2 1 Ja n 19 99 Schrödinger flows on Grassmannians

The geometric non-linear Schrödinger equation (GNLS) on the complex Grassmannian manifold M is the evolution equation on the space C(R,M) of paths on M : Jγγt = ∇γxγx, where ∇ is the Levi-Civita connection of the Kähler metric and J is the complex structure. GNLS is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kähler form on M . It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schrödinger equation (MNLS). We construct via gauge transformations an isomorphism from C(R,M) to the phase space of the MNLS equation so that the GNLS flow corresponds to the MNLS flow. The existence of global solutions to the Cauchy problem for GNLS and the hierarchy of commuting flows follows from the correspondence. Direct geometric constructions show the flows are given by geometric partial differential equations, and the space of conservation laws has a structure of a non-abelian Poisson group. We also construct a hierarchy of symplectic structures for GNLS. Under pullback, the known order k symplectic structures correspond to the order k − 2 symplectic structures that we find. The shift by two is a surprise, and is due to the fact that the group structures depend on gauge choice. 1 Research supported in part by NSF Grant DMS 9626130 2 Research supported in part by Sid Richardson Regents’ Chair Funds, University of Texas system 1