The Hamiltonian structure of the Camassa-Holm equation (Q1355787)

First, the author presents a detailed derivation of the Camassa-Holm equation \(u_t- u_{xxt}+ 3uu_x= 2u_x u_{xx}+ uu_{xxx}\) which describes the dispersive solitary waves on shallow water. Then, by using the properties of the Green-Naghdi shallow water equations, it is shown that the Camassa-Holm equation can be written in two Hamiltonian forms and possesses an infinite number of functionally independent conservations laws.