Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions

Abstract

A new set of canonically conjugate variables 1s introduced for the penoclic Korteweg-de Vries equation and the periodic Toda lattice. These vanables are used for reducing both equations to a nonlinear system which can be integrated in terms of theta functions. It becomes clear that the discrete and the continuous problems are, in a sense, Jsomorph1c. Action variables are defined by loop integrals, and the basic oscillation frequencies are computed. In the infinite-period limit, these action variables tend to the ones used in the canonical description of the inverse-scattering solution method.