Liouville Correspondences between Integrable Hierarchies

[摘要] 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.

[关键词] Liouville transformation;Miura transformation;bi-Hamiltonian structure;conservation law;Novikov equation;Degasperis–Procesi equation;Sawada–Kotera equation;Kaup–Kupershmidt equation