Exponential asymptotics for discrete Painlevé equations Steven Luu A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy School of Mathematics and Statistics University of Sydney May 2018 CONTENTS Abstract............................................................................. vi Statement of Originality.............................................................. viii Acknowledgements .................................................................. ix Chapter 1. Introduction ............................................................ 1 1.1. Painlevé Equations. 1 1.2. Discrete Painlevé equations from the continuous Painlevé equations . 3 1.3. Discrete Painlevé Equations . 5 1.3.1. Types of difference equations . 8 1.3.2. Applications of discrete Painlevé equations . 8 1.4. Some asymptotic results of the Painlevé equations . 8 1.5. Thesis Outline. 9 Chapter 2. Exponential Asymptotics ................................................ 11 2.1. Classical asymptotic power series . 13 2.1.1. Truncation of asymptotic series expansions . 16 2.2. Superasymptotics . 17 2.2.1. Darboux’s Theorem . 21 2.2.2. Optimally-truncated error via Borel summation . 23 2.3. Hyperasymptotics . 26 2.3.1. Applications of Hyperasymptotics . 28 2.4. Stokes Phenomenon . 29 2.5. Example: the hyper-Airy equation . 32 2.5.1. Rescaling the hyper-Airy equation . 32 2.5.2. Asymptotic series expansion . 33 2.5.3. Late-order terms analysis . 34 2.5.3.1. Calculating the singulant, χ ................................................ 35 2.5.3.2. Calculating the prefactor, G ................................................ 35 2.5.3.3. Calculating the value of γi ................................................. 36 2.5.3.4. Calculating the value of Λ .................................................. 37 2.5.4. Optimal truncation . 38 2.5.5. Analysis of the remainder using exponential asymptotics. 39 2.5.5.1. Stokes smoothing . 41 iii iv CONTENTS 2.5.6. Stokes structure . 43 2.6. Further Ideas . 46 Chapter 3. Additive difference equations ............................................ 48 3.1. Introduction . 48 3.2. Discrete Painlevé II . 51 3.3. Scalings of discrete Painlevé II . 52 3.4. Vanishing asymptotics of discrete Painlevé II . 53 3.4.1. Asymptotic series expansions . 53 3.4.2. Late-order terms analysis . 55 3.4.2.1. Calculating the singulant, χ ................................................ 56 3.4.2.2. Calculating the prefactor, F ................................................ 57 3.4.2.3. Calculating the value of k .................................................. 57 3.4.2.4. Calculating the value of Λ .................................................. 58 3.4.3. Vanishing behaviour remainder analysis . 60 3.4.3.1. Stokes smoothing . 61 3.4.4. Asymptotic expansions in terms of the original variables . 63 3.4.5. Stokes structure . 64 3.5. Non-vanishing asymptotics of discrete Painlevé II. 67 3.5.1. Asymptotic analysis of non-vanishing solutions . 68 3.5.2. Singularities of the leading order behaviour . 68 3.5.3. Late-order terms analysis . 70 3.5.3.1. Calculating the singulant, η ................................................ 71 3.5.4. Stokes structure . 72 3.6. Numerical computation for discrete Painlevé II . 76 3.7. Conclusions . 77 Chapter 4. Multiplicative difference equations ....................................... 79 4.1. Introduction . 81 4.1.1. q-Borel summation methods . 83 4.1.2. q-Stokes phenomenon . 84 4.1.3. Describing Stokes phenomena for q-difference equations. 85 4.2. Riemann sheets : Reverting the transformations. 86 4.3. A q-analogue of the Airy equation . 88 4.3.1. Asymptotic analysis of the q-Airy equation . 90 4.3.2. q-periodic functions . 91 4.3.3. WKB phase factor analysis for q-Airy. 93 4.3.4. WKB amplitude factor analysis for q-Airy . 94 4.3.5. Complete asymptotic power series . 95 4.3.6. Stokes structure of q-Airy. 97 4.4. The first q-Painlevé equation . 102 4.4.1. Known results for q-Painlevé I . 102 4.4.2. Asymptotic analysis of the first q-Painlevé equation . 105 CONTENTS v 4.4.3. Asymptotic Series Expansion . 106 4.5. Type A solutions of q-Painlevé I. 110 4.5.1. Late-Order Terms Behaviour . 111 4.5.1.1. Calculating the singulant, χ ................................................ 112 4.5.1.2. Calculating the prefactor, U.................................................112 4.5.1.3. Calculating the value of γ1..................................................114 4.5.2. q-Painlevé I inner problem . 115 4.5.3. Analysis of the remainder using exponential asymptotics. 118 4.5.3.1. Analysis for the homogeneous remainder equation. 119 4.5.3.2. Stokes Smoothing for q-PI ................................................. 120 4.5.4. Stoke structure for type A solutions . 122 4.6. Type B solutions. 132 4.6.1. Stoke Geometry for type B solutions . 134 4.7. Connection between type A and type B solutions and the nonzero and vanishing asymptotic solutions of q-Painlevé I . ..