# Miura Reciprocal transformation for two integrable hierarchies in 1+1 dimensions

We here present two different hierarchies of PDEs in 1+1 dimensions whose first and second member are the shallow water wave Camassa-Holm and Qiao equations, correspondingly. These two hierarchies can be transformed by reciprocal methods into the Calogero-Bogoyanlevski-Schiff equation (CBS) and its modified version (mCBS), respectively. Considering that there exists a Miura transformation between the CBS and mCBS, we shall obtain a relation between the initial hierarchies by means of a composition of a Miura and reciprocal transforms.

Article:
Miura Reciprocal transformation for two integrable hierarchies in 1+1 dimensions
Authors:
P. G. Estevez, C. Sardon
Journal:
ArXiv preprint
Year:
2013
URL:
http://arxiv.org/abs/1301.3636