Classification of bi-Hamiltonian pairs extended by isometries

The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

EQUILIBRIUM THERMODYNAMICS OF SIMPLE HYDRODYNAMIC TYPE SYSTEM

A generating thermodynamic functional for the one-dimensional, one-component hydrodynamic type system with the Hamiltonian density au2+bu4is determined. Correlation functions of this model are considered.