Reciprocal transformations of generalized negative flows in integrable hierarchies

Abstract

We construct some reciprocal transformations for the first negative flows of the Korteweg–de Vries, the modified Korteweg–de Vries, the Ablowitz–Kaup–Newell–Segur, the Fordy–Gibbons, the Boussinesq and the modified Boussinesq hierarchies or their reductions. With the help of these reciprocal transformations, the first negative flows of the above hierarchies are converted to the Camassa–Holm type and their short-wave limit, the generalized short pulse type as well as some new equations.