On an extension of the generalized BGW tau-function

Abstract

For an arbitrary solution to the Burgers--KdV hierarchy, we define the tau-tuple $({\tau }_1,{\tau }_2)$ of the solution. We show that the product ${\tau }_1{\tau }_2$ admits Buryak's residue formula. Therefore, according to Alexandrov's theorem, ${\tau }_1{\tau }_2$ is a tau-function of the KP hierarchy. We then derive a formula for the affine coordinates for the point of the Sato Grassmannian corresponding to the tau-function ${\tau }_1{\tau }_2$ explicitly in terms of those for ${\tau }_1$. Applications to the analogous open extension of the generalized BGW tau-function and to the open partition function are given.

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