On the hierarchy of the Landau-Lifshitz equation (Q1068400)

The Landau-Lifshitz equation \(\vec S_ t=\vec S\times \vec S_{xx}+\vec S\times J\vec S\), \(J=diag(J_ 1,J_ 2,J_ 3)\), \(| \vec S| =1\), is considered. The complete integrability of this equation, describing spin waves in an anisotropic ferromagenet, was proved by E. K. Sklyanin. He found infinitely many constants of motion via the expansion of the logarithm of the scattering data with respect to the spectral parameter. However it is difficult to get the explicit formulae from this expansion. The author obtains here the explicit form of all constants of motion using a Lie algebra approach which appeared successful in the cases of Benjamin-Ono and Kadomtsev-Petviashvili equations. The conserved densities are given by the recursion formula \(\bar H_{n+1}(\vec S)=\{\bar H_ n(\vec S),\bar T(\vec S)\}\) where T(\(\vec S)=1/2x((\vec S,J\vec S)-(\vec S_ x,\vec S_ x))\), and \(\{\), \(\}\) denotes a certain Poisson bracket in the space of densities.