Dressing transformations and regularization of the BKP hierarchy

This thesis comprises a study of the regularity properties of the BKP hierarchy of mathematical physics using the theory of the hierarchy due to M. Sato. Recall that Sato has shown that the hierarchy is best understood by coordinatizing via the so-called wave function {dollar}Psi(t){dollar}--a formal version of spectral data for a linear integro-differential equation. These wave functions form an infinite dimensional version of a Grassmann manifold and the BKP solutions become integral curves of a simple vector field on this Grassmannian.; In constructing this manifold one must complete the BKP flows and thereby allow singular solutions to the equations. We answer the question of how to desingularize these solutions in a canonical way. There are three major points. Firstly, the Grassmannian is a homogeneous space for an infinite dimensional orthogonal group and comes equipped with a stratification by finite codimensional cells; these cells control the level of singularity of a solution. Furthermore the strata are indexed by the collection of all distinct partitions of integers, and the coefficients in Laurent expansions of singular solutions can be expressed in terms of combinatorial objects related to said distinct partitions. The second point is that the action of the Clifford algebra suggests a natural method of desingularization. Specifically, there are special one parameter subgroups (the Backlund transformations) which are transverse to all strata at the identity and locally about the identity lie in a less singular stratum. One iterates these geometric transformations until one has a wave function which lies in the non-singular stratum. The last and most difficult point is to synthesize points one and two; that is to say one wants an analytic understanding of the process of regularization by Backlund transformations. Here the crucial point is to understand the Laurent expansion of the identities between wave functions implied by their relation via Backlund transformation. This involves proving an extremely complicated identity involving the combinatorial objects alluded to in point one.