The action-angle transformation for interacting solitons and the dynamics of eigenfunctions for soliton equations Benno Fuchssteiner Boris Konopelchenko ∗ University of Paderborn D 4790 Paderborn Sandra Carillo Dipartimento di M.M.M. per le Sc. Appl. Universita La Sapienza I 00161 Roma Abstract A method is presented which allows the explicit construction of the gradients of action and angle variables related to the dynamics of soliton eigenfunction equations. The interacting soliton equations, as well as the Lax-pair eigenfunctions, related to a number of known completely integrable systems are taken as examples to illustrate the method. Among these are the Korteweg de Vries, the mKdV, the nonlinear Schr¨ odinger equation and the ZS-AKNS-system. 1 Introduction Completely integrable flows on infinite dimensional manifolds are generally called soliton equations because, under suitable boundary conditions at infinity, solu- tions often decompose asymptotically into traveling waves. These asymptotically emerging traveling waves are termed solitons and if a solution decomposes com- pletely into solitons, it is termed a multisoliton. Here, by complete decomposition we mean that there is some suitable energy-norm such that all the energy is car- ried by the asymptotic solitons. Interaction of these solitons has been widely studied and, indeed, the very name soliton has been chosen referring to them since reductions of these systems ∗ Permanent address: Institute of Nuclear Physics, Novosibirsk -90, 630090, USSR This author is grateful to the Deutsche Forschungsgemeinschaft for providing support and to Paderborn University for the kind hospitality 1 Rendiconti di Matematica, Serie VII, 11, p.351-376, 1991