On bi-Hamiltonian deformations of exact pencils of hydrodynamic type

Abstract

In this paper we are interested in non trivial bi-Hamiltonian deformations of the Poisson pencil $ω_λ=ω_2+λω_1=uδ'(x-y)+\f{1}{2}u_xδ(x-y)+λδ'(x-y)$. 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.

34 pages, revised version. Proof of Theorem 16 completely rewritten due to an error in the first version

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• Bihamiltonian cohomology of KdV brackets
• Deformations of non semisimple Poisson pencils of hydrodynamic type
• On integrable conservation laws
• The bi-Hamiltonian cohomology of a scalar Poisson pencil
• Inherited structures in deformations of Poisson pencils