The associated Camassa-Holm equation and the KdV equation

Abstract

Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Holm equation by a reciprocal transformation, was introduced by Schiff, who derived Bäcklund transformations by a loop group technique and used these to obtain some simple soliton and rational solutions. We show how the ACH equation is related to Schrödinger operators and KdV, and describe how to construct solutions of ACH from tau-functions of the KdV hierarchy. Rational, N-soliton and elliptic solutions are considered, as well as exact solutions given by a particular case of the third Painlevé transcendent.