Exact Poisson pencils, $\tau$-structures and topological hierarchies

TL;DR

This paper explores the role of exact Poisson pencils within Dubrovin-Zhang's framework for integrable PDEs, revealing that their exactness in the semisimple case corresponds to constant central invariants.

Contribution

It establishes a precise link between the exactness of Poisson pencils and the constancy of central invariants in integrable hierarchies.

Findings

Exact Poisson pencils are characterized by constant central invariants.

In the semisimple case, exactness and constancy of invariants are equivalent.

The work deepens understanding of the geometric structures underlying integrable PDEs.

Abstract

We discuss, in the framework of Dubrovin-Zhang's perturbative approach to integrable evolutionary PDEs in 1+1 dimensions, the role of a special class of Poisson pencils, called exact Poisson pencils. In particular we show that, in the semisimple case, exactness of the pencil is equivalent to the constancy of the so-called "central invariants" of the theory that were introduced by Dubrovin, Liu and Zhang.