Isomonodromic deformatiion with an irregular singularity and hyperelliptic curve

Abstract

In this paper, we extend the result of Kitaev and Korotkin to the case where a monodromy-preserving deformation has an irregular singularity. For the monodromy-preserving deformation, we obtain the $\tau$-function whose deformation parameters are the positions of regular singularities and the parameter $t$ of an irregular singularity. Furthermore, the $\tau$-function is expressed by the hyperelliptic $\Theta$ function moving the argument $\z$ and the period $\B,$ where $t$ and the positions of regular singularities move $z$ and $\B,$ respectively.