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

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.

arXiv comment23 pages

References in corpus (3)

• An integrable shallow water equation with peaked solitons
• Integrability of invariant metrics on the diffeomorphism group of the circle
• Lie Groups and mechanics: an introduction

Cited by in corpus (3)

• On the N=2 Supersymmetric Camassa-Holm and Hunter-Saxton Equations
• The Frobenius-Virasoro algebra and Euler equations
• Euler Equations Related to the Generalized Neveu-Schwarz Algebra