GEOMETRY AND ACTION-ANGLE VARIABLES OF MULTI SOLITON SYSTEMS

Abstract

For all completely integrable nonlinear hamiltonian systems which have a localized hereditary recursion operator, a complete action-angle variable representation is given for the multisoliton manifolds. Here multisoliton manifolds are defined as reductions with respect to suitable linear sums of symmetry generators. The embedding of these multisoliton manifolds, into the manifold of all solutions, is described in terms of the construction of its tangent bundle. The basis vectors of the respective tangent spaces are given by local densities. This local geometrical description of the tangent bundle turns out to be independent of the special structure of the particular equation under consideration. The principal tool for finding the necessary geometrical quantities are the canonical commutation relations for the so called mastersymmetries. These relations reflect the hereditary structure. All mastersymmetries turn out to be elements of the tangent space. Although the mastersymmetries, in the case under consideration, principally cannot be hamiltonian, suitable integrating factors are found which make them hamiltonian on the reduced manifold. So, up to suitable linear combinations, the mastersymmetries are shown to correspond to the angle variables. The action-angle-structure found in this way is put into one-to-one correspondence with the spectrum of the recursion operator. The spectrum of this operator is shown to be of multiplicity two and all its eigenvectors are explicitly constructed. Again, this construction is of a canonical nature, i.e., independent of the particular equation under consideration. For vanishing boundary conditions the given action-angle-structure is compared to the asymptotic data (speeds and phases), and the gradients of these global asymptotic data are given in terms of local quantities. It turns out that for all times during the evolution the derivatives of the field function with respect to any particular asymptotic datum yields an eigenvector of the recursion operator. Thus a method is given for reconstructing the spectral resolution of the recursion operator by partial derivatives. This method yields new methods of solution for other equations (for example the singularity equation and the Harry Dym equation). The superposition formula for phase shifts is shown to hold in all generality for the systems under consideration. Several examples are given. An extensive comparison of the present results with the work of others is carried out.