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.
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The authors thank the referees for valuable comments and suggestions. J. Kang acknowledges the support and hospitality during her stay at the University of Minnesota, where this work was completed. The work of J. Kang is supported by NSF-China Grant 11471260 and the Foundation of Shannxi Education Committee-12JK0850. The work of X.C. Liu is supported by NSF-China Grant 11401471 and Ph.D. Programs Foundation of Ministry of Education of China-20136101120017 and the Grant under Shannxi Province 2013JQ1001. The work of P.J. Olver is partially supported by NSF Grant DMS-1108894. The work of C.Z. Qu is supported by the NSF-China Grant-11471174 and NSF of Ningbo Grant-2014A610018.
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- Hamiltonian conservation law
- Liouville transformation
- Local conservation law
- Modified Camassa–Holm hierarchy
- Modified KdV hierarchy
- Scaling homogeneity
- Tri-Hamiltonian duality