Geometric aspects of non-homogeneous 1+0 operators
Marta Dell'Atti, Alessandra Rizzo, Pierandrea Vergallo
TL;DR
The paper develops a geometric framework for non-homogeneous Hamiltonian operators of type $(1+0)$, formed by a Dubrovin–Novikov first-order part plus an ultralocal zero-order part. It provides a complete Casimir classification in degenerate and non-degenerate cases for $n=2$ and $n=3$ components, and analyzes bi-Hamiltonian structures by introducing bi-pencils that couple compatible first-order pencils with ultralocal pencils; it also connects these structures to Nijenhuis geometry, offering both concrete 2-component classifications and preliminary 3-component results, including a KdV-like example. The work highlights how non-homogeneous operators admit rich geometric interpreations (bi-pencils, Killing–Yano pencils, and Nijenhuis tensors) and points to future directions in higher components and deeper Nijenhuis-theoretic descriptions. Overall, it advances a unified, geometry-driven view of $(1+0)$ Hamiltonian pairs and their integrability properties for multicomponent hydrodynamic-type systems.
Abstract
Led by the key example of the Korteweg-de Vries equation, we study pairs of Hamiltonian operators which are non-homogeneous and are given by the sum of a first-order operator and an ultralocal structure. We present a complete classification of the Casimir functions associated with the degenerate operators in two and three components. We define tensorial criteria to establish the compatibility of two non-homogeneous operators and show a classification of pairs for systems in two components, with some preliminary results for three components as well. Lastly, we study pairs composed of non-degenerateop erators only, introducing the definition of bi-pencils. First results show that the considered operators can be related to Nijenhuis geometry, proving a compatibility result in this direction in the framework of Lie algebras.