A Grassmannian version of the Darboux transformation
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 Grassmannian approach to the study of soliton equations, in particular to the Kadomtsev-Petviashvilli hierarchy, was one of several key contributions of the Kyoto school to soliton theory. The work under review proposes an interesting construction of the Darboux method that works at the level of \(\tau\)-functions. It also generalizes an earlier construction of Adler-van Moerbeke. One of the key features of such new description is that it allows the construction of rational solutions of the KdV hierarchy and seems to fit also well the framework of the bispectral problem. The work concludes by applying the results to the bispectral problem and computing the corresponding \(\tau\)-functions. It also discusses a number of connections with important objects in soliton theory such as the Virasoro algebra, Fermion-Boson equivalence, and W-symmetries.
scientific article; zbMATH DE number 1309201