Exact solutions of the associated Camassa-Holm equation

Abstract

Recently the associated Camassa-Holm (ACH) equation, related to the Fuchssteiner-Fokas-Camassa-Holm equation by a hodograph 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 the KdV hierarchy, and use this connection to obtain exact solutions (rational and N-soliton solutions). More generally, we show that solutions of ACH on a constant background can be obtained directly from the tau-functions of known solutions of the KdV hierarchy on a zero background. We also present exact solutions given by a particular case of the third Painlevé transcendent.