Group Theoretic Formulation of the Segal-Wilson Approach to Integrable Systems with Applications

A general group theoretic formulation of integrable systems is presented. The approach generalizes the discussion of the KdV equations of Segal and Wilson based on ideas of Sato. The starting point is the construction of commuting flows on the group via left multiplication with elements from an abelian subgroup. The initial data are then coded by elements, called abstract scattering data, in a certain coset space. The resulting equations of motion are then derived from a suitably formulated Maurer-Cartan equation (zero curvature condition) given an abstract Birkhoff factorization. The resulting equations of motion are of the Zakharov-Shabat type. In the case of flows periodic in x-space, the integrals of motion have a natural group theoretic interpretation. A first example is provided by the generalized nonlinear Schrödinger equation, first studied by Fordy and Kulish with integrals of motion which may be local or nonlocal. A suitable reduction gives the mKdV equations of Drinfeld and Sokolov. On the level of abstract scattering data the generalized Miura transformation from solutions of the mKdV equations to the KdV type equations is then just the canonical map from a coset space to a double coset space.

This group theoretic approach is related to the algebraic geometric discussion of integrable systems via an affine map from the abelian group describing flows restricted to a suitable set of abstract scattering data, called algebraic geometric, onto a connected component of the Picard variety.