Stokes phenomena in discrete Painlevé I

1. Introduction

We consider the asymptotic limit $n\to \mathrm{\infty }$ of the first discrete Painlevé equation (dP_I)

and apply methods of exponential asymptotics to describe solutions which demonstrate asymptotically pole-free behaviour within certain sectors. This nonlinear equation also appears in the study of orthogonal polynomials, known there as the Freud equation [1–3]. Previous studies [4–6] considered the far-field behaviour of solutions to dP_I, finding the asymptotic solution in terms of elliptic functions. Joshi [5] and Vereshchagin [6] also determined discrete analogues of asymptotically pole-free solutions behaviours by Boutroux for the continuous first Painlevé equation (P_I). In this paper, we find the full formal series solutions including exponentially small terms, study the influence of Stokes phenomena on these solutions and deduce their regions of validity. The resultant asymptotic solutions share several features with the asymptotically pole-free solutions to P_I known as tronquée and tritronquée solutions.

Equation (1.1) tends to P_I in the scaling limit w_n=1−h(2y(x)−c/2), $x=\sqrt{h}n-b/2-c^2/24$, with α=h^3/2a, β=−γ+3+h²b, γ=6+hc, as $h\to 0$. P_I is one of the six Painlevé equations which play a key role in a range of physical problems, in particular arising as universal models in reductions of the nonlinear Schrödinger equation [7], the Korteweg–de Vries equation [8,9] and the Boussinesq equation [10], and appear as models in such diverse fields as statistical mechanics [11], nonlinear optics [12], electrochemistry [13] and plasma physics [8]. The first discrete Painlevé equation also arises in physical applications, such as the study of quantum gravity [14,15].

Motivated by these applications, the study of the Painlevé equations in asymptotic limits has been widely pursued. However, the corresponding study of discrete Painlevé equations remains sparse. In this paper, we apply techniques of exponential asymptotics for differential-difference and difference equations developed by King & Chapman [16] in their analysis of the Frenkel–Kontorova model for atomic lattices to dP_I, in order to obtain asymptotic approximations which are accurate up to exponentially small correction terms.

We note that there have been some previous investigations into exponential asymptotics for difference equations, such as that of [17,18]. The study by Olde Daalhuis [17] performed a Borel-plane analysis to consider the behaviour of second-order linear difference equations and to determine exponentially improved solutions in terms of an inverse factorial series. In fact, these Borel-plane methods could be applied to the current investigation, and controlled error estimates for the asymptotics could be obtained in this fashion. The primary reason for using the Stokes-smoothing technique described in [16] is that the analysis may be performed directly on the nonlinear difference equation, rather than requiring the solution to be formulated in terms of an integral expression; this feature permits the analysis to be easily extended to a wide range of more complicated problems.

(a) Background

The asymptotic behaviour of solutions to the Painlevé equations was studied by Boutroux [19], who considered the behaviour of the first Painlevé equation (P_I),

in the limit $|x|\to \mathrm{\infty }$. Boutroux identified special solutions which are asymptotically free of poles in particular sectors of the complex plane of angular width 4π/5, and furthermore discovered five unique solutions that are pole-free in sectors of angular width 8π/5. These solutions are known as tronquée and tritronquée solutions, respectively, and tend to $w\sim \pm \sqrt{-x/6}$ as $|x|\to \mathrm{\infty }$ within the asymptotically pole-free sectors. In [20], the real tritronquée solution was shown to have no poles on the whole semi-axis x<0. Several asymptotic studies of P_I have been carried out in [21–23]. These asymptotic approximations have played essential roles in many of the physical problems described earlier [7,13].

(b) Discrete Painlevé equations

In this study, we are interested in the asymptotic behaviour of solutions of the first discrete Painlevé equation. The discrete Painlevé equations are a set of integrable nonlinear difference equations, which tend to the ordinary Painlevé equations when the continuum limit is taken.

The discrete Painlevé equations first appeared in the study of solutions to the continuous Painlevé equations. It was discovered that these equations possess Bäcklund transformations, which map one solution to another solution with different parameter values, or even to a solution of a different Painlevé equation (e.g. [10,24]). These transformations have been used to construct hierarchies of solutions to the Painlevé equations, often in terms of rational or special functions.

It was noted by Fokas et al. [14,15] that dP_I arises in the study of Bäcklund transformations associated with the fourth Painlevé equation (P_IV). This equation has two parameters, a and b, and is given by

If we associate a choice of parameters (a_n,b_n) of P_IV with a solution denoted by w_n(z), a hierarchy of solutions may be found corresponding to parameter choices where μ is an arbitrary complex-valued constant. The recurrence relation for w_n is subsequently given by which is simply a version of dP_I with appropriately chosen parameters. A variant of this relationship was used by Bassom et al. [25] to describe hierarchies of solutions to P_IV in terms of parabolic cylinder functions, as well as in [10] to construct solutions in terms of rational functions, as well as Okamoto and Hermite polynomials. By studying the asymptotic behaviour of the discrete Painlevé equations, we are able to better understand the behaviour of these solution families.

The first discrete Painlevé equation has also featured in the study of problems arising outside of the field of integrable systems, particularly relating to the study of orthogonal and orthonormal polynomials [2,3]. A set of orthonormal polynomials p_n(x), where n≥0, is defined with respect to some weight function w(x) as satisfying

where δ_m,n=1 if m=n and is equal to zero otherwise. In general, these polynomials satisfy a recurrence relation where a_n and b_n themselves satisfy particular recurrence relations, described for example in [2]. In the case of orthonormal polynomials with respect to the Freud weight, $w(x)=\exp (-\kappa x^4-\mu x^2)$, we find that b_n=0 (this holds for all even weight functions) and that $u_n=a_n^2$ satisfies which, again, is dP_I, with appropriately chosen parameters. By analysing the solutions to dP_I, we can determine the behaviour of families of orthogonal polynomials, even in the limit that their degree becomes large. It is this relationship to orthonormal polynomials that underpins the appearance of dP_I in the study of quantum gravity models [14,15]. We also note in passing that the variation of w_n with respect to the parameters κ and μ can be computed directly from the fourth continuous Painlevé equation with appropriate choice of parameters [2], thus confirming the existence of another connection between dP_I and P_IV.

In recent decades, however, discrete integrable systems such as dP_I have become an object of study in themselves. In part, this is due to these equations being convenient examples of nonlinear difference equations which, due to their integrability, are more tractable than many equations within this class. Hence, they provide a useful testing ground for mathematical techniques intended for wider applicability.

The primary motivation for studying these equations, however, is to better understand their solutions. These equations are important discretizations of the continuous Painlevé equations; their particular significance is due to the fact that they retain the integrability property upon discretization. They arise as independent entities in applications and possess interesting solutions, whose properties permit us to understand the behaviour of classes of functions, such as the orthonormal polynomials described above. While addressing such questions about all solutions lies beyond the scope of the present study, it motivates our interest in understanding the asymptotic behaviour of the discrete Painlevé equations.

(c) Exponential asymptotics and Stokes lines

The formal series we study in this paper are shown to be divergent (see §2) and contain exponentially small terms that are invisible to conventional asymptotic approaches. The central idea of exponential asymptotic methods is that a divergent asymptotic series, when truncated optimally, produces an error term that is exponentially small in the asymptotic limit [26]. The problem may then be rescaled to directly determine the behaviour of this remainder term, allowing the exponentially small component of the solution to be determined separately to the asymptotic series itself. This idea was introduced by Berry [27–29] and Berry & Howls [30], who used these methods to determine the behaviour of special functions such as the Airy function.

As we uncover this exponentially small behaviour, we will discover two varieties of important curves, identified originally by Stokes [31] and known as Stokes lines and anti-Stokes lines. These curves relate to the behaviour of exponentially small components of the solutions. As Stokes lines are crossed, the exponentially small behaviour experiences a smooth, rapid change in value in the neighbourhood of the curve itself; in many cases, the behaviour may switch off entirely on one side of the Stokes line. Hence, Stokes lines allow us to determine which exponentially small contributions may be present in certain regions of the plane. Anti-Stokes lines determine directions where exponentially small behaviour changes to become exponentially large, and therefore mark the boundaries of the region in which the asymptotic series expression for the solution is valid.

In this investigation, we will apply an exponential asymptotic method developed by Olde Daalhuis et al. [32] for linear differential equations and extended by Chapman et al. [33] for application to nonlinear differential equations. We provide a brief outline of the process here; however, more detailed explanation of the methodology may be found in these studies.

To optimally truncate an asymptotic series, we require a general form for the asymptotic series coefficients, which is often algebraically intractable. In singular perturbation problems, Dingle [34] noted that successive terms in the asymptotic series expansion are obtained by repeated differentiation of an earlier term in the series. Hence, the late-order terms a_m of the resultant asymptotic series typically diverge as the ratio between a factorial and the increasing power of a function χ, such as

where Γ is the gamma function defined in [35], and A, γ and χ are functions that do not depend on m. In particular, the singulantχ is equal to zero when the leading-order behaviour is singular, ensuring that this singularity is present in all higher order terms. Chapman et al. [33] therefore propose a factorial-over-power ansatz (1.6) for the late-order terms, which may then be used to optimally truncate the asymptotic expansion.

The next step, described by Olde Daalhuis et al. [32], is to substitute the truncated series expression back into the original problem, obtaining an equation for the exponentially small remainder term after truncation. The positions of Stokes lines are found by determining the curves along which the remainder term varies rapidly, and the behaviour of this remainder is found by solving the remainder equation in the neighbourhood of these Stokes lines.

Away from the Stokes line, the governing behaviour is obtained by solving the homogeneous version of the remainder equations, generally giving an expression of the form A e^−χ/ϵ (this is a consequence of the analysis in Chapman et al. [33], although it is also shown as a part of the Stokes smoothing analysis contained herein). Stokes switching will occur when the switching exponential is maximally subdominant; this occurs when the singulant is purely real and positive. Anti-Stokes lines may be found when the exponential term switches from being small to large; this occurs when the singulant is purely imaginary. These are therefore convenient conditions to determine the possible location of Stokes and anti-Stokes lines, as observed by Dingle [34], and we apply them to the series solutions of dP_I.

(d) Paper outline

In §2, we find the full formal series expansions of the (pole-free) solutions of dP_I and provide recurrence relations for the coefficients. It is important to note that we find two types of series solutions, called Type A and B solutions. We also describe the late-order terms in these series. We determine the Stokes structures of the Type A solutions in §3, which involves descriptions of the exponentially small terms in the solutions. These are used to find the regions in which the solutions are pole-free. We carry out the analysis of Type B solutions in §4. Finally, we discuss the results and conclusions of the paper in §5.

2. Asymptotic series expansions

In this section, we expand the solution as a power series in the limit $n\to \mathrm{\infty }$ and determine a recurrence relation for the coefficients of the series. Finally, we determine a general expression for the late-order terms of the series.

Consider the first discrete Painlevé equation

Now, simply scaling into the far field requires the behaviour to be smooth in this limit; however, we also wish to consider asymptotically pole-free solutions that allow for the odd and even terms to vary rapidly. Hence, we consider separately the odd and even terms in the series, such that where y_n and z_n do not necessarily have the same asymptotic behaviour as $n\to \mathrm{\infty }$. Equation (2.2a) may now be represented as a system, giving and

In subsequent analysis, we assume $n\in \mathbb{C}$ with iteration n↦n+1 acting on domains and takes an initial value as given by a function f(n) in a domain that includes the initial point. This is a standard interpretation [36,37] and underlies the theory of difference calculus. This allows us to analytically continue the solution in the complex n-plane.

We now apply the method of multiple scales in terms of the fast variable n, and a slow variable s=ϵn. We scale y_n and z_n and express them such that

which gives the rescaled equations and

We assume that u(s,ϵ) and v(s,ϵ) are analytic in the variable s and expand the solutions to (2.4a,b) as a Taylor series in s to give the system

and

We now expand the solutions u(s,ϵ) and v(s,ϵ) as a formal power series in ϵ^1/2, such that

Equations (2.5a,b) therefore become

and

Matching orders of ϵ as $ϵ\to 0$ gives

This gives two sets of solutions. The first of these is given by while the second is where the upper and lower choice of signs correspond. In subsequent analysis, we will refer to solutions with the leading-order behaviour given by (2.9) as Type A, while those with leading-order behaviour given by (2.10) as Type B.

We see that in both solution types, u₀ and v₀ contain a singularity at s=0, and that this singularity increases in strength by one half in each subsequent term of the asymptotic series. Consequently, we expect that the series will diverge, and exponentially small Stokes switching behaviour will be present in the solution.

In general, we find

and

In order to determine the optimal truncation point, we must determine the form of the terms in the asymptotic series as $n\to \mathrm{\infty }$. We therefore follow [33] and define a late-order ansatz in the limit $n\to \mathrm{\infty }$:

where χ is the singulant, satisfying χ(0)=0, U and V are the prefactors and k is a constant.

Applying this ansatz to (2.11a,b) and matching leading orders of m as $m\to \mathrm{\infty }$ gives

and Continuing to the subsequent order as $m\to \mathrm{\infty }$, we find u₁=v₁=γ/2 and u₁=v₁=γ/6 for Type A and B solutions, respectively, which are always satisfied. At the next order, we find and

To determine the behaviour of the prefactor and singulant terms, we must consider Type A and B solutions separately and solve (2.13a,b) to find the singulant χ(s), and then (2.14a,b) to find the prefactors U(s) and V (s).

3. Stokes structure of Type A solutions

In this section, we solve the late-order term equations for Type A solutions. We use the resultant expressions to determine the Stokes structure, as well as the form of the exponentially small components of the solution. From these results, we determine the region of the complex plane which is asymptotically pole-free.

(a) Late-order terms

The singulant equations given in (2.13a,b) reduce to

Recalling that we are considering the late-order terms in the limit $m\to \mathrm{\infty }$, the leading-order behaviour of χ may be found by taking the series to be infinite, introducing error into the singulant behaviour that is only exponentially small in this limit. This gives The solutions to this expression take the form where M is any integer. In general, the late-order behaviour of u_n and v_n will be the sum of terms associated with the singulants given in (3.3), with each term associated with a particular choice of sign and M. However, in a similar fashion to the analysis carried out in [16], we note that this sum will be dominated in the $n\to \mathrm{\infty }$ limit by the two terms associated with M=0. Hence, we may restrict our attention to these components in our analysis.

Recalling that u₀ and v₀ are singular at s=0, and hence that χ(0)=0, we find that the relevant singulants are given by

To find the prefactor associated with these singulants, we solve (2.14a)–(2.14b). We take the limit of the summation terms as $n\to \mathrm{\infty }$. Equation (2.14b) becomes

Applying the form of the singulant given in (3.3), we find that this reduces to U′=0. Hence we set the prefactor U(s)=Λ, where Λ takes constant value. Recalling that there are two distinct singulant contributions, we conclude that there are two constants associated with U, denoted as Λ₁ and Λ₂. A similar analysis performed on (2.14a) produces V ′=0. Hence, we find that there are also two constants associated with V , which we will denote as Λ₃ and Λ₄.

Finally, we note that for the ansatz (2.12) to be consistent with the leading-order behaviour given in (2.9), we must have k=−1/2. This gives a singularity of the appropriate strength in the limit $n\to 0$. Hence, the late-order terms are given by

The only component of the late-order terms that remains to be found is the value of the constants Λ_i, which may be determined numerically. We illustrate this on an example problem in appendix A(a).

(b) Stokes structure

Having calculated χ₁ and χ₂, we recall from §1c that the behaviour of the exponential term will be proportional to $\exp (-\chi /ϵ)$; this is also demonstrated in the full Stokes switching analysis contained within appendix B. This term will be exponentially small when Re(χ)>0, and exponentially large when Re(χ)<0. We also recall that Stokes lines exist where Im(χ)=0 and Re(χ)>0, while anti-Stokes lines follow Re(χ)=0. Figure 1 illustrates the behaviour of the singulants in each of the four quadrants within the complex s-plane.

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We see that there is an anti-Stokes line following the positive real axis, while Stokes lines follow the negative and positive imaginary axis for χ₁ and χ₂, respectively. To determine the behaviour that occurs as the Stokes lines are crossed, we note that the remainder term is of algebraic order along anti-Stokes lines. However, this would make it comparable in size to the leading-order behaviour on the positive real axis, which is not possible. Hence, we conclude that the remainder terms associated with χ₁ and χ₂ are not present along the real axis at all, and hence cannot be active in the first or fourth quadrant.

Consequently, we conclude that the exponential contributions associated with χ₁ and χ₂ are present on the left-hand side of their Stokes lines, but are switched off as they cross from the left to the right-hand side (into the fourth and first quadrant, respectively). Figure 2 illustrates the regions in which the exponential contributions are active.

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Importantly, we see that as the exponential contributions associated with χ₁ and χ₂ are present only in the third and second quadrants, respectively, in which they are exponentially small (as Re(χ)>0), they do not cause any difficulty for the validity of the asymptotic expansion. Hence, the solutions satisfying (2.9) are valid on the entire complex plane, with the exception of the branch cut. We therefore conclude that the solutions are asymptotically pole-free in the entire far-field, aside from the negative real axis.

Finally, now that we know the nature of the switching behaviour as the Stokes lines are crossed, we are in a position to calculate the precise form of the remainder term associated with the late-order terms, and hence determine the exponentially small contribution present in the problem.

(c) Stokes smoothing

To determine the behaviour in the neighbourhood of the Stokes line, we return to (2.5a,b). Now, we express u and v as asymptotic expansions truncated after N terms, with remainder R_N and S_N respectively, such that

If N is chosen so that the series are truncated after their smallest terms, the remainder terms will be exponentially small. By considering the behaviour of these remainder terms, we can obtain the exponentially small corrections to the asymptotic series and use these to determine the region in which the asymptotic series provides a valid approximation of the exact solution.

A full analysis of the remainder terms R_N and S_N is given in appendix B. This analysis is quite technical; we therefore summarize the results in the remainder of this section.

From appendix B, we find that the optimal truncation point is given by N_opt=2|χ|/ϵ+ω, where 0≤ω<1 is chosen so that N is integer-valued. The remainder terms given in (B5) and (B6) are shown to take the form

and where ${\mathcal{S}}_i$ is the Stokes-switching parameter that varies rapidly in the neighbourhood of Stokes lines. From the analysis in appendix B, we find that the value of ${\mathcal{S}}_1$ changes smoothly from ${\mathcal{S}}_1=4$ to ${\mathcal{S}}_1=0$ when crossing from the third to the fourth quadrant, and the value of ${\mathcal{S}}_2$ changes smoothly from ${\mathcal{S}}_2=0$ to ${\mathcal{S}}_2=4$ when crossing from the second to the first quadrant. Hence, the exponential behaviour of the asymptotic approximation has been completely determined and is shown in figure 2b.

Consequently, the optimally truncated asymptotic series for u(s,ϵ) and v(s,ϵ) takes the form

and where the leading orders of u_n and v_n are give in (2.9), and the late-order behaviour is given in (3.5). This expression is therefore an accurate asymptotic approximation up to exponentially small terms, valid in the entire complex s-plane.

4. Stokes structure of Type B solutions

In this section, we determine the late-order asymptotic terms associated with Type B. These are then used to determine the exponentially small contributions to the asymptotic solution, and subsequently the Stokes structure. From these results, we determine the region of validity for the general solution and show that for particular choices of the two Stokes-switching parameters, the region of validity may be extended.

(a) Late-order terms

When considering Type B solutions, we find that the asymptotic series expansion for u and v are identical, and hence that u_m=v_m for all values of m. We therefore set V (s)=U(s).

Now, using (2.10), both singulant equations (2.13a,b) reduce to

where we have again taken the upper bound of the summation terms to be infinite. From the leading-order behaviour (2.10), we know that the solution is singular at s=0, and hence that χ=0 at s=0. Solving (4.1) with this boundary condition gives where M is some integer. As in Type A solutions, the late-order behaviour for u_n will take the form of an infinite sum of factorial-over-power ansatz terms, each associated with a unique choice of χ. However, at late orders, this sum will be dominated by the four terms associated with M=0 and M=−1, which have the smallest absolute value. Hence, the four terms that contribute to the exponential behaviour of the solution are We observe that, unlike Type A, there are four sets of late-order behaviour to be considered in order to understand the Stokes structure of the associated solution. By solving (2.14a,b), we again find that U′=0. However, in this case, there are four distinct singulant expressions, and hence four constants associated with the corresponding late-order terms (denoted Λ₁ - Λ₄). We also find that, as before, k=−1/2.

Hence, we find that the late-order terms may be expressed as

where Λ_i is associated with χ_i.

The numerical process to determine Λ_i is more complicated than that demonstrated in appendix A, as we must now compute four values of Λ_i, rather than two. An example numerical calculation is illustrated in appendix Ab.

(b) Stokes structure

Having calculated the singulants, we may determine the Stokes structure in a similar fashion to §3b, noting that Stokes lines follow curves on which Im(χ)=0 and Re(χ)=0, and anti-Stokes lines follow curves satisfying Re(χ)=0. The behaviour of the four different singulants is illustrated in figure 3.

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From this figure, we see that the Stokes structure of the solution is substantially more complicated than in Type A solutions. Not only are there four different exponential contributions to consider, but there are two different sets of Stokes lines, each associated with two of the singulants.

There is, however, also a more interesting difference. From figure 1, we were able to conclude that neither exponential contribution could be present on the positive real axis in Type A solutions, as they would not be exponentially small. Hence, as the jump across Stokes lines is prescribed in (3.7) and (3.8), the asymptotic behaviour is specified uniquely, with no free parameters. However, from figure 3, we see that this is no longer true for Type B solutions. While we can see that χ₂ and χ₃ cannot be present on the positive real axis, as they would be exponentially large in the asymptotic limit, the same cannot be said about χ₁ and χ₄. As Re(χ_1,4)>0 on the positive real axis, the associated exponential contribution is exponentially small, and the asymptotic series is still valid when such contributions are present. Hence, the value of ${\mathcal{S}}_{1,4}$ on the real axis may be freely specified, meaning that the solution contains two free parameters.

Consequently, we determine that the exponential contributions active in each region of the plane are those shown in figure 4a. We see that, assuming that ${\mathcal{S}}_{1,4}$ is non-zero on the real axis, the region in which the asymptotic expansion associated with (2.10) is valid is a sector between the anti-Stokes lines associated with χ_1,2 and χ_3,4, in which the exponentials associated with χ₂ and χ₃ are inactive, and those associated with χ₁ and χ₄ are exponentially small. The angle between the edge of the sector and the real axis (denoted by θ in figure 4) is given by

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However, we note that it is possible to choose ${\mathcal{S}}_{1,4}=0$ on the real axis. Figure 4b illustrates this behaviour. In this case, the region in which the asymptotic expansion is valid is substantially larger than the general case shown in figure 4a; there is no exponential contribution associated with χ₁ or χ₄ to become large as the anti-Stokes lines are crossed. Instead, the angle θ is now given by

(c) Stokes smoothing

Using a nearly identical analysis to §3c, we are able to show that for Type B solutions, the quantity switched across each Stokes line takes the form

where R_N is again the exponential contribution to the solution, and $\mathcal{S}$ is the Stokes switching parameter. Using matched asymptotic expansions, we find that We note that Hence, from (4.3), the full exponential behaviour takes the form where ${\mathcal{S}}_i$ switches rapidly across the relevant Stokes line such that the difference is given by We also note for completion that ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ must equal zero on one side of the Stokes line, and hence are uniquely determined, while this condition is not required for ${\mathcal{S}}_1$ and ${\mathcal{S}}_4$. However, if both ${\mathcal{S}}_1^{+}$ and ${\mathcal{S}}_4^{+}$ are zero, we obtain the special Stokes structure illustrated in figure 4b.

Consequently, the optimally truncated asymptotic series for u(s,ϵ), and hence v(s,ϵ), is shown to take the form

where the leading orders of u_n are give in (2.10), and the late-order behaviour is given in (4.2). This expression is an accurate asymptotic approximation up to exponentially small terms. Unlike the Type A solution, however, this approximation of the Type B solution is only valid within a wedge of the complex plane illustrated in figure 4. We also recall from §4b that ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ are uniquely determined, while ${\mathcal{S}}_1$ and ${\mathcal{S}}_4$ contain a Stokes-switching parameter that may be freely specified; this behaviour is not present in the asymptotic approximation of Type A solutions.

5. Conclusion

In this paper, we used methods of exponential asymptotics to compute and investigate two types of asymptotic solutions to the first discrete Painlevé equation, associated with the leading-order behaviours described in (2.9) and (2.10). We identified the Stokes structure of the solution for each type and used this information to deduce the sectors displaying asymptotically pole-free behaviour in the far-field limit. These asymptotic solutions were given as the sum of a truncated asymptotic power series and an exponentially subdominant remainder term, shown in (3.9) and (3.10) for Type A solutions, and (4.5) for Type B solutions.

In §3, we showed that the asymptotic solution associated with (2.9) is a valid approximation in the far field of the entire complex plane, aside from a branch-cut along the negative real axis, and hence the solution is asymptotically pole-free. This solution has no exponentially small behaviour on the positive real axis, although there is exponentially small behaviour present in the left-half complex plane. This implies that this asymptotic behaviour is uniquely defined, with no free parameter present in the solution.

The asymptotic solution associated with (2.10), discussed in §4, is generally asymptotically pole-free within a narrow wedge in the complex plane containing the positive real axis, seen in figure 4a. Unlike the first asymptotic solution, however, there are two free parameters in the asymptotic approximation, associated with Stokes switching. If these parameters are chosen appropriately, we discover asymptotic solutions with larger ranges of validity, such as that shown in figure 4b. Outside of the asymptotically pole-free regions, the asymptotic series solution is no longer valid, and we instead see the elliptic function behaviour described by Joshi [5]. In fact, the oscillatory behaviour that is present along anti-Stokes lines is a degenerate form of this behaviour.

We note with interest that the two classes of asymptotically pole-free solutions to dP_I, Types A and B, demonstrate power series-type behaviour in the far field; however, they differ in that Type A solutions are uniquely specified with no free parameters, while Type B solutions contain free parameters associated with the Stokes multipliers in the solution. Similar asymptotic behaviour is seen in the tritronquée and tronquée solutions to P_I, the former of which is uniquely specified, while the latter contains free parameters associated with Stokes switching. Consequently, the two classes of asymptotic behaviour described in this study may be regarded as corresponding to tritronquée and tronquée solutions of P_I, respectively.

The asymptotic solutions to dP_I also provide insight into the behaviour of solution hierarchies to the equations described in (1.3) and (1.5). The first of these describes solution hierarchies to the fourth Painlevé equation. For example, Type B solutions (2.10) correspond to solution families which grow as $w_n(z)\sim \pm \sqrt{-n/3}-z/3+\mathcal{O}(n^{-1/2})$ in the large-n limit, with the parameters described in (1.2) also growing appropriately. These solution hierarchies are of particular mathematical interest due to their power series dependence on n in the asymptotic limit, which is particular to the asymptotic classes of dP_I solutions described in the present study.

Finally, we emphasize that this methodology may be applied to a wide range of difference equations, as it does not require the problem to be integrable, nor does it require that solution be expressed in terms of an integral equation. This is demonstrated in appendix C, where we outline how this procedure could be applied to determine the Stokes switching behaviour of a non-integrable difference equation. Consequently, although we applied the method to determine information about the exponentially small components of the solution to dP_I, it could easily be adapted to analyse asymptotic solutions of broad classes of difference equations.

One particular case of interest is the behaviour of orthonormal polynomials satisfying (1.4) for a variety of different weight functions, such as the semi-classical weights described in [2]. In general, these polynomial families may be computed using non-integrable difference equations. However, as there is no aspect of the methodology which depends on the integrability of the equation, these difference equations are still amenable to the asymptotic methods described in the present study.

Note added in proof

We thank Professor R. Schiappa for informing us about his related work on instantons in string theory [38,39].

Data accessibility

We have no supporting data aside from that contained within the text.

Authors contributions

N.J. and C.J.L. collaborated on the mathematical analysis. C.J.L. drafted the manuscript. All authors gave final approval for publication.

Funding statement

N.J. and C.J.L. were supported by Australian Laureate Fellowship grant no. FL120100094 from the Australian Research Council.

Conflict of interests

We have no competing interests.

Footnotes