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.

Linear Aspects in the Theory of Solitons and Nonlinear Integrable Equations

“…It turns out that indeed for the N-soliton case a description in terms of local densities is possible, but that this description becomes more difficult if continuous parts of the recursion operator become important. Altogether the analysis is involved and tedious (see [16] for details, see also [34]). The essential tool for carrying out this analysis are the so called mastersymmetries (see [11]).…”

“…There are two different and simple ways to give a direct construction of these additional eigenvectors. One is given by use of partial derivatives of the field function u with respect to asymptotic data (see [16]). Here we present another method, a method which I chose because it leads in a natural way to some results of the next chapter.…”

Hamiltonian Structure and Integrability

“…(ii) Given some symplectic J, then elements K ∈ L such that JK is closed are called inverse-hamiltonian 8 (with respect to J). …”

“…One only needs some kind of Poincaré Lemma to show that. 8 Observe that for invertible Θ or J the notions hamiltonian and inverse-hamiltonian do coincide. This is the reason why in the finite dimensional theory the notion inverse-hamiltonian does not appear, since there symplectic forms are usually assumed to be non-degenerate.…”

“…Instead of attempting here a general definition we shall show that, under reasonable conditions, the existence of a compatible bi-hamiltonian pair leads to complete integrability on finite dimensional reductions given by the corresponding symmetry group generators. We will give a survey on the results which can be obtained in that direction, for details the reader is referred to [8].…”