Riemann theta function solutions of the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy

Abstract

The Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of Lenard recursion equations. By using the characteristic polynomial of Lax matrix for the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy, we introduce a three-sheeted Riemann surface Km−1 of arithmetic genus m−1, from which we derive the associated Baker–Akhiezer function, the meromorphic function on it. Based on the theory of Riemann surfaces, we construct the explicit Riemann theta function representations of the Baker–Akhiezer function, the meromorphic function, and solutions for the Caudrey–Dodd–Gibbon–Sawada–Kotera hierarchy.