Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

We study an explicit correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy. A Liouville transformation between the isospectral problems of the two hierarchies also relates their respective recursion operators, and serves to establish the Liouville correspondence between their flows and Hamiltonian conservation laws. In addition, a novel transformation mapping the modified Camassa-Holm equation to the Camassa-Holm equation is found. Furthermore, it is shown that the Hamiltonian conservation laws in the negative direction of the modified Camassa-Holm hierarchy are both local in the field variables and homogeneous under rescaling.