B\"acklund transformations for the Camassa-Holm equation

Abstract

The B\"acklund transformation (BT) for the Camassa-Holm (CH) equation is presented and discussed. Unlike the vast majority of BTs studied in the past, for CH the transformation acts on both the dependent and (one of) the independent variables. Superposition principles are given for the action of double BTs on the variables of the CH and the potential CH equations. Applications of the BT and its superposition principles are presented, specifically the construction of travelling wave solutions, a new method to construct multi-soliton, multi-cuspon and soliton-cuspon solutions, and a derivation of generating functions for the local symmetries and conservation laws of the CH hierarchy.