Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations

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 the assumption that \(\mathcal P_k\) has Darboux coordinates (Darboux–Poisson bracket) and is nondegenerate. For such brackets, we obtain general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples), and provide the description of the Casimirs, using a purely algebraic procedure. In the two-component case, we completely classify such brackets up to a 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 the two-component case, our HS equation differs from the well-known HS2 equation.