On the polynomial \(\tau\)-functions for the KP hierarchy and the bispectral property
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.
scientific article; zbMATH DE number 64269
scientific article