Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 035, 26 pages      arXiv:1702.01227      https://doi.org/10.3842/SIGMA.2017.035

Liouville Correspondences between Integrable Hierarchies

Jing Kang a, Xiaochuan Liu a, Peter J. Olver b and Changzheng Qu c
a) Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi'an 710069, P.R. China
b) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
c) Center for Nonlinear Studies and Department of Mathematics, Ningbo University, Ningbo 315211, P.R. China

Received February 07, 2017, in final form May 22, 2017; Published online May 28, 2017

Abstract
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.

Key words: Liouville transformation; Miura transformation; bi-Hamiltonian structure; conservation law; Novikov equation; Degasperis-Procesi equation; Sawada-Kotera equation; Kaup-Kupershmidt equation.

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