Classification of degenerate non-homogeneous Hamiltonian operators

TL;DR

This paper classifies degenerate non-homogeneous Hamiltonian operators, crucial for understanding certain scalar equations, focusing on systems with 2 and 3 components.

Contribution

It provides a complete classification of degenerate Hamiltonian operators in low-dimensional systems, advancing the understanding of their structure and applications.

Findings

Complete classification of 2-component degenerate operators

Complete classification of 3-component degenerate operators

Insights into the structure of Hamiltonian scalar equations

Abstract

In this paper, the authors investigate non-homogeneous Hamiltonian operators composed of a first-order Dubrovin-Novikov operator and an ultralocal one. The study of such operators turns out to be fundamental for the inverted system of equations associated with a class of Hamiltonian scalar equations. Often, the involved operators are degenerate in the first-order term. For this reason, a complete classification of the operators with degenerate leading coefficient in systems of 2 and 3 components is presented.