Compatible pairs of Hamiltonian operators of the first and third orders
Paolo Lorenzoni, Stanislav Opanasenko, Raffaele Vitolo
TL;DR
This work addresses the problem of constructing bi-Hamiltonian hierarchies by coupling a first-order weakly nonlocal Hamiltonian operator $P$ with a third-order homogeneous operator $R_3$. It derives a complete, algebraic set of compatibility conditions via the variational Schouten bracket, showing the nonlocal tail $w^i_{\alpha j}u^j_x$ consists of Hamiltonian fluxes for $R_3$ and that the metric $g^{ij}$ obeys a Structure Formula $g^{ij}=\psi^i_\gamma Z^{\gamma j}+\psi^j_\gamma Z^{\gamma i}-c^{\alpha\beta}w^i_\alpha w^j_\beta$, with $Z^{\gamma j}$ linear in the fields. The fluxes $w^i_\alpha$ and the auxiliary potentials organize into Hamiltonian systems with respect to $R_3$, which must commute, enabling a finite-dimensional parametrization of admissible $P$ via algebraic relations. The authors illustrate the theory with a broad array of KdV-type and WDVV-type examples, including new WDVV cases in dimensions $N=4$ and $N=5$, thus demonstrating the practical applicability and computational feasibility of the method for high-dimensional systems. Overall, the work provides a rigorous, algebraic framework for identifying compatible pairs and offers a structured path to generate new bi-Hamiltonian hierarchies from higher-order and nonlocal operators.
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.