2019 Volume 9 Issue 5
Article Contents

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.

# EXPLICIT PEAKON SOLUTIONS TO A FAMILY OF WAVE-BREAKING EQUATIONS

• 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.

MSC: 34C23, 34C60, 35B65, 35Q35

###### 通讯作者: 陈斌, bchen63@163.com
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沈阳化工大学材料科学与工程学院 沈阳 110142

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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.