Nonlinear $q$-Stokes phenomena for $q$-Painlevé I

Abstract

We consider the asymptotic behaviour of solutions of the first $q$-difference Painlevé equation in the limits $|q|\to 1$ and $n\to \infty$. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the Stokes curves are described by curves known as $q$-spirals.

arXiv comment31 pages, 29 figures

References in corpus (5)

• Riemann--Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement
• Multiple orthogonal polynomials associated with an exponential cubic weight
• Unique positive solution for an alternative discrete Painlevé I equation
• Stokes Phenomena in Discrete Painlevé I
• Analytic solutions of $q$-$P(A_1)$ near its critical points

Cited by in corpus (1)

• Capturing the cascade: a transseries approach to delayed bifurcations