EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS

Lijun Zhang; Jianming Zhang; Yuzhen Bai; Robert Hakl

doi:10.11948/20190061

2019 Volume 9 Issue 5

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Lijun Zhang, Jianming Zhang, Yuzhen Bai, Robert Hakl. EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1987-1998. doi: 10.11948/20190061

Citation:

Lijun Zhang, Jianming Zhang, Yuzhen Bai, Robert Hakl. EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(5): 1987-1998. doi: 10.11948/20190061

EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology Qingdao, Shandong 266590, China
2.
Department of Mathematics, school of science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China
3.
School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, China
4.
Institute of Mathematics, Branch in Brno, Czech Academy of Sciences, Zizkova, Brno, Czech Republic

Corresponding author: li-jun0608@163.com(L. Zhang)

Abstract

Abstract

The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations.
- wave-breaking equations /
- singular wave solutions /
- peakon solutions /
- singular line
MSC: 34C23, 34C60, 35B65, 35Q35

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References

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