Bi-Hamiltonian structures of KdV type

TL;DR

This paper classifies compatible bi-Hamiltonian structures related to the KdV equation, revealing new integrable systems with non-trivial invariants, expanding the understanding of Hamiltonian operators in integrable systems.

Contribution

It introduces a classification of compatible Hamiltonian operator trios, leading to new bi-Hamiltonian systems beyond known examples.

Findings

Recovered known bi-Hamiltonian examples

Identified new integrable systems with non-zero invariants

Established a framework for classifying Hamiltonian operator trios

Abstract

Combining an old idea of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators we classify compatible trios of two-component homogeneous Hamiltonian operators. The trios yield pairs of compatible bi-Hamiltonian operators whose structure is a direct generalization of the bi-Hamiltonian pair of the KdV equation. The bi-Hamiltonian pairs give rise to multi-parametric families of bi-Hamiltonian systems. We recover known examples and we find new integrable systems whose central invariants are non-zero; this shows that new examples are not Miura-trivial.