The Boundary Condition for Some Isomonodromy Equations

Abstract

In this article, we study a special class of Jimbo-Miwa-Mori-Sato isomonodromy equations, which can be seen as a higher-dimensional generalization of Painlev\'e VI. We first construct its convergent $n\times n$ matrix series solutions satisfying certain boundary condition. We then use the Riemann-Hilbert approach to prove that the resulting solutions are almost all the solutions. Along the way, we find a shrinking phenomenon of the eigenvalues of the submatrices of the generic matrix solutions in the long time behaviour.