Birkhoff strata, Bäcklund transformations, and regularization of isospectral operators. (English) Zbl 0814.35114

Adler, Mark; van Moerbeke, Pierre

Adv. Math. 108, No. 1, 140-204 (1994).

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

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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

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, Cauchy identities, Fay identities, Bäcklund-Darboux transformations, etc.) which are of independent interests.

Reviewer: J.Chrastina (Brno)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds

Keywords:

isospectral equations; wave functions; Young diagrams; Grassmannian; Schur polynomial