Liouville correspondences between integrable hierarchies

In this paper, we study explicit correspondences between the integrable Novikov and Sawada–Kotera hierarchies, and between the Degasperis–Procesi and Kaup–Kupershmidt hierarchies. We show how a pair of Liouville transformations between the isospectral problems of the Novikov and Sawada–Kotera equations, and the isospectral problems of the Degasperis–Procesi and Kaup–Kupershmidt equations relate the corresponding hierarchies, in both positive and negative directions, as well as their associated conservation laws. Combining these results with the Miura transformation relating the Sawada–Kotera and Kaup– Kupershmidt equations, we further construct an implicit relationship which associates the Novikov and Degasperis–Procesi equations.