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On tau-functions for the KdV hierarchy
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For an arbitrary solution to the KdV hierarchy, the generating series of logarithmic derivatives of the tau-function of the solution can be expressed by the basic matrix resolvent via algebraic manipulations. Based on this we develop in this paper two new formulae for the generating series by introducing a pair of wave functions of the solution. Applications to the Witten--Kontsevich tau-function, to the generalized Br\'ezin--Gross--Witten (BGW) tau-function, as well as to a modular deformation of the generalized BGW tau-function which we call the Lam\'e tau-function are also given.
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Virasoro Constraints for Drinfeld-Sokolov hierarchies and equations of Painlev\'{e} type
A unified tau cover and Virasoro constraints for Drinfeld-Sokolov hierarchies over arbitrary affine Kac-Moody algebras yield Painleve-type equations with affine Weyl group actions.
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