On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

TL;DR

This paper studies non-trivial bi-Hamiltonian deformations of a specific Poisson pencil of hydrodynamic type, proving their unobstructed nature up to eighth order and revealing geometric properties related to symplectic leaves.

Contribution

It constructs higher-order deformations of the Poisson pencil, demonstrating their polynomial form and establishing geometric conditions for their existence.

Findings

Deformations are unobstructed up to eighth order.

Linear relations encode tangency to symplectic leaves.

Deformations maintain polynomial structure in derivatives.

Abstract

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $\omega_{\lambda}=\omega_2+\lambda \omega_1=u\delta'(x-y)+\f{1}{2}u_x\delta(x-y)+\lambda\delta'(x-y)$. Deformations are generated by a sequence of vector fields $\{X_2, X_4,...\}$, where each $X_{2k}$ is homogenous of degree $2k$ with respect to a grading induced by rescaling. Constructing recursively the vector fields $X_{2k}$ one obtains two types of relations involving their unknown coefficients: one set of linear relations and an other one which involves quadratic relations. We prove that the set of linear relations has a geometric meaning: using Miura-quasitriviality the set of linear relations expresses the tangency of the vector fields $X_{2k}$ to the symplectic leaves of $\omega_1$ and this tangency condition is equivalent to the exactness of the pencil $\omega_{\lambda}$. Moreover, extending the results of [17], we construct the non trivial deformations of the Poisson pencil $\omega_{\lambda}$, up to the eighth order in the deformation parameter, showing therefore that deformations are unobstructed and that both Poisson structures are polynomial in the derivatives of $u$ up to that order.