Inherited structures in deformations of Poisson pencils

In this paper we study some properties of bi-Hamiltonian deformations of Poisson pencils of hydrodynamic type. More specifically, we are interested in determining those structures of the fully deformed pencils that are inherited through the interaction between structural properties of the dispersionless pencils (in particular exactness or homogeneity) and suitable finiteness conditions on the central invariants (like polynomiality). This approach enables us to gain some information about each term of the deformation to all orders in $\epsilon$. Concretely, we show that deformations of exact Poisson pencils of hydrodynamic type with polynomial central invariants can be put, via a Miura transformation, in a special form, that provides us with a tool to map a fully deformed Poisson pencil with polynomial central invariants of a given degree to a fully deformed Poisson pencil with constant central invariants to all orders in $\epsilon$. In particular, this construction is applied to the so called $r$-KdV-CH hierarchy that encompasses all known examples with non-constant central invariants. As far as homogeneous Poisson pencils of hydrodynamic type is concerned, we prove that they can also be put in a special form, if the central invariants are homogeneous polynomials. Through this we can compute the homogeneity degree about the tensorial component appearing in each order in $\epsilon$, namely the coefficient of the highest order derivative of the $\delta$.

💡 Research Summary

The paper investigates bi‑Hamiltonian deformations of Poisson pencils of hydrodynamic type, focusing on how certain structural properties of the dispersionless (ε = 0) limit are inherited by the fully deformed pencil when the central invariants satisfy finiteness conditions, in particular polynomiality. The authors consider two distinguished classes of dispersionless pencils: (i) exact pencils, for which there exists a Liouville vector field e such that ℒₑ ω₁ = 0 and ℒₑ ω₂ = ω₁, and (ii) homogeneous pencils, for which an Euler vector field E satisfies ℒ_E ω₁ = (d − 2) ω₁ and ℒ_E ω₂ = (d − 1) ω₂, where d is the charge of the underlying Frobenius manifold.

In the exact case the Lie derivative with respect to e commutes with the Poisson cohomology differential d_{ω₁} and satisfies ℒₑ ∘ d_{ω₂} = d_{ω₂} ∘ ℒₑ + d_{ω₁}. Consequently, if the central invariants are constant, the whole deformed pencil remains exact; if they are polynomial of degree k, the authors prove that a Miura transformation can be chosen so that the deformed pencil assumes a “normal form” in which each ε‑order term is homogeneous with respect to e with weight k + 1. This normal form provides an explicit map that sends a pencil with polynomial central invariants of degree k to a pencil with constant central invariants, preserving the exactness property at every order in ε.

For homogeneous pencils the analogous relations are ℒ_E ∘ d_{ω₁} = (d − 2) d_{ω₁} ∘ ℒ_E and ℒ_E ∘ d_{ω₂} = (d − 1) d_{ω₂} ∘ ℒ_E, leading to a combined identity ℒ_E ∘ (d_{ω₁} d_{ω₂}) = (2d − 3) (d_{ω₁} d_{ω₂}) ∘ ℒ_E. When the central invariants are homogeneous polynomials of degree D, each ε^{2k}‑order contribution Q^{(2k)} to the deformed pencil satisfies
ℒ_E Q^{(2k)} =