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

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.

31 pages, final version to appear in Physica D: Nonlinear Phenomena

References in corpus (1)

• On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

Cited by in corpus (12)

• $F$-manifolds, multi-flat structures and Painlevé transcendents
• Bihamiltonian Cohomologies and Integrable Hierarchies II: the Tau Structures
• Computing with Hamiltonian operators
• A Dubrovin-Frobenius manifold structure of NLS type on the orbit space of $B_n$
• Deformations of non semisimple Poisson pencils of hydrodynamic type
• Dispersionless (3+1)-dimensional integrable hierarchies
• Variational Bihamiltonian Cohomologies and Integrable Hierarchies II: Virasoro symmetries
• On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations
• Poisson pencils: reduction, exactness, and invariants
• Inherited structures in deformations of Poisson pencils
• Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds
• Low dimensional bihamiltonian structures of topological type