| Volume 9, Number 5, 2019, Pages 1987-1998 DOI：10.11948/20190061 |
| Explicit Peakon solutions to a family of wave-breaking equations |
| Lijun Zhang,Jianming Zhang,Yuzhen Bai,Robert Hakl |
| Keywords:wave-breaking equations, singular wave solutions, peakon solutions, singular line. |
| 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. |
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