Classification of 1+0 two-dimensional Hamiltonian operators
Alessandra Rizzo
TL;DR
The paper addresses the classification of Hamiltonian operators in two spatial dimensions that are sums of a first-order operator and a Poisson tensor (non-homogeneous, 1+0 type) for 2- and 3-component systems, including degenerate and nondegenerate leading coefficients. It applies the multidimensional hydrodynamic-type framework, requiring Hamiltonianity of both the first-order part $P^{ij}$ and the ultralocal part $\omega^{ij}$, and enforces the compatibility conditions (via $T^{ijk\alpha}$) to obtain complete local classifications. For 2D with two components, it completes the degenerate classification by starting from Savoldi’s degenerate homogeneous forms; for 2D with three components, it treats both nondegenerate (canonical forms $P_3$, $P_4$) and degenerate (ranked by $g_\lambda$, forms $P_5$–$P_{24}$) cases, delivering explicit Hamiltonianity conditions on the ultralocal part. The results provide a rigorous foundation for identifying Hamiltonian (and potential bi-Hamiltonian) structures in non-homogeneous quasilinear systems, enabling systematic integrability analysis and future extensions via cotangent coverings and higher-dimensional classifications.
Abstract
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three components, analyzing both the cases of degenerate and non degenerate leading coefficients.