Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators
The primary problem is as follows. A differential operator \(P = (d/dx)^ n + q_ 2 (x,t) (d/dx)^{n-1} + \cdots + q_ n(x,t)\) with \(x \in \mathbb{C}\), \(t = (t_ 1, t_ 2, \dots) \in \mathbb{C}^ \infty\) is subjected to the isospectral evolution \(\partial P/ \partial t_ k = [P^{k/n}_ +, P]\), and we ask what is the behaviour of its blow-up locus and how does one desingularize \(P\) near it. The answer is elementary for the KdV case \((n=2)\) and the authors thoroughly analyze analogous problems for the pseudodifferential operator \(L = P^{1/n}\) where \(L = d/dx + \sum^{- \infty}_{-1} a_ j (x,t) (d/dx)^ j\) is governed by \(\partial L/ \partial t_ k = [(L^ k)_ +, L]\). Following Sato, \(L = Sd/dxS^{- 1}\) where \[ S = \sum^ \infty_ 0 {p_ n (-\widetilde \partial) \tau (t) \over \tau (t)} (d/dt)^{-n} \] is operator involving a parameter function \(\tau\). The vanishing \(\tau(t^*) = 0\) indicates the singularities. Then, in terms of the wave function \(\psi (t,z)\) satisfying \(L \psi = z \psi\) and \(\partial \psi/ \partial t_ n = (L^ n)_ + \psi\), an infinite-dimensional Grassmannian is introduced with cellular decomposition (the Birkhof strata) parametrized by certain Young diagrams which permits to describe the vanishing of \(\tau\) hence measures the depth of the singularity of \(L\). The paper is rather comprehensive and involves a large amount of technical tools (Schur polynomials, Cauchy identities, Fay identities, Bäcklund-Darboux transformations, etc.) which are of independent interests.
scientific article; zbMATH DE number 706490