Equations with Peakon Solutions in the Negative Order Camassa-Holm Hierarchy

摘要

The negative order Camassa-Holm (CH) hierarchy consists of nonlinear evolution equations associated with the CH spectral problem. In this paper, we show that all the negative order CH equations admit peakon solutions; the Lax pair of the -order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders and are obtained. Indeed, for , the peakons of -order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for -order CH equation when is odd are much different from the CH peakons; we present the case as an example.