A Grassmannian version of the Darboux transformation
type: Item
instance of: scholarly article
title: A Grassmannian version of the Darboux transformation
zbMATH Open document ID: 0936.35157
DOI: 10.1016/S0007-4497(99)80019-0
author: Jean Fastré
published in: Bulletin des Sciences Mathématiques
publication date: 2000-05-09T00:00:00Z
review text: In soliton theory, a Darboux transformation (DT) refers to the following process: First factorize a given second order operator \(L=(d/dx)^2-q\) as a product of two first order operators. Then, swap the order of these operators to obtain a new operator \(\widetilde{L}=(d/dx)^2-\widetilde{q}\). The new potential \(\widetilde{q}\) is given by \(\widetilde{q} = q - 2 (d/dx)^{2} (\log \phi) \), where \(\phi \) is an element in the null-space of \(L\). This procedure can be traced back at least to \textit{G.~Darboux}'s work [Leçons sur la théorie générale des surfaces {II}, Livre IV, Chapitre IX (1972; Zbl 0257.53001) and C. R. Acad. Sci., Paris 94, 1456-1459 (1882; JFM 14.0264.01)]. It has also been associated with the names of Moutard and Crum. Darboux transformations have been used extensively in soliton theory. One of the reasons for the importance of DT in this context is the following: If one starts with a potential \(q\) that is explicitly solvable, then one can construct by successive DT iterations an infinite family of potentials with the same property. Furthermore, if the original potential depends smoothly on a time variable and satisfies the Korteweg-de Vries (KdV) equation associated to isospectral deformations of the underlying operator \(L\), then suitable constants of integration can be found so that after each step of the DT process one gets a solution of KdV. An example where this method was employed with success can be found in Adler-Moser's construction of the rational solutions of the KdV hierarchy [\textit{M. Adler} and \textit{J. Moser}, Commun. Math. Phys. 61, 1-30 (1978; 428.35067)]. Another instance, which is directly related to the paper under review, can be found in the construction of the bispectral Schrödinger operators by \textit{J. J. Duistermaat} and \textit{F. A. Grünbaum} [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)]. In the work under review, the Darboux transformation is studied at the level of the Sato Grassmannian. The G…
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K30
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 37K40
zbMATH DE Number: 1309201
zbMATH Keywords: Grassmannian
zbMATH Keywords: Darboux transformation
zbMATH Keywords: Virasoro algebra
zbMATH Keywords: bispectral problem
zbMATH Keywords: JFM 14.0264.01
reviewed by: J. P. Zubelli
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1309201