Bi-Hamiltonian structures of WDVV-type

TL;DR

This paper classifies nonlinear PDEs with bi-Hamiltonian structures similar to WDVV equations, providing new examples and an algorithm for computing Hamiltonian operators, advancing understanding of integrable systems.

Contribution

It introduces a classification of such PDEs in low-component cases, provides evidence for uniqueness in four components, and develops an algorithm for Hamiltonian operator computation.

Findings

Classified 2- and 3-component equations with bi-Hamiltonian structures.

Provided evidence for uniqueness of integrable systems in four components.

Constructed a six-component system with the desired bi-Hamiltonian structure.

Abstract

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin form, which are compatible. Using various equivalence groups, we classify such equations in two-component and three-component cases. In a four-component case, we add further evidence to the conjecture that there exists only one integrable system of the above type. Finally, we give an example of the six-component system with required bi-Hamiltonian structure. To streamline the symbolic computation we develop an algorithm to find the aforementioned Hamiltonian operators, which includes putting forward a conjecture on the structure of the metric parameterising the first-order Hamiltonian operator.