Bihamiltonian Cohomologies and Integrable Hierarchies II: the Tau Structures

Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures.

💡 Research Summary

The paper investigates a distinguished class of bihamiltonian structures of hydrodynamic type, namely flat exact semisimple bihamiltonian structures, and establishes a deep connection with Frobenius manifolds, principal hierarchies, and tau‑structures.

1. Flat exact bihamiltonian structures.
A bihamiltonian pair ((P_{1},P_{2})) of hydrodynamic type is called exact if there exists a vector field (Z) such that (