Poisson bracket on 1-forms and evolutionary partial differential equations

Abstract

We introduce a bracket on 1-forms defined on ${\mathcal{J}}^{\infty }(S^1,{\mathbb{R}}^n)$, 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.

35 pages

References in corpus (2)

• From Darboux-Egorov system to bi-flat $F$-manifolds
• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains

Cited by in corpus (4)

• Darboux-Egorov system, bi-flat $F$-manifolds and Painlevé VI
• Deformations of non semisimple Poisson pencils of hydrodynamic type
• On integrable conservation laws
• Purely non-local Hamiltonian formalism, Kohno connections and $\vee$-systems