Quicksilver Solutions of a q‐Difference First Painlevé Equation

Abstract

In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé 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 Painlevé equation (qPI), whose phase space (space of initial values) is a rational surface of type A7(1). We describe four families of almost stationary behaviors, 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 qPI, is also applicable to other q‐difference Painlevé equations.

Geometric aspects of Painlevé equations

“…By applying w to (5.26), we have 42) which give the precise form of w( f ) and w(g) observed in (5.5).…”

Nonlinear q-Stokes phenomena for q-Painlevé I

“…Motivated by Boutroux's study of the first Painlevé equation [7], the asymptotic behaviour of (1) in the limit |x| → ∞ has been considered in [28,30]. Joshi [28] showed that there exists a true solution satisfying w → 0 as |x| → ∞, which is asymptotic to a divergent series. However, [28] does not describe the Stokes switching behaviour that is typically associated with (divergent) asymptotic series.…”

“…Joshi [28] showed that there exists a true solution satisfying w → 0 as |x| → ∞, which is asymptotic to a divergent series. However, [28] does not describe the Stokes switching behaviour that is typically associated with (divergent) asymptotic series.…”

“…The asymptotic study of variants of the sixth q-Painlevé equation have been investigated by Mano [41] and Joshi and Roffelson [33] using the q-analogue of the isomonodromic deformation approach. The first q-Painlevé equation has also been investigated in [28,30]. In particular, Joshi [28] proves the existence of true solutions of (1), which are asymptotic to a divergent asymptotic power series in the limit |x| → ∞.…”

“…The first q-Painlevé equation has also been investigated in [28,30]. In particular, Joshi [28] proves the existence of true solutions of (1), which are asymptotic to a divergent asymptotic power series in the limit |x| → ∞. Such expansions are known to exhibit Stokes phenomena in the complex plane.…”