Abstract
Sato's theory of infinite dimensional Grassmannians, has been applied to explain the geometry of the K-P equation ([S; DJKM]), it has been used as a tool to study blow up behaviors and to regularize the solutions near the blow up [A-vM2]. The point is that realizing the K-P flow as a holomorphic flow of planes, enables one to follow what happens to the limiting planes as the equation in the original bad coordinates blows up. The blow-up behaviors are characterized by the various strata the orbit of planes visits in the Grassmannian. In this paper such ideas are applied to the N-periodic Toda flow (on periodic Jacobi matrices) which translates into a flow on the space of N-periodic flags of planes in the Grassmannians. Indeed here the N-periodic Toda flow amounts to N coupled KP equations with special interactions between time flows [U-T]. How such matrices blow up has been studied in [FI; F1-Ha; A-vM1] for arbitrary Lie algebras and Kac-Moody Lie algebras, whereas this paper focusses on regularizing the flow near the blow up locus; that is, on finding the boundary of isospectral sets. If N-periodic Jacobi matrices
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