An Overview of Geometric Asymptotic Analysis of Continuous and Discrete Painlev\'e Equations

The classical Painlev\'e equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamoto's geometric space of initial values for the first and second Painlev\'e equations. In this paper, we review this method and indicate how it may be extended to the discrete Painlev\'e equations.Comment: 14 pages, 2 figures; presented at conferences "Various Aspects on the Painlev\'e Equations" RIMS Kyoto, Japan (26-30 Nov 2012) and "Recent progress in the theory of Painlev\'e equations" IRMA Strasbourg, France (4-8 Nov 2013

A Local Asymptotic Analysis of the First Discrete Painlev\'e Equation as the Discrete Independent Variable Approaches Infinity

The first discrete Painlev\'e equation (dPI), which appears in a model of quantum gravity, is an integrable nonlinear nonautonomous difference equation which yields the well known first Painlev\'e equation (PI) in a continuum limit. The asymptotic study of its solutions as the discrete time-step $n\to\infty$ is important both for physical application and for checking the accuracy of its role as a numerical discretization of PI. Here we show that the asymptotic analysis carried out by Boutroux (1913) for PI as its independent variable approaches infinity can also be achieved for dPI as its discrete independent variable approaches the same limit.Comment: 21 pages in LaTeX2e, to appear in \textit{Methods and Applications of Analysis

The Second Painlev\'e Equation in the Large-Parameter Limit I: Local Asymptotic Analysis

In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter $\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these are asymptotic behaviours by finding explicit error bounds. Moreover, we show that they are connected and complete in the sense that they correspond to all possible values of initial data given at a point in the local region.Comment: 30 pages in LaTeX2e. Submitte

Singular dynamics of a qq-difference Painlev\'e equation in its initial-value space

We construct the initial-value space of a $q$-discrete first Painlev\'e equation explicitly and describe the behaviours of its solutions $w(n)$ in this space as $n\to\infty$, with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.Comment: 23 pages, 5 figure

The Coalescence Limit of the Second Painlev\'E Equation

In this paper, we study a well known asymptotic limit in which the second Painlev\'e equation (P_II) becomes the first Painlev\'e equation (P_I). The limit preserves the Painlev\'e property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movable simple poles of opposite residue of the generic solution of P_{II} must coalesce in the limit to become movable double poles of the solutions of P_I, even though the limit naively carried out on the Laurent expansion of any solution of P_{II} makes no sense. Here we show rigorously that a coalescence of poles occurs. Moreover we show that locally all analytic solutions of P_I arise as limits of solutions of P_{II}.Comment: 16 pages in LaTeX (1 figure included

Elliptic Painlev\'e equations from next-nearest-neighbor translations on the E8(1)E_8^{(1)} lattice

The well known elliptic discrete Painlev\'e equation of Sakai is constructed by a standard translation on the $E_8^{(1)}$ lattice, given by nearest neighbor vectors. In this paper, we give a new elliptic discrete Painlev\'e equation obtained by translations along next-nearest-neighbor vectors. This equation is a generic (8-parameter) version of a 2-parameter elliptic difference equation found by reduction from Adler's partial difference equation, the so-called Q4 equation. We also provide a projective reduction of the well known equation of Sakai.Comment: 14 pages, 1 figur

Asymptotic behaviour of the fifth Painlev\'e transcendents in the space of initial values

We study the asymptotic behaviour of the solutions of the fifth Painlev\'e equation as the independent variable approaches zero and infinity in the space of initial values. We show that the limit set of each solution is compact and connected and, moreover, that any solution with the essential singularity at zero has an infinite number of poles and zeroes, and any solution with the essential singularity at infinity has infinite number of poles and takes value $1$ infinitely many times.Comment: 36 pages, 5 figures. arXiv admin note: text overlap with arXiv:1412.354

Asymptotic Behaviours Given by Elliptic Functions in PIP_I--PVP_V

Following the study of complex elliptic-function-type asymptotic behaviours of the Painlev\'e equations by Boutroux and Joshi and Kruskal for $P_I$ and $P_{II}$, we provide new results for elliptic-function-type behaviours admitted by $P_{III}$, $P_{IV}$, and $P_{V}$, in the limit as the independent variable $z$ approaches infinity. We show how the Hamiltonian $E_{\rm J}$ of each equation $\rm P_{\rm J}$, $\rm J=I, \ldots , V$, varies across a local period parallelogram of the leading-order behaviour, by applying the method of averaging in the complex $z$-plane. Surprisingly, our results show that all the equations $P_I-P_{V}$ share the same modulation of $E$ to the first two orders

Analytic solutions of qq-P(A1)P(A_1) near its critical points

For transcendental functions that solve non-linear $q$-difference equations, the best descriptions available are the ones obtained by expansion near critical points at the origin and infinity. We describe such solutions of a $q$-discrete Painlev\'e equation, with 7 parameters whose initial value space is a rational surface of type $A_1^{(1)}$. The resultant expansions are shown to approach series expansions of the classical sixth Painlev\'e equation in the continuum limit

Elliptic asymptotics in qq-discrete Painlev\'{e} equations

We study the asymptotic behaviour of two multiplicative- ($q$-) discrete Painlev\'e equations as their respective independent variable goes to infinity. It is shown that the generic asymptotic behaviours are given by elliptic functions. We extend the method of averaging to these equations to show that the energies are slowly varying. The Picard-Fuchs equation is derived for a special case of $q$-P$_{\rm III}$Comment: 1 figur