakon solutions $u$ and $v$ given by (2.11) with $A_1 = 0$, $A_2 = 1$, $A_3 = 3$ and $c = 0$
Two-peakon solutions $u$ and $v$ given by (2.12) with $A_1 = 0$, $A_2 = -1$, $A_3 = -3$ and $c = \frac{1}{10}$
Conservation laws
 CH equation DP equation Novikov equation Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved? yes no yes Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved? yes yes no
 CH equation DP equation Novikov equation Is $\int_{\mathbb{R}}\, (u m)(t, x) d x$ conserved? yes no yes Is $\int_{\mathbb{R}}\, m(t, x) d x$ conserved? yes yes no
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