Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

TL;DR

This paper surveys bi-Hamiltonian systems on the dual of the Lie algebra of circle vector fields, focusing on affine structures and their role in integrable shallow water wave equations.

Contribution

It provides a comprehensive overview of bi-Hamiltonian structures involving affine modifications and their connection to integrable shallow water models.

Findings

Identification of affine Lie-Poisson structures in shallow water equations

Analysis of the integrability of Euler equations with these structures

Clarification of the role of constant structures in bi-Hamiltonian systems

Abstract

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. We investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures called affine or modified Lie-Poisson structures are involved in the integrability of certain Euler equations that arise as models of shallow water waves.