Asymptotics of Discrete Painlevé V transcendents via the Riemann–Hilbert Approach

Authors


Yuqiu Zhao, Department of Mathematics, Sun Yat-sen University, GuangZhou 510275, China; e-mail: stszyq@mail.sysu.edu.cn

Abstract

We study a system of discrete Painlevé V equations via the Riemann–Hilbert approach. We begin with an isomonodromy problem for dPV, which admits a discrete Riemann–Hilbert problem formulation. The asymptotics of the discrete Riemann–Hilbert problem is derived via the nonlinear steepest descent method of Deift and Zhou. In the analysis, a parametrix is constructed in terms of specific Painlevé V transcendents. As a result, the asymptotics of the dPV transcendents are represented in terms of the PV transcendents. In the special case, our result confirms a conjecture of Borodin, that the difference Schlesinger equations converge to the differential Schlesinger equations at the solution level.

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