Darboux transformations of the Camassa-Holm type systems

Abstract

We propose a unified approach to establish Darboux-Bäcklund transformations of the Camassa-Holm (CH) type equations. The method simplifies the approach presented by Xia, Zhou and Qiao [J. Math. Phys. 57, 103502 (2016)] to obtain soliton solutions of the CH equation by the Darboux transformation, because it does not need to solve spectral problem of the CH equation at λ=0 which is a singular point in the auxiliary problem, and avoids computing asymptotic properties of the wave functions. We also study many other CH type equations such as the 2-CH, the modified CH, the Degasperis-Procesi and the Novikov via the method. Especially, we present three Bäcklund transformations of the 2-CH system.