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 B\"{a}cklund 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 Schr\"{o}dinger 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 Painlev\'{e} transcendent.