Elliptic asymptotic representation of the fifth Painlev\'e transcendents

Abstract

For the fifth Painlev\'e transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part may be understood to depend on the phase shift as a single integration constant, which is parametrised by monodromy data for the associated isomonodromy deformation. The other integration constant is contained in the error term or in a correction function. This paper contains corrections of the Stokes graph and of the related results in the early version.