On solutions for the b-family of peakon equations

Abstract

We investigate a family of peakon equations, labelled by two parameters $b$ and $\kappa$, all of which admit one-peakon solutions in a unified form. The well known Camassa-Holm equation and Degasperis-Procesi equation are derived from the b-family peakon equations by choosing $b=2$ and 3, respectively. For all values of $b$, their two peakon-type solutions are shown in difference form and in weak sense. At the same time, the dynamic behaviors are shown in difference phenomenons, such as the peaked waves collide elastically in the case $b=2$, while the peaked waves collide inelastically in other cases.