Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators
type: Item
instance of: scholarly article
title: Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators
zbMATH Open document ID: 0814.35114
published in: Advances in Mathematics
publication date: 1995-01-23T00:00:00Z
review text: 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.
reviewed by: Jan Chrastina
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J72
zbMATH DE Number: 706490
zbMATH Keywords: isospectral equations
zbMATH Keywords: wave functions
zbMATH Keywords: Young diagrams
zbMATH Keywords: Grassmannian
zbMATH Keywords: Schur polynomial
author: Mark Adler
author: Pierre van Moerbeke
MaRDI profile type: Publication
full work available at URL: aima.1994.1070
OpenAlex ID: W1977031017
DOI: 10.1006/AIMA.1994.1070
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description: scientific article; zbMATH DE number 706490