Compatible pairs of Hamiltonian operators of the first and third orders

TL;DR

This paper derives algebraic compatibility conditions for pairs of Hamiltonian operators of first and third order, revealing new insights into bi-Hamiltonian structures in integrable PDEs and providing novel examples, including WDVV equations.

Contribution

It establishes algebraic compatibility conditions for first and third order Hamiltonian operators, advancing the understanding of bi-Hamiltonian structures in integrable systems.

Findings

Compatibility conditions are purely algebraic.

First-order operator determined by commuting conservation laws.

New examples related to WDVV equations.

Abstract

We compute general compatibility conditions between a weakly nonlocal homogeneous Hamiltonian operator and a third-order homogeneous Hamiltonian operator. Such operators determine a bi-Hamiltonian structure for many integrable PDEs (Korteweg--De Vries, Camassa--Holm, dispersive water waves, Dym, WDVV and others). Remarkably, the full set of conditions is purely algebraic and the first-order operator is completely determined by commuting systems of conservation laws that are Hamiltonian with respect to a third-order operator. We illustrate the above results with several examples, some of which, concerning WDVV equations, are new.