Hamiltonian structures for general PDEs

TL;DR

This paper introduces a new geometric framework for constructing Hamiltonian operators applicable to a wide class of partial differential equations, demonstrated through classical examples like KdV and Camassa-Holm.

Contribution

It proposes a novel geometric approach to derive Hamiltonian structures for general PDEs, extending beyond traditional methods.

Findings

Framework successfully applied to KdV, Camassa-Holm, and Kupershmidt's deformation

Provides a systematic way to construct Hamiltonian operators for non-evolutionary PDEs

Enhances understanding of geometric structures underlying PDE Hamiltonian formulations

Abstract

We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and Kupershmidt's deformation of a bi-Hamiltonian system.