On the polynomial \(\tau\)-functions for the KP hierarchy and the bispectral property
type: Item
instance of: scholarly article
title: On the polynomial \(\tau\)-functions for the KP hierarchy and the bispectral property
zbMATH Open document ID: 0757.35074
DOI: 10.1007/BF00430001
published in: Letters in Mathematical Physics
publication date: 1992-09-27T00:00:00Z
review text: The author proves that if \(\tau(x_ 1,x_ 2,\dots)\) is a Schur function then the function \[ w(x_ 1,x_ 2,\dots;k)=\exp\left(\sum_{j=1}^ \infty x_ j k^ j\right)\tau(x_ 1-(1/k),x_ 2-(1/2k^ 2),\dots)/\tau(x_ 1,\dots) \] satisfies the two differential equations \(Q(x_ 1,\partial_{x_ 1})w=kw\) and \(A(k,\partial_ k)w=\theta(x_ 1)w\) (the so-called bispectral property) where the operator \(Q\) gives rise to the Kadomtsev-Petviashvili hierarchy of integrable evolution equations and \(\theta(x_ 1)\) is a polynomial of \(x_ 1\). This result is a generalization of that obtained by \textit{J. J. Duistermaat} and \textit{F. A. Grünbaum} [Commun. Math. Phys. 103, 177-240 (1986; Zbl 0625.34007)] for the Korteweg-de Vries equation.
reviewed by: E. D. Belokolos
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 34L25
zbMATH DE Number: 64269
zbMATH Keywords: rational solutions of soliton equations
zbMATH Keywords: bispectral property
zbMATH Keywords: Korteweg-de Vries equation
author: J. P. Zubelli
MaRDI profile type: Publication
cites work: Differential equations in the spectral parameter
cites work: Solitons and infinite dimensional Lie algebras
cites work: Q3947818
cites work: Rational solutions of nonlinear evolution equations, vertex operators, and bispectrality
cites work: Differential equations in the spectral parameter, Darboux transformations and a hierarchy of master symmetries for KdV
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description: scientific article; zbMATH DE number 64269