Quicksilver Solutions of a q-difference first Painlev\'e equation

TL;DR

This paper introduces new unstable solutions called quicksilver solutions for a q-difference version of the first Painlevé equation, analyzing their asymptotic behavior and divergence properties.

Contribution

It presents the first formal power series and asymptotic analysis of quicksilver solutions for a q-Painlevé equation, extending methods to other similar equations.

Findings

Derived formal power series for solutions

Showed divergence of the series and existence of true solutions

Applicable method to other q-Painlevé equations

Abstract

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a $q$-difference Painlev\'e equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlev\'e equation ($q\Pon$), whose phase space (space of initial values) is a rational surface of type $A_7^{(1)}$. We describe four families of almost stationary behaviours, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain $q$-domain. The method, while demonstrated for $q\Pon$, is also applicable to other $q$-difference Painlev\'e equations.