On the global asymptotic analysis of a q-discrete Painlevé equation

In this thesis we make effective the global asymptotic analysis of a nonlinear q-difference Painlevé equation, whose initial value space is a rational surface of type A1(1) according to Sakai's classification. This equation can be thought of as an eight parameter generalisation of the celebrated sixth Painlevé equation, where the reduction to the differential equation goes via the continuum limit of its symmetric form. The first part of the thesis is concerned with the local asymptotic analysis of solutions near the critical points of the q-difference equation, the origin and infinity. A conjecturally complete list of possible asymptotic behaviours is found near both critical points. It is shown that, upon taking the continuum limit, the list essentially coincides with that of critical behaviours of solutions of the sixth Painlevé equation, obtained by Guzzetti. In the second part of the thesis, the integrability of the equation under consideration is exploited, to solve the nonlinear connection problem, which entails explicitly relating the critical behaviours of solutions near the two different critical points. This is done by employing a q-analog of the isomonodromic deformation method to a q-difference Lax pair devised by Yamada. The direct monodromy problem is solved, both for critical behaviours near the origin and infinity, by showing that near the critical points, the connection problem of Yamada's system factorises in two copies of a simpler connection problem, which can be solved explicitly. Comparison of the results, leads to explicit parametric connection formulae for critical behaviours of the q-difference Painlevé equation

Exponential asymptotics for discrete Painlevé equations

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

Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation

Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation

Special functions arising from discrete Painlevé equations: A survey

AbstractThis article is a survey on recent studies on special solutions of the discrete Painlevé equations, especially on hypergeometric solutions of the q-Painlevé equations. The main part of this survey is based on the joint work [K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Hypergeometric solutions to the q-Painlevé equations, IMRN 2004 47 (2004) 2497–2521, K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta, Y. Yamada, Construction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005 24 (2005) 1439–1463] with Kajiwara, Masuda, Ohta and Yamada. After recalling some basic facts concerning Painlevé equations for comparison, we give an overview of the present status of studies on difference (discrete) Painlevé equations as a source of special functions

Three approaches to detecting discrete integrability

A class of discrete equations is considered from three perspectives corresponding to three measures of the complexity of solutions: the (hyper-) order of meromorphic solutions in the sense of Nevanlinna, the degree growth of iterates over a function field and the height growth of iterates over the rational numbers. In each case, low complexity implies a form of singularity confinement which results in a known discrete Painlev\'e equation

Integrable Discrete Equations Derived by Similarity Reduction of the Extended Discrete KP Hierarchy

We consider the extended discrete KP hierarchy and show that similarity reduction of its subhierarchies lead to purely discrete equations with dependence on some number of parameters together with equations governing deformations with respect to these parameters. It is written down discrete equations which naturally generalize the first discrete Painlevé equation dPI in a sense that autonomous version of these equations admit the limit to the first Painlevé equation. It is shown that each of these equations describes Bäcklund transformations of Veselov-Shabat periodic dressing lattices with odd period known also as Noumi-Yamada systems of type A2(n-1)⁽¹⁾

A q-discrete Analogue of the Third Painlevé Equation and its Linear Problem

In this thesis we investigate the rational and Riccati type special solutions for particular parameter values of a q-discrete analogue of the third Painlevé equation, with rational surface A(1) 5 and affine Weyl group (A2 + A1)(1). The general solutions of this equation are highly transcendental in nature. We work closely with an associated system of discrete linear equations, which we refer to as ‘the linear problem.’We demonstrate that the linear problem can be solved both in terms of q-Gamma functions and series expansions for different parameter values of our discrete Painlevé equation. By developing a Schlesinger transformation, which transforms a series expansion of the linear problem for one parameter value to another series expansion of the linear problem for another parameter value, we are able to develop determinantal representations of the rational and Riccati type special solutions. These determinantal forms appear different to those discovered previously by Kajiwara. This technique has only been used to develop the determinantal forms of two other continuous and discrete Painlevé equations and hence the results presented here further indicate its potential

Studies on the geometry of Painlevé equations

This thesis is a collection of work within the geometric framework for Painlevé equations. This approach was initiated by the Japanese school, and is based on studying Painlevé equations (differential or discrete) via certain rational surfaces associated with affine root systems. Our work is grouped into two main themes: on the one hand making use of tools and techniques from the geometric framework to study problems from applications where Painlevé equations appear, and on the other hand developing and extending the geometric framework itself. Differential and discrete Painlevé equations arise in a wide range of areas of mathematics and physics, and we present a general procedure for solving the identification problem for Painlevé equations. That is, if a differential or discrete system is suspected to be equivalent to one of Painlevé type, we outline a method, based on constructing the associated surfaces explicitly, for identifying the system with a standard example, in which case known results can be used, and demonstrate it in the case of equations appearing in the theory of orthogonal polynomials. Our results on the geometric framework itself begin with an observation of a new class of discrete equations that can described through the geometric theory, beyond those originally defined by Sakai in terms of translation symmetries of families of surfaces. To be precise, we build on previous studies of equations corresponding to non-translation symmetries of infinite order (so-called projective reductions, with fewer parameters than translations of the same surface type) and show that Sakai’s theory allows for integrable discrete equations to be constructed from any element of infinite order in the symmetry group and still have the full parameter freedom for their surface type. We then also make the first steps toward a geometric theory of delay-differential Painlevé equations by giving a description of singularity confinement in this setting in terms of mappings between jet spaces