Degenerate first-order Hamiltonian operators of hydrodynamic type in 2D

TL;DR

This paper classifies degenerate first-order Hamiltonian operators of hydrodynamic type in 2D and investigates the integrability of related 2+1 Hamiltonian systems using hydrodynamic reductions.

Contribution

It provides a complete classification of degenerate two- and three-component Hamiltonian operators in 2D and analyzes their associated integrable systems.

Findings

Complete classification of degenerate 2D Hamiltonian operators

Identification of conditions for degeneracy in operators

Analysis of integrability of systems derived from classified operators

Abstract

First-order Hamiltonian operators of hydrodynamic type were introduced by Drubrovin and Novikov in 1983. In 2D, they are generated by a pair of contravariant metrics $g$, $\tilde{g}$ and a pair of differential-geometric objects $b$, $\tilde{b}$. If the determinant of the pencil $g+\lambda \tilde{g}$ vanishes for all $\lambda$, the operator is called degenerate. In this paper we provide a complete classification of degenerate two- and three-component Hamiltonian operators. Moreover, we study the integrability, by the method of hydrodynamic reductions, of 2+1 Hamiltonian systems arising from the structures we classified.