The monoidal transformation by Painlevé divisor and resolution of the poles of the Toda lattice

Abstract

One of the feature of the integrable systems is that all forward things are determined by its initial conditions. It is also well known that the Toda lattice has poles in finite time. Thus the behavior of the blowing up of the Toda lattice have to be governed by its initial conditions. Through the Painlevé analysis it is revealed that the blowing up themselves are characterized by the Weyl group [Flaschka and Haine]. In this paper we study how the behavior at the blowing up point is governed by its initial condition. For this purpose we realize the Painlevé divisor as the analytic variety. By virtue of this description we can compactify level set by using monoidal transformation by Painlevé divisor. The study of Painlevé divisor and compactification in this paper bring us the more precise informations on the blowing up of the Toda lattice than which has ever been obtained.