Third order nonlinear Hamiltonian systems: some remarks on 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 obtained in [2,8], which give a method to construct the action-angle transformation on the so-called “multi-soliton” manifold [15], 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.