The bi-Hamiltonian cohomology of a scalar Poisson pencil

TL;DR

This paper calculates the bi-Hamiltonian cohomology for scalar Poisson pencils, revealing specific isomorphisms and vanishing results, which enhances understanding of integrable systems and their cohomological properties.

Contribution

It introduces a spectral sequence method to compute bi-Hamiltonian cohomology for dispersionless Poisson pencils, extending results known from the KdV case.

Findings

Cohomology is isomorphic to  for (p,d)=(0,0)

Cohomology is isomorphic to C^\u221e() for (p,d)=(1,1), (2,1), (2,3), (3,3)

Cohomology vanishes otherwise

Abstract

We compute the bi-Hamiltonian cohomology of an arbitrary dispersionless Poisson pencil in a single dependent variable using a spectral sequence method. As in the KdV case, we obtain that $BH^p_d(\hat{F}, d_1,d_2)$ is isomorphic to $\mathbb{R}$ for $(p,d)=(0,0)$, to $C^\infty (\mathbb{R})$ for $(p,d)=(1,1)$, $(2,1)$, $(2,3)$, $(3,3)$, and vanishes otherwise.