Exponential asymptotics for discrete Painlevé equations

Abstract

In this thesis we study Stokes phenomena behaviour present in the solutions of both additive and multiplicative difference equations. Specifically, we undertake an asymptotic study of the second discrete Painlevé equation (dPII) as the independent variable approaches infinity, and consider the asymptotic behaviour of solutions of the q-Airy equation and the first q-Painlevé equation in the limits q□(→┴ ) 1 and n□(→┴ ) ∞. Exponential asymptotic methods are used to investigate Stokes phenomena and obtain uniform asymptotic expansions of solutions of these equations. In the first part of this thesis, we obtain two types of asymptotic expansions which describe vanishing and non-vanishing type solution behaviour of dPII. In particular, we show that both types of solution behaviour can be expressed as the sum of an optimally-truncated asymptotic series and an exponentially subdominant correction term. We then determine the Stokes structure and investigate Stokes behaviour present in these solutions. From this information we show that the asymptotic expansions contain one free parameter hidden beyond-all-orders and determine regions of the complex plane in which these asymptotic descriptions are valid. Furthermore, we deduce special asymptotic solutions which are valid in extended regions and draw parallels between these asymptotic solutions to the tronquée and tri-tronquée solutions of the second Painlevé equation. In the second part of this thesis, we then extend the exponential asymptotic method to q-difference equations. In our analysis of both the q-Airy and first q-Painlevé equations, we find that the Stokes structure is described by curves referred to as q-spirals. As a consequence, we discover that the Stokes structure for solutions of q-difference equations separate the complex plane into sectorial regions bounded by arcs of spirals rather than traditional rays.