Elliptic asymptotic representation of the fifth Painlevé transcendents

Abstract

For the fifth Painlevé 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.

Figures (10)

• Figure 2.1: Loops $l_0$, $l_1$, $l_{\infty}$ for $-\pi/2 <\arg x <\pi/2$
• Figure 2.2: Cycles $\mathbf{a},$$\mathbf{b}$ on $\Pi^*$
• Figure 3.1: Loops $\hat{l}_0$ and $\hat{l}_1$ on the $\lambda$-plane
• Figure 3.2: Cuts on $\mathbb{P}_+$ and the limit Stokes graph on $\mathbb{P}_+ ^{\infty}$
• Figure 4.1: Limit Stokes graph

Theorems & Definitions (10)

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