Normal forms of dispersive scalar Poisson brackets with two independent variables

TL;DR

This paper classifies dispersive scalar Poisson brackets with two independent variables, revealing an infinite set of invariants distinguishing their Miura equivalence classes, unlike the single-variable case.

Contribution

It provides a complete classification of these brackets up to Miura transformations, introducing the concept of numerical invariants for the first time.

Findings

Miura classes are parametrized by infinitely many constants.

Explicit formulas for initial numerical invariants are derived.

Contrasts with the single-variable case where triviality was established.

Abstract

We classify the dispersive Poisson brackets with one dependent variable and two independent variables, with leading order of hydrodynamic type, up to Miura transformations. We show that, in contrast to the case of a single independent variable for which a well known triviality result exists, the Miura equivalence classes are parametrised by an infinite number of constants, which we call numerical invariants of the brackets. We obtain explicit formulas for the first few numerical invariants.