Miura-reciprocal transformations and localizable Poisson pencils

TL;DR

This paper establishes a correspondence between deformations of localizable semisimple Poisson pencils under Miura-reciprocal transformations and deformations under Miura transformations, showing each class contains a local representative.

Contribution

It demonstrates that equivalence classes of deformations under Miura-reciprocal transformations have local representatives and correspond to classes under Miura transformations.

Findings

Equivalence classes contain local representatives.

One-to-one correspondence between classes under different transformation groups.

Clarifies the structure of deformations of Poisson pencils.

Abstract

We show that the equivalence classes of deformations of localizable semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura-reciprocal group contain a local representative and are in one-to-one correspondence with the equivalence classes of deformations of local semisimple Poisson pencils of hydrodynamic type with respect to the action of the Miura group.