Geometry of inhomogeneous Poisson brackets, multicomponent Harry Dym hierarchies and multicomponent Hunter-Saxton equations

TL;DR

This paper introduces a new class of multicomponent local Poisson structures, classifies them, and derives novel Harry Dym hierarchies and Hunter-Saxton equations, expanding understanding of integrable systems.

Contribution

It defines a natural class of multicomponent Poisson brackets, finds their normal forms, classifies two-component cases, and derives new integrable hierarchies and equations.

Findings

Classified two-component Poisson brackets up to point transformations.

Derived new Harry Dym hierarchies for multiple components.

Formulated new Hunter-Saxton equations differing from classical forms.

Abstract

We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the point transformation. From the description of Casimirs we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In two component case our HS equation differs from the well-known HS2 equation.