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Advances in Mathematical Physics
Volume 2018 (2018), Article ID 1679625, 11 pages
Research Article

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

School of Mathematical Sciences, Tongji University, Shanghai 200092, China

Correspondence should be addressed to Fengfeng Dong

Received 26 December 2017; Accepted 1 February 2018; Published 28 February 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 Fengfeng Dong and Lingjun Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.