Poisson bracket on 1-forms and evolutionary partial differential equations

TL;DR

This paper introduces a new Poisson bracket on 1-forms in the context of infinite jet spaces and demonstrates its application to Hamiltonian structures in integrable hierarchies, connecting generalized and standard Hamiltonian formalisms.

Contribution

It defines a Poisson bracket on 1-forms on jet spaces and applies it to show Hamiltonian properties of hierarchies in F-manifolds, bridging generalized and classical Hamiltonian theories.

Findings

Defined a Poisson bracket on 1-forms on jet spaces.

Proved the bracket satisfies Poisson properties.

Connected generalized Hamiltonian structures to standard ones under certain metric conditions.

Abstract

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, \mathbb{R}^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that certain hierarchies appearing in the framework of $F$-manifolds with compatible flat connection $(M, \nabla, \circ)$ are Hamiltonian in a generalized sense. Moreover, we show that if a metric $g$ compatible with $\nabla$ is also invariant with respect to $\circ$, then this generalized Hamiltonian set-up reduces to the standard one.