The conservation of the Hamiltonian structures in the deformations of the Whitham systems

TL;DR

This paper explores how Hamiltonian structures are preserved when deforming Whitham systems related to the KdV equation, providing a scheme for averaging Poisson structures in such deformations.

Contribution

It demonstrates that classical Hamiltonian brackets lead to deformed Dubrovin-Novikov brackets in the deformed Whitham system, offering a general averaging scheme for Poisson structures.

Findings

Hamiltonian properties are conserved in the deformed Whitham system.

Both Gardner-Zakharov-Faddeev and Magri brackets produce deformed Dubrovin-Novikov brackets.

The approach provides a general scheme for averaging Poisson structures.

Abstract

We consider the construction of the deformed Whitham system for the KdV-equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then, that both the Gardner - Zakharov - Faddeev and the Magri brackets give the deformed Dubrovin - Novikov brackets (the brackets of Dubrovin - Zhang type) for the deformed Whitham system constructed by our procedure. The general approach used in the paper gives a scheme for the averaging of the Poisson structures in the general situation.