Third order nonlinear Hamiltonian systems: Some remarks on the the action-angle transformation

Multi-soliton solutions of third order nonlinear evolution equations admitting a recursion operator as well as a Lax operator are here considered. Specifically, results previouly obtained by the author, which give a method to construct the action-angle transformation on the so-called “multi-soliton” manifold are briefly discussed. Crucial to achieve such a result is the nonlinear link between the eigenvectors of the Lax and the recursion operator. Furthermore, the action-angle transformation can be recognized to be an infinitesimal symmetry generator of the corresponding interacting soliton equation; thus, it can be also obtained via the direct analysis of the structural properties of the underlying dynamics. Work partially supported by the G.N.F.M. of C.N.R.and by the M.U.R.S.T. project Geometria e Fisica