Equations with peakon solutions in the negative order Camassa-Holm hierarchy (Q1629153)

Summary: 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 \(N\)-order CH equation given by the hierarchy is compatible with its peakon solutions. Special peakon-antipeakon solutions for equations of orders \(- 3\) and \(- 4\) are obtained. Indeed, for \(N \leq - 2\), the peakons of \(N\)-order CH equation can be constructed explicitly by the inverse scattering approach using Stieltjes continued fractions. The properties of peakons for \(N\)-order CH equation when \(N\) is odd are much different from the CH peakons; we present the case \(N = - 3\) as an example.