Poisson cohomology for nonhomogeneous Hamiltonian operators

Compute the Poisson–Lichnerowicz cohomology groups H(P) for nonhomogeneous Hamiltonian operators associated with multidimensional evolutionary PDEs, including operators of the 1+0 form P^{ij}(u)=g^{ij\alpha}(u)\,\partial_\alpha+b^{ij\alpha}_k(u)\,u^k_\alpha+\omega^{ij}(u), since the Poisson cohomology for nonhomogeneous structures remains uncomputed.

Background

The paper studies multidimensional first-order quasi-linear systems and their Hamiltonian structures via Poisson vertex algebras and cotangent coverings. A key theme is the relationship between necessary compatibility conditions and Poisson cohomology: compatibility selects symmetries of the Hamiltonian operator, and non-Hamiltonian symmetries correspond to nontrivial first Poisson cohomology.

While the Poisson cohomology for homogeneous operators (notably of hydrodynamic type) has been investigated in prior work, the authors note that the cohomology for nonhomogeneous operators has not yet been computed. They provide partial insights by relating H^1 for certain 1+0 operators to bi-Hamiltonian cohomology of the homogeneous components and illustrating the N=D=2 case, but the general computation of Poisson cohomology for nonhomogeneous structures remains open.