Research reports in Physics -Nonlinear Dynamics- Springer Verlag Berlin-Heidelberg-New York S. Carillo and O.Ragnisco, eds. pages 127-130, 1990 The Action-Angle Transformation for the Korteweg-deVries equation Sandra Carillo Dipartimento di M. M. M. per le Sc. Appl., Universtit´ a La Sapienza, Roma, Italy Benno Fuchssteiner Department of Mathematics, University of Paderborn, D 4790 Paderborn, Germany Abstract: For the multi-soliton solutions of the KdV (Korteweg-de Vries equation) a map from the action variables to the angle variables is constructed. The analysis pre- sented here is valid for nonlinear evolution equations admitting a recursion operator as well as a Lax operator. The method is based on the nonlinear link between the eigenvectors of these two operators. Since the action-angle map is recognized to be an infinitesimal symmetry generator of the corresponding interacting soliton equation the result also follows directly from the structural properties of the underlying dynamics. In case of the KdV this symmetry group generator can be found from the fact that it must generate a group in the kernel of the Miura transformation. As an example, to illustrate a more general method, we consider the KdV equation u t = u xxx +6uu x (1) whose hereditary recursion operator [5,8] is given by the Lenard operator [1] Φ(u)= D 2 +2u x D −1 +4u (2) where, as usual, D −1 denotes integration from −∞ to x. For this equation we consider only solutions vanishing rapidly at infinity. Further examples and detailed computations concerning the construction of action-angle transformations in general are comprised in [7]. As usual [4] we characterize the N-soliton manifold by M N = {u | there are c n s. t. N  n=1 c n K n (u)=0} . (3) According to the general theory [9,10,12], the recursion operator of the given nonlinear system follows to be doubly degenerated when restricted to M N . Furthermore, when the gradients of canonical action-angle variables are mapped by the hamiltonian formulation onto vector fields, then the eigenvectors of the recursion operator are obtained. Keeping in mind that the hamiltonian formulation of the KdV is given by the differential operator 1