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

B. Kolev
2007 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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.
doi:10.1098/rsta.2007.2012 pmid:17360267 fatcat:hlbgtg4vozhdpm3xfjh7kwlxoa