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

TL;DR

This paper investigates the asymptotic behavior of certain $q$-discrete Painlevé equations, revealing that their solutions tend to elliptic functions at infinity, and extends averaging methods to analyze their energy variations.

Contribution

It introduces an extension of the averaging method to $q$-discrete Painlevé equations and derives the Picard-Fuchs equation for a specific case, advancing understanding of their asymptotics.

Findings

Elliptic functions describe the generic asymptotics

Energies vary slowly in these equations

Derived the Picard-Fuchs equation for $q$-P$_{ m III}$

Abstract

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$_{\rm III}$

Equations

$$\text{q-}\text{PI}:$$

$$\text{q-}\text{PIII}:$$

$$\overline{u}\underline{u}=\frac{\gamma \delta (u-\alpha \Xi )(u-\beta \Xi )}{(u-\gamma )(u-\delta )},$$

$$u(\varphi )=a,u(\varphi +1)=p,$$

$$u(\varphi +{\Omega }_i^{(J)})=u(\varphi )=a,$$

$$E_{\mathrm{I}}:=\frac{{\overline{u}}^2{u}^2+\overline{u}+u}{\overline{u}u}.$$

$${\Delta }_i{E}_{\mathrm{I}}=\frac{{\zeta }_0}{e^{\eta }}\left(\frac{{\omega }_i}{\eta U_{\varphi +1}{U}_{\varphi }}+\frac{1}{\eta }\sum _{k=\varphi +1}^{\varphi +{\Omega }_i-1}\frac{1}{U_{k+1}{U}_k}+\mathcal{O}\left(\frac{1}{{\eta }^2}\right)\right).$$

$${\Omega }_i={\omega }_i-\frac{s(\varphi +{\omega }_i)}{U^{\prime }(\varphi )}+o(s),$$

$$s_m$$

$$E_{\mathrm{III}}:=\frac{{\overline{u}}^2{u}^2-(\gamma +\delta )\overline{u}u\left(\overline{u}+u\right)+\gamma \delta \left({\overline{u}}^2+u^2\right)-(\alpha +\beta )\gamma \delta \left(\overline{u}+u\right)+\alpha \beta \gamma \delta }{\overline{u}u}.$$

$${\Delta }_i{E}_{\mathrm{III}}$$

$$-\frac{2\alpha \beta \gamma \delta }{\eta }\frac{{\omega }_i}{U_{\varphi +1}{U}_{\varphi }}+\frac{2\alpha \beta \gamma \delta }{\eta }\sum _{k=\varphi }^{\varphi +{\Omega }_i}\frac{1}{U_{k+1}{U}_k}+\mathcal{O}\left(\frac{1}{{\eta }^2}\right).$$

$${\Omega }_i={\omega }_i-\frac{s(\varphi +{\omega }_i)}{U^{\prime }(\varphi )}+o(s),$$

$$s_m$$

$$-\frac{2\alpha \beta \gamma \delta }{3\eta }\frac{3l+4}{F_{l+1}{F}_l}+\frac{2\alpha \beta \gamma \delta }{\eta }\frac{U_{l+1}{U}_l}{F_{l+1}{F}_l}\sum _{k=\varphi }^l\frac{1}{U_{k+1}{U}_k}$$

$$-\frac{(\alpha +\beta )\gamma \delta }{\eta }\frac{U_{l+1}{U}_l}{F_{l+1}{F}_l}\sum _{k=\varphi }^l\left(\frac{1}{U_{k+1}}+\frac{1}{U_k}\right)),$$

$$\xi ={\xi }_0{e}^{-\frac{1}{2}}{e}^{\eta }\left(1+\frac{3m+1}{3\eta }+\frac{18m^2-6m-7}{36{\eta }^2}+\mathcal{O}\left(\frac{1}{{\eta }^3}\right)\right).$$

$$s(m+\omega +1)-s(m+\omega )\frac{G(m+\omega +1)}{G(m+\omega )}=\lambda G(m+1)R(m+\omega ).$$

$$u(\varphi +\Omega +1)$$

$$\Omega =\omega +{\left(-\frac{s}{U^{\prime }}+\frac{s{s}^{\prime }}{(U^{\prime }{)}^2}-\frac{s^2{U}^{\prime \prime }}{2(U^{\prime }{)}^3}\right)|}_{n=\varphi +\omega }+\mathcal{O}(s^3).$$

$$y_{tt}=\frac{y_t^2}{y}+e^t(\alpha y^2+\beta )+e^{2t}\frac{\gamma y^4+\delta }{y},$$

$$\overline{u}u\underline{u}=1-\frac{1}{\xi u}.$$

$$\overline{u}u\underline{u}\sim 1,$$

$$\overline{U}U\underline{U}=1,$$

$$E_0=\frac{{\overline{U}}^2{U}^2+\overline{U}+U}{\overline{U}U},$$

$$E:=\frac{{\overline{u}}^2{u}^2+\overline{u}+u}{\overline{u}u},$$

$$\Delta E=\left(\frac{1}{\overline{u}}-\frac{1}{\underline{u}}\right)\frac{1}{\xi u}.$$

$$\overline{u}u\underline{u}=1-\frac{{\zeta }_0}{e^{\eta }u}\left(1-\frac{3m+1}{3\eta }+\frac{18m^2+30m+11}{36{\eta }^2}+\mathcal{O}\left(\frac{1}{{\eta }^3}\right)\right),$$

$$E=E_c+\frac{{\zeta }_0}{e^{\eta }}\left(\frac{1}{\overline{u}u}-\frac{3m+4}{3\eta \overline{u}u}+\sum _{k=\varphi }^m\frac{1}{\eta u_{k+1}{u}_k}+\mathcal{O}\left(\frac{1}{{\eta }^2}\right)\right),$$

$$\overline{U}U\underline{U}=1,$$

$$\overline{U}\underline{U}s+\overline{U}U\underline{s}+U\underline{U}\overline{s}+\overline{U}s\underline{s}+\underline{U}\overline{s}s+U\overline{s}\underline{s}+\overline{s}s\underline{s}=-\frac{1}{\xi }\sum _{k=0}^{\infty }\frac{(-s{)}^k}{U^{k+1}}.$$

$$\{\begin{array}{l}U(\varphi )=a\\ s(\varphi )=0\end{array},\{\begin{array}{l}U(\varphi +1)=p\\ s(\varphi +1)=0\end{array}.$$

$${\overline{U}}^2{U}^2-E_0\overline{U}U+\overline{U}+U=0,$$

Full text

This research was supported by an Australian Laureate Fellowship # FL 120100094 from the Australian Research Council.

EL’s research was supported by a postgraduate research award from the University of Sydney.

1. Introduction

Despite widespread interest in the asymptotic analysis of the Painlevé equations, many corresponding questions on asymptotic behaviours of discrete Painlevé equations remain open. Asymptotic results for $q$-discrete Painlevé equations are particularly scarce (with only a few known cases), even though they arise in cluster algebra [Okub2013] and in problems related to gap probability functions [Kniz2016].

In this paper, we analyse the asymptotic behaviours of general solutions of $q$-discrete Painlevé equations and show that they are asymptotic to elliptic functions, in a way that closely resembles the behaviours of their continuous counterparts. This study is motivated by our recent study of the elliptic asymptotics of the first to the fifth Painlevé equations [JL2018]. We restrict our attention to the limit when the independent variable $\xi=\xi_{0}q^{n}$ approaches infinity with $|q|>1$ and $\Re(n)>0$.

We focus on the following $q$-discrete Painlevé equations:

[TABLE]

where $w=w(\xi)=w_{n}$, $\overline{w}=w(q\xi)=w_{n+1}$, $\underline{w}=w(\xi/q)=w_{n-1}$ with $\xi=\xi_{0}q^{n}$ for $\xi_{0}\in\mathbb{C}\backslash\{0\}$. Further information about the derivation and properties of these equations, including their respective continuum limits to the first and third Painlevé equations, are provided in Section 1.2. In an instance of differing nomenclature, these equations are also referred to in the literature by the root systems characterising their initial value space and symmetry group [Sakai2001]. The equation labelled $q$-P${}_{\text{I}}$ has initial value space $A_{7}^{(1)}$ and symmetry group $A_{1}^{(1)}$, while that labelled $q$-P${}_{\text{III}}$ has initial value space $A_{3}^{(1)}$ and symmetry group $D_{5}^{(1)}$.

Our main results are stated as Theorems 1.4 and 1.5 in Section 1.1. We write $q=1+\frac{1}{\eta}$ and assume that $n$ lies in a domain near infinity by taking $n=\eta^{2}+m$ where $m$ is bounded, with $|\eta|\to\infty$ and $\Re(n)>0$. We start by transforming variables in such a way that the equations become autonomous and of order unity to leading order as $|\eta|\to\infty$. For $q$-P${}_{\text{I}}$, no transformation is needed, hence $u=w$. For $q$-P${}_{\text{III}}$ the scaling $u=e^{\frac{1}{2}-\eta}w$ is applied, hence Equation (1.2) becomes

[TABLE]

where $c=e^{\eta-\frac{1}{2}}\gamma$, and $d=e^{\eta-\frac{1}{2}}\delta$, and renaming the parameters $\alpha:=a\;\!\xi_{0}$, $\beta:=b\;\!\xi_{0}$, and $\Xi:=\frac{\xi}{\xi_{0}e^{\eta-\frac{1}{2}}}$ in the limit $|\eta|\to\infty$, for more details see Section 3.1. In the following section, we use these transformations to set up our initial value problems and express our main results.

We show that each leading order equation has an invariant, which enables us to identify an energy-like quantity called $E$ (defined in Equations (1.5) and (1.9)), which is related to the modulus of the leading-order elliptic function. We will refer to $E$ as energy for short. Techniques from the calculus of differences are used to show that the modulation of the energy is small. In this paper we apply this approach to only two cases of $q$-discrete Painlevé equations, however, given appropriate scaling, we expect our approach to be extendable to other $q$-discrete Painlevé equations.

Below we state our main results in detail, provide a description of background and give an outline of the paper.

1.1. Main result

First we provide some preliminary definitions and remarks to describe the initial value problems and clarify our results. We shall assume that all the variables and parameters in Equations (1.1) and (1.2) are complex, and that the discrete dependent variable is at least once differentiable as a function of its independent variable. Our major results are stated as Theorems 1.4 and 1.5.

Recalling that $\xi=\xi_{0}\>\!q^{n}$, the iterations of $q$-P${}_{\text{I}}$, $q$-P${}_{\text{III}}$ can be thought of as iterating $n$ on a line $\ell$ in $\mathbb{C}$ or iterating $\xi$ on a spiral $\Gamma\subset\mathbb{C}$. Given an initial point $\xi_{0}$ on $\Gamma$, we assume that initial values $u_{0}$, $u_{1}$ are given as analytic functions in a domain $\mathcal{D}_{0}$ containing $\xi_{0}$ and $q\xi_{0}$ as interior points. We assume $q\in\mathbb{C}\backslash\{0,1\}$ is such that subsequent iterations give solutions in overlapping domains with points on $\Gamma$ as interior points. Alternatively, we can regard the iteration as giving successive overlapping domains $\mathcal{D}$ with points on the horizontal line $\ell$ as interior points. With this in mind, we define admissible initial conditions for each respective discrete equation below.

Definition 1.1**:**

Given real $\delta>0$, define $\mathcal{D}_{\delta}=\{|m|<\delta\}$, and assume that $\phi$ and $\phi+1$ lie in $\mathcal{D}_{\delta}$. For such $\phi$, assume $a$, $p$ are given initial values for a solution $u$ of $q$-P${}_{\text{I}}$ or $q$-P${}_{\text{III}}$, such that

[TABLE]

where $p\neq a$ and $a\neq 0$ for $q$-P${}_{\text{I}}$ and $q$-P${}_{\text{III}}$, $p\neq 0$ for $q$-P${}_{\text{III}}$, $p\neq\frac{1}{\xi_{0}q^{\phi}}$ for $q$-P${}_{\text{I}}$, and $p\neq\alpha$, $\beta$, $\gamma$, or $\delta$ for $q$-P${}_{\text{III}}$. We define the set of numbers $\phi$, $a$, $p$ satisfying the above conditions to be admissible.

Definition 1.2**:**

Let $\phi$, $a$, and $p$ be admissible data. We define $\Omega_{i}^{(J)}$ to be the distance to the next occurrence of the initial value $a$ in the direction of $\omega_{i}^{(J)}$, the periods for the leading order elliptic behaviour,

[TABLE]

for $i=1,\,2$ and $J=\rm I,III$. We call this quantity an approximate-period. See Figure 1.1.

Remark 1.3**:**

Note that for any admissible initial values in $\mathcal{D}_{\delta}$, the mapping given by $q$-P${}_{\text{I}}$ and $q$-P${}_{\text{III}}$ is analytic. Therefore, the derivative of the solution cannot be identically zero in the image domain. Since we are working in a bounded domain in $m$, without loss of generality, we assume that $\mathcal{D}_{\delta}$ and its images do not contain a zero of the derivative of $u$. It follows that we can apply the inverse function theorem to Equation (1.4) in the resultant domain to show that $\Omega_{i}^{(J)}$ exists.

Below we state our main theorems, in which we use the dependent variable $u$ and independent variable $m$. We also drop the superscripts $\rm(I)$ and $\rm(III)$ when there is no ambiguity.

Theorem 1.4** (q-P${}_{\text{I}}$):**

Let $\phi$, $a$, $p$ be admissible data and assume $u=U+s$ is a solution of $q$-P${}_{\text{I}}$, satisfying $u(\phi)=a,u(\phi+1)=p$, where $s\ll U$ as $|\eta|\to\infty$. Define the energy

[TABLE]

Then for sufficiently large $\eta$, there exist approximate-periods, $\Omega_{i}$, $i=1,2$, such that $\Delta_{i}E_{\rm I}:=E_{\rm I}(\phi+\Omega_{i})-E_{\rm I}(\phi)$ take the form:

[TABLE]

Furthermore, the approximate-periods $\Omega_{i}$ are given by:

[TABLE]

where ${}^{\prime}=\frac{d\,}{dm}$, and

[TABLE]

where $F_{m}=U_{m+1}U_{m}^{2}-U_{m}^{2}U_{m-1}$.

Theorem 1.5** (q-P${}_{\text{III}}$):**

Let $\phi$, $a$, $p$ be admissible data and assume $u=U+s$ is a solution of $q$-P${}_{\text{III}}$, satisfying $u(\phi)=a,u(\phi+1)=p$, where $s\ll U$ as $|\eta|\to\infty$. Define the energy $E_{\rm III}$:

[TABLE]

Then for sufficiently large $\eta$, there exist approximate-periods, $\Omega_{i}$, $i=1,2$, such that $\Delta_{i}E_{\rm III}:=E_{\rm III}(\phi+\Omega_{i})-E_{\rm III}(\phi)$ take the form:

[TABLE]

Furthermore, the approximate-periods $\Omega_{i}$ are given by:

[TABLE]

where ${}^{\prime}=\frac{d\,}{dm}$, and

[TABLE]

where $F_{m}=(U_{m+1}-U_{m-1})(U_{m}-\gamma)(U_{m}-\delta)$.

We will use the following lemmas in the analysis of both $q$-P${}_{\text{I}}$ and $q$-P${}_{\text{III}}$.

Lemma 1.6**:**

Suppose that $\eta$ is large, then for $\xi=\xi_{0}q^{n}$ ($\xi_{0}\in\mathbb{C}\backslash\{0\}$), where $|q|>1$ and $\Re(n)>0$ with $q=1+\frac{1}{\eta}$ and $n=\eta^{2}+m$ where $m$ is bounded, $\xi$ takes the following expansion in the limit $|\eta|\to\infty$:

[TABLE]

Proof.

The proof follows by direct computation. ∎

Lemma 1.7**:**

Suppose $s(m)=\lambda\;\!G(m)\sum_{k=\phi}^{m-1}R(k)$, where $\lambda$ is independent of $m$, and $G$ is periodic in $\omega$, then

[TABLE]

Proof.

This can be shown by direct expansion of the sum $s(m)$. ∎

Lemma 1.8**:**

The expression $u(\phi+\Omega+1)$ has the following expansion:

[TABLE]

where $u=U+s$, $s\ll U$, in the limit where the independent variable goes to infinity. Furthermore, $\Omega$ is of the following form:

[TABLE]

Proof.

See Appendix LABEL:ubexpan:pf. ∎

Proofs of Theorems 1.4 and 1.5 can be found in Section 2 and Section 3 respectively.

1.2. Background

Discrete Painlevé equations have a long history, beginning with the theory of orthogonal polynomials [Sho1939]. In particular, Shohat derived a nonlinear recurrence relation that was later recognised as a discrete Painlevé equation by Fokas et al [FIK1992]. Around the same time, such discrete equations appeared in models of quantum gravity [GM1990, PS1990].

Our equations of interest, $q$-P${}_{\text{I}}$ and $q$-P${}_{\text{III}}$, were derived and studied by Ramani et al [RG1996, RGH1991]. These authors also showed that the conventional form of Equation (1.1)111See [Josh2015] for the transformation of $q$-P${}_{\text{I}}$ to its conventional form. has a continuum limit to the first Painlevé equation $y_{tt}=6y^{2}+t$, while Equation (1.2) has a continuum limit to the third Painlevé equation [RGH1991]

[TABLE]

which is a transformed version of the conventional form of P${}_{\text{III}}$ (see [Ince1956, Chapter 14] for the transformation).

These equations also arise in other fields of mathematical physics. Equation (1.1) arises in cluster algebras as an equation related to the non-autonomous version of Somos-4 [Okub2013]. Recently, the asymmetric version of Equation (1.2), known as $q$-P${}_{\text{VI}}$, has been studied from the point of view of conformal block theory [JNS2017].

These perspectives lead to a natural question about the behaviours of solutions. Nishioka [Nish2010] showed that the solutions of $q$-P${}_{\text{I}}$ are highly transcendental functions in the sense that they cannot be expressed in terms of any earlier known functions or solutions of linear equations with such functions as coefficients. A similar result is believed to be true for $q$-P${}_{\text{III}}$, since $q$-P${}_{\text{I}}$ arises as a limiting equation from $q$-P${}_{\text{III}}$.

However, there appears to be no explicit information about such transcendental solutions, except for studies of asymptotic behaviours, which have been developed for a few cases of $q$-discrete Painlevé equations in the complex plane [Mano2010, RofThesis, JRo2016, Josh2015, Ohya2010]. In these studies, solutions asymptotic to power series expansions in the independent variable were studied for $q$-P${}_{\text{VI}}$, $q$-P$(A_{1})$, and $q$-P${}_{\text{I}}$. For $q$-P${}_{\text{I}}$, unstable solutions termed quicksilver solutions were identified [Josh2015]. The geometric description of the space of initial values in the asymptotic limit $|\xi|\to\infty$ was given in [JL2016]. In the latter study, the invariant for the autonomous leading-order equation was also considered. This contrasts with the studies of classical Painlevé equations, where general solutions asymptotic to elliptic functions are well known [JL2018].

Elliptic-function-type asymptotic behaviours have been studied in the case of the additive first discrete Painlevé equation, d-P${}_{\text{I}}$ [Vere1996, Josh1997]. Moreover, the Stokes phenomenon has been studied for the solution behaviours of additive discrete P${}_{\text{I}}$ and P${}_{\text{II}}$ equations [JL2015, JLL2017].

A general theory concerning the asymptotic analysis of difference equations can be found in [HS1964, HS1965]. Stokes phenomena in difference equations were also studied by Dingle [Ding1973]. However, many of the techniques developed to study asymptotic behaviours of ODEs have not yet been extended to difference equations. In particular the method of averaging has not been applied, prior to the present work, to study $q$-difference equations. The standard method of averaging is an approximation method often used for weakly nonlinear dynamical systems. It allows one to infer properties of the original dynamic system by understanding the dynamics of the averaged system. Although the standard averaging theorem addresses continuous functions in a real domain, it can be applied to any continuous almost periodic function in the complex plane. These extensions have been studied and implemented in [JoshiThesis, Mahm1998, Hust1970, HB2009, SA2003]. For its statement, proof and more details, we refer to [Ver1996, SVM2007]. As in the continuous setting, we find the method of averaging to be well-suited for deriving the asymptotic behaviour of almost periodic discrete equations.

1.3. Outline of the paper

In Section 2, we give a detailed study of $q$-P${}_{\text{I}}$. The analysis is divided into three subsections which respectively include the leading-order analysis, next-to-leading order analysis and analysis of the variation of the invariant. In Section 3, we study $q$-P${}_{\text{III}}$ in a similar way, with an extra subsection that treats a special case. This special case allows an explicit formulation of the leading order solution, its periods, and its Picard-Fuchs equation.

2. The first $q$-discrete Painlevé equation

In this section, we perform our asymptotic analysis on $q$-P${}_{\text{I}}$. No scaling is needed in the limit $|\xi|\to\infty$, but to remain consistent with the notation used for $q$-P${}_{\text{III}}$ later we rewrite the equation in the form:

[TABLE]

*We start by analysing the leading-order behaviour in Section 2.1. The next-to-leading-order behaviour is then calculated in Section 2.2. In Section 2.3 we prove that the modulation of the energy varies slowly as $|\xi|\rightarrow\infty$. The subscript $\rm I$ denoting $q$-**P${}_{\text{I}}$*will be dropped in this section for simplicity.

2.1. Leading-order analysis

In the limit $|\xi|\to\infty$, Equation (2.1) becomes

[TABLE]

to leading order. We define the leading-order $U$ as the solution of

[TABLE]

which can be integrated (actually summed) once to give

[TABLE]

where $E_{0}$ is a constant. It can be readily shown that $U$ is periodic with period 3.

This leads to the asymptotic expansion: $u=U+s$, where $s\ll U$ as $|\xi|\to\infty$. Motivated by this, we define an energy-like quantity, which is constant to leading-order as shown in the following lemma.

Lemma 2.1**:**

Define

[TABLE]

where $u$ satisfies Equation (2.1). Then, $E$ is constant to leading order in the limit $|\xi|\to\infty$.

Proof.

The proof follows by direct calculation using the integration factor $\frac{1}{\underline{u}}-\frac{1}{\overline{u}}$. ∎

Corollary 2.2**:**

The change of $E$, $\Delta E=E(m+1)-E(m)$, is given by

[TABLE]

The corollary follows directly from the proof of the lemma.

The expansion of $\xi$ as $|\eta|\to\infty$ is given by Lemma 1.6 and, therefore, we have the following result.

Lemma 2.3**:**

Equation (2.1) has the following expansion as $|\eta|\to\infty$:

[TABLE]

where $\zeta_{0}:=\frac{e^{1/2}}{\xi_{0}}$.

Corollary 2.4**:**

Corollary 2.2 and Lemma 1.6 imply that $E$ has the following expansion

[TABLE]

where $E_{c}$ is the integration constant in $m$.

Note that $E_{c}$ is constant as a function of $m$, but may have an expansion in powers of $\eta$. Using $u=U+s$, we can expand Equation (2.1) in powers of $\eta$. This leads to the following lemma.

Lemma 2.5**:**

For admissible initial conditions, Equation (2.1) becomes

[TABLE]

Here, $U$ and its iterations are of order unity, while $s$ and its iterations are of order $e^{-\eta}$.

Proof.

Equations (2.9) and (2.10) can be generated by direct substitution of $u=U+s$ into (2.1) and equating terms of equal order. The admissible initial conditions are used to allow (2.10) to remain bounded. $U$ and its iterations are solved by Weierstrass $\wp$-functions (see Lemma 2.7 below), and hence are of order unity. That $s$ and its iterations are of order $e^{-\eta}$ is shown by using dominant balance analysis on (2.10). ∎

Corollary 2.6**:**

In terms of the asymptotic expansion, the initial values now require:

[TABLE]

Lemma 2.7**:**

The leading-order equation (2.9) satisfied by $U$ is solved by a Weierstrass $\wp$-function.

Proof.

Equation (2.9) can be summed once to

[TABLE]

where $E_{0}$ is the integration constant. The transformations $U=x+\frac{E_{0}^{2}}{12}$ and $\overline{U}=y+\frac{E_{0}^{2}}{12}$ convert (2.12) into the following biquadratic polynomial,

[TABLE]

where

[TABLE]

Equation (2.13) can be parametrised by $x=\wp(z)$ and $y=\wp(z\pm\rho)$, where

[TABLE]

We note here that $\rho$ is not unique since the inverse of the Weierstrass $\wp$-function gives two values in any given period parallelogram. Therefore the leading order solution $U$ takes the following form:

[TABLE]

The forward iteration is $U(m+1)=\wp(z\pm\rho)+\frac{E_{0}^{2}}{12}$. The sign of $\rho$ which determines the step in $m$ depends on the initial values. ∎

The above results in the differential equation satisfied by $U$, which in turn allows the expression of its period integrals.

Corollary 2.8**:**

From Lemma 2.7, we can conclude that $U$ satisfies the following differential equation with respect to $z$,

[TABLE]

Therefore, we define the periods by:

[TABLE]

where $j=1,2$ and $C_{j}$ are linearly independent contours.

Remark 2.9**:**

Equation (2.12) has $4$ degenerate points at $E_{0}=3$, $3(-1)^{2/3}$, $3(-1)^{4/3}$, and $\infty$. This number can be reduced to $2$ under the identification $K^{1/3}=E_{0}$. In this case $K=27$ and $\infty$ are the degenerate points.

The elliptic integral (2.16) has three branch points in the finite plane and one at infinity, which results in two linearly independent periods. At the degenerate points, the integrand in (2.16) develops a simple pole and it is no longer possible to define two linearly independent contours. Hence $U$ becomes singly periodic in these cases. We note that the sequence $U$ and its analytic continuation in terms of the Weirestrass $\wp$ function retain the periodicity of 3 implied by Equation (2.3).

Corollary 2.10**:**

The following properties can be derived from (2.12):

[TABLE]

where in deriving (2.17) we assumed that $E_{0}$ is not a function of $U$ and $\overline{U}$.

2.2. Next-to-leading order analysis

Lemma 2.11**:**

Suppose that $s_{l}$ is the leading order behaviour of $s$, where $s_{l}$ satisfies the limiting equation (2.10) as $|\eta|\to\infty$, then $s_{l}$ is given by

[TABLE]

where $s_{l}(\phi):=0$ and $F:=\overline{U}U^{2}-U^{2}\underline{U}$.

Proof.

We use the asymptotic expansion $u=U+s$ to expand (2.5) and (2.8). Equating the order of $s_{l}$ (i.e. order of $e^{-\eta}$) terms gives the following equation:

[TABLE]

where $E_{1}$ is derived from the initial conditions

[TABLE]

Equation (2.20) can be integrated by recognising that two of its coefficients are exact differences, that is, we have

[TABLE]

where we have used Equation (2.9) in Equation (2.22) and (2.23) uses a forward iteration of (2.9). Using the definition of $F$ and dividing (2.20) by $\overline{F}\>\!F$ give

[TABLE]

Summing the above yields the desired result, where the integration constant turns out to be zero. Note that for $s_{l}$ to be well-defined, we have assumed (w.l.o.g.) that $F$ is non-zero in the domain $\mathcal{D}_{\delta}$. ∎

Remark 2.12**:**

$F$* can also be expressed as:*

[TABLE]

by using (2.18).

Notice that $F$ and each component of $s_{l}$ are periodic in $\omega$. Moreover, we note that $\frac{d\overline{U}}{dU}=\frac{\overline{F}}{F}$.

Corollary 2.13**:**

The approximate-period $\Omega$ in the direction of $\omega$ has the following expansion:

[TABLE]

where $s_{l}$ can be represented by (2.19) and $U^{\prime}=\frac{d\>\!U}{d\>\!m}$, with

[TABLE]

which follows by transforming (2.15) into $m$.

2.3. Evolution of the energy

In this section we derive the local change of the energy across an approximate-period.

Lemma 2.14**:**

In the case of $q$-P${}_{\text{I}}$, $\lambda=\frac{\zeta_{0}}{e^{\eta}}$, $G=F$, and $R(k)=\frac{1}{F_{k+1}F_{k}}\left(1-\frac{U_{k+1}U_{k}}{U_{\phi+1}U_{\phi}}\right)$, hence

[TABLE]

Proof.

This can be shown by using Lemmas 1.7 and 1.8. ∎

*We are now in a position to evaluate the change of the energy across an approximate-period. In the proof below, we use (2.8) to represent $E$. *

Proposition 2.15**:**

The evolution of the energy $E$ of $q$-P${}_{\text{I}}$ is given by

[TABLE]

Proof.

We start with the expansion (2.8) for $E$. Consider the difference:

[TABLE]

Consider the first term from (2.29), which can be reduced by using Corollary 2.14 to give

[TABLE]

where we have also used property (2.17). The second term from (2.29) can be simplified into:

[TABLE]

We then replace $\Omega$ in the above equation by using (2.25). The third term can be evaluated by recognising that it is a telescoping sum. Combining the results gives the desired result. ∎

This concludes our proof of Theorem 1.4.

3. The third $q$-discrete Painlevé equation

In this section, we study the generic behaviour of $q$-P${}_{\text{III}}$ in the limit $\xi$ goes to infinity. We focus on deriving the leading and next-to-leading order behaviour. We then show that the energy is slowly varying. A special case is considered in order to find the explicit formulation of the Picard-Fuchs equation for the period integral of the leading-order behaviour. Again, we drop the subscript denoting $q$-P${}_{\text{III}}$.

3.1. Leading-order analysis

First we transform Equation (1.2) into a form in which the asymptotic behaviour in the limit $|\xi|\to\infty$ can be readily studied.

Lemma 3.1**:**

Using the result of Lemma 1.6 and under the same conditions, the scaling:

[TABLE]

transforms (1.2) into the following form,

[TABLE]

where $\alpha:=a\;\!\xi_{0}$, $\beta:=b\;\!\xi_{0}$, and $\Xi:=\frac{\xi}{\xi_{0}e^{\eta-\frac{1}{2}}}$, for $|\eta|\to\infty$.

Proof.

Lemma 1.6 can be used to show that the controlling factor of $\xi$ is $e^{\eta}$ which allows us to postulate the following ansatz:

[TABLE]

The method of dominant balance is used to solve for $\rho$, $\sigma$, and $\theta$ under the conditions that the transformations are not singular. The resulting limiting equation is of non-degenerative QRT type in general, and a maximal number of terms is maintained. Under these conditions, we have:

[TABLE]

This allows us to peel off the dominant behaviour of $q$-P${}_{\text{III}}$ and gives the resulting equation in the stated Lemma. New parameters are defined so that the leading behaviour of $\Xi$ is $1$:

[TABLE]

∎

Remark 3.2**:**

In the rest of this section, the notation $\Xi_{1}:=\Xi-1$ is often used, where $\Xi_{1}$ represents the small perturbation terms.

In the limit $|\eta|\to\infty$, Equation (3.1) becomes:

[TABLE]

to leading order. We define the leading-order $U$ as the solution of

[TABLE]

which can be integrated (actually summed) once to give

[TABLE]

where $E_{0}$ is a constant, also known as the invariant. This leads to the asymptotic expansion: $u=U+s$, where $s\ll U$ as $|\eta|\to\infty$. Motivated by this, we define the energy in the following lemma.

Lemma 3.3**:**

Define

[TABLE]

where $u$ satisfies Equation (3.1). Then, $E$ is constant to leading order in the limit $|\xi|\to\infty$.

Proof.

This desired result yields after multiplying both sides of Equation (3.1) by $(u-\gamma)(u-\delta)$, then using the integration factor $\frac{1}{u\underline{u}}-\frac{1}{\overline{u}u}$. ∎

Corollary 3.4**:**

The change of $E$, $\Delta E=E(m+1)-E(m)$, is given by

[TABLE]

Equivalently, $E$ can also be expanded as

[TABLE]

where $E_{c}$ is the integration constant in $m$.