Theoretical and Mathematical Physics, Vol. 105, No. 2, 1995 FACTORIZATION AND POISSON CORRESPONDENCES A.P. Fordy, A.B. Shabat, and A.P. Veselov The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general/actorization problem. Generalizations related to energy-dependent SchrSdinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows. Dedicated to the memory of Ira Dorfman 1. Introduction Multivalued mappings or correspondences appear in a very natural way in symplectic geometry. The standard way to determine a canonical transformation (p, q) ~ (P, Q) uses the generating function S(q, Q): OS OS P = -~q (q' Q)' P = - oq (q' Q)" (1) One can see that (1) does not determine a mapping but a correspondence, because one has to solve the first equation with respect to Q. This multi-valuedness brings some new features into the dynamics, particularly into the integrability theory for such mappings (see the review [1]). A general approach to integrable discrete Lagrangian systems was proposed in [2]. It is based on the factorization of matrix polynomials and can be applied to various systems like the Heisenberg lattice, ellipsoidal billiard, and discrete top. The basic scheme is as follows: There exists a matrix polynomial L(A) with matrix coefficients depending on the coordinates (p, q) or, more generally, on the point of the corresponding Poisson manifold, such that the dynamics of the system can be described as L(A) = A(A)B(A) ~-~ L'(A) = B(A)A(A), for an appropriate factorization problem. The fact that such a procedure leads to a symplectic (or Poisson) mapping is not obvious at all, but it is true in many important cases. The approach is not restricted to finite-dimensional examples. For the SchrSdinger operator L = -D 2 + u(x), the Darboux transformation [3] is known to be related to factorization [4]. In the context of the KdV equation, this also gives rise to the Miura map, which is known to be a Poisson map between the MKdV and KdV hierarchies. This construction has been generalized to higher-order Lax operators in [5, 6]. The Darboux transformation for the SchrSdinger operator corresponds to an interchange of factors and gives rise to a Poisson correspondence which leaves the (local) Poisson bracket of the MKdV hierarchy invariant. We should mention here the papers of Flaschka and McLaughlin [7] and Semenov-Tian-Shansky [8], where the Poisson properties of the Darboux-B~icklund transformation and the dressing transformation group were discussed, and the recent paper [9] where some explicit examples of Poisson mappings generated by shifts in nonlinear lattices are considered. We would like to emphasize that the Darboux transformation (DT) is Department of Applied Mathematical Studies and Centre for Nonlinear Studies, University of Leeds, Leeds LS2 9JT, UK, e-mail amt6apf@leeds.ac.uk; Landau Institute for Theoretical Physics, ul. Kosygina 2, 117334 Moscow, Russia, e-mail shabat@itp.ac.ru; Department of Mathematics and Mechanics, Moscow State University, 119899 Moscow, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 225-245, Novem- ber, 1995. Original article submitted November 29, 1994. 0040-5779/95/1052-1369512.50 (~) 1996 Plenum Publishing Corporation 1369