Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type

TL;DR

This paper characterizes compatible Dubrovin-Novikov Hamiltonian operators using Lie derivatives, linking them to flat manifolds and integrable bi-Hamiltonian hierarchies, and provides explicit solutions via inverse scattering.

Contribution

It establishes a geometric criterion for compatibility of Hamiltonian operators and connects this to flat manifolds and integrable systems of hydrodynamic type.

Findings

Compatibility is characterized by Lie derivatives along vector fields.

Defines a class of special flat manifolds related to compatible operators.

Provides explicit solutions for integrable systems using inverse scattering.

Abstract

We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result gives a natural invariant definition of the class of special flat manifolds corresponding to all the class of compatible Dubrovin--Novikov Hamiltonian operators (the Frobenius--Dubrovin manifolds naturally belong to this class). There is an integrable bi-Hamiltonian hierarchy corresponding to every flat manifold of this class. The integrable systems are also studied in the present paper. This class of integrable systems is explicitly given by solutions of the nonlinear system of equations, which is integrated by the method of inverse scattering problem.