Tronquée Solutions of the Painlevé Equation PI

Abstract

We analyze the one-parameter family of tronquée solutions of the Painlevé equation PI in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one free parameter multiplying exponentially small corrections to the Borel summed power series. We link the position of the poles in the first array to the free parameter and find the asymptotic expansion of the pole positions in this first array (in inverse powers of the independent variable). We show that the tritronquées are given by the condition that the parameter be zero. We show how this analysis in conjunction with the asymptotic study of the pole sector of the tritronquée in Costin et al (A direct method to find stokes multipliers in closed form for integrable systems. Trans Amer Math Soc, to appear, arXiv:1205.0775) leads to a closed form expression for the Stokes multiplier directly from the Painlevé property, not relying on isomonodromic or related type of results.

Appendices

Appendices

Sections 5.1–5.3 contain an outline of some results found in [10, 13] and illustration of these results on some simple examples. Section 5.4 contains a brief overview on the development of the subject.

1.1 Representation of Solutions as Transseries

Very few differential equations can be explicitly solved, and even when this is possible, their expression may be too complicated for easily extracting useful information about solutions; by contrast, formal solutions can often be obtained algorithmically as asymptotic expansions, from which properties of solutions such as rate of decay/increase, or approximations, can be easily read. Sometimes, free parameters are “hidden” beyond all orders of a classical asymptotic series; in such cases, transseries are instrumental in uncovering these parameters.

It is well known that equations written in terms of analytic functions have convergent power series solutions at any regular point of the equation; convergent expansions for solutions also exist at regular singularities (generically). But at irregular singularities, the asymptotic expansions of solutions are often divergent.

To illustrate, consider the simple equation

$$\begin{aligned} y'+y=\frac{1}{x^2}. \end{aligned}$$

(59)

The point $x=\infty$ is an irregular singular point of the equation (indeed, the substitution $x=1/z$ maps $\infty$ to $0$ and brings (59) to the form $-z^2\tfrac{\hbox {d}y}{\hbox {d}z}+y=z^2$ for which $z=0$ is an irregular singularity).

It is easy to see that there exists a unique asymptotic power series solution for $x\rightarrow \infty$,

$$\begin{aligned} \tilde{y}_0(x)=\sum _{n=2}^\infty \frac{(n-1)!}{x^n}, \end{aligned}$$

(60)

and it is divergent. On the other hand, the general solution of (59) is

$$\begin{aligned} y(x;C)=y_0(x)+C\mathrm{e}^{-x},\quad \mathrm{where}\, y_0(x)=\mathrm{e}^{-x}\int _{x_1}^x\frac{\mathrm{e}^s}{s^2}\, ds\,\sim \,\tilde{y}_0(x)\ \mathrm{as }\,x\rightarrow +\infty \end{aligned}$$

(61)

and $C$ is a free parameter.

This simple example illustrates main phenomena at irregular singularities: the power series solutions are divergent, there is loss of information (in (61) there is a one parameter family of solutions asymptotic to the same power series), and asymptoticity holds only in sectors ($y_0(x)\sim \tilde{y}_0$ only for $x\rightarrow \infty$ with $|\arg x|<\tfrac{\pi }{2}$).

In view of (60),(61), it is natural to consider that the complete formal solution of (59) is

$$\begin{aligned} \tilde{y}(x)=\tilde{y}_0(x)+C\mathrm{e}^{-x},\quad \mathrm{for }\ x\rightarrow +\infty . \end{aligned}$$

(62)

The formal expression (62), which satisfies (59), is not an asymptotic series in the sense of Poincaré if $|\arg x|<\tfrac{\pi }{2}$, as the term $C\mathrm{e}^{-x}$ is much smaller than all the powers of $x$ in $\tilde{y}_0(x)$: it is a term beyond all orders of the main series. The formal solution (62) is the simplest example of a transseries.

Consider next a nonlinear example, namely (59) plus a nonlinear term,

$$\begin{aligned} y'+y=\frac{1}{x^2}+y^4. \end{aligned}$$

(63)

Again, equation (63) has a unique power series solution $\tilde{y}_0(x)=\frac{1}{x^2}+\frac{2}{x^3}+\frac{6}{x^4}+\cdots$, which can be shown to be divergent too. To find possible further terms in a formal expansion, we search for a perturbation: substituting $y=\tilde{y}_0+\delta$ in (63) and using the fact that $\tilde{y}_0$ is already a formal solution, we get $\delta '+\delta \sim 2\tilde{y}_0\delta$, giving $\delta \sim C\mathrm{e}^{-x}\tilde{y}_1(x)$, where $\tilde{y}_1(x)$ is a (divergent) power series. Since $\tilde{y}_0+C\mathrm{e}^{-x}\tilde{y}_1(x)$ does not solve the equation, further corrections are required, yielding a complete formal solution of (63) in the form

$$\begin{aligned} \tilde{y}=\tilde{y}(x;C)=\, \tilde{y}_0(x) +C\mathrm{{e}}^{-x}\tilde{y}_1(x) + C^2\mathrm{{e}}^{-2x}\tilde{y}_2(x)+\cdots , \end{aligned}$$

(64)

where $\tilde{y}_k(x)$ are divergent power series and $C$ is an arbitrary parameter. The formal solution (64) is a transseries for $x\rightarrow \infty$ along directions in the complex plane for which the terms can be well ordered decreasingly, namely for $x\in \mathrm{e}^{ia}{\mathbb {R}}_+$ with $|a|<\tfrac{\pi }{2}$.

Scalar equations more general than (63), of the form

$$\begin{aligned} y'+\left( \lambda -\frac{\alpha }{x}\right) y=g(x^{-1},y) \quad \mathrm{with}\,g=O(x^{-2})+O(y^2) \quad \mathrm{for } x\rightarrow \infty ,\, y\rightarrow 0, \end{aligned}$$

(65)

have formal series solution of the form

$$\begin{aligned} \tilde{y}=\tilde{y}(x;C)=\,\tilde{y}_0(x)+\sum _ {k=1}^\infty C^k\mathrm{{e}}^{-\lambda kx}\tilde{y}_k(x), \quad \mathrm{with }\ \tilde{y}_k(x)=x^{k\alpha }\tilde{s}_k(x) \end{aligned}$$

(66)

(with $\tilde{s}_k(x)$ an integer power series in $x^{-1}$), which is a transseries for $x\rightarrow \infty$ along any direction for which $|\arg (\lambda x)|<\tfrac{\pi }{2}$.

Systems have transseries solutions which are similar: generic equations can be brought to the normal form (in [10] this is eq. (1.1))

$$\begin{aligned}&\displaystyle { {\mathbf {y}}'+\left( \Lambda -\frac{1}{x}A\right) {\mathbf {y}}=\mathbf {g}(x^{-1},{\mathbf {y}})},\nonumber \\&\displaystyle { \mathrm{with }\mathbf {g}=O(x^{-2})+O(|{\mathbf {y}}|^2) \quad \mathrm{for } x\rightarrow \infty ,\, |{\mathbf {y}}|\rightarrow 0}\nonumber \\&\displaystyle { \mathrm{and}\Lambda =\mathrm {diag}(\lambda _1,\ldots ,\lambda _d),\ A=\mathrm {diag}(\alpha _1,\ldots ,\alpha _d)}. \end{aligned}$$

(67)

Under appropriate nonresonance conditions,² systems (67) have formal solutions

$$\begin{aligned} \tilde{{\mathbf {y}}}=\tilde{{\mathbf {y}}}(x;{\mathbf {C}})=\,\tilde{{\mathbf {y}}}_{\mathbf {0}}(x) +\sum _{{\mathbf {k}}\in {\mathbb {N}}^d\setminus {\mathbf {0}}} {\mathbf {C}}^{\mathbf {k}}\mathrm{{e}}^{-\varvec{\lambda }\cdot {\mathbf {k}}x} \tilde{{\mathbf {y}}}_{\mathbf {k}}(x), \quad \mathrm{with }\tilde{{\mathbf {y}}}_{\mathbf {k}}(x)=x^{\varvec{\alpha }\cdot {\mathbf {k}}}\tilde{{\mathbf {s}}}_{\mathbf {k}}(x), \end{aligned}$$

(68)

which is a transseries for $x\rightarrow \infty$ along any direction along which the terms can be well ordered, meaning that all the exponentials are decaying, therefore along any direction in the sector

$$\begin{aligned} S_\mathrm{trans}=\{x\in {\mathbb {C}}\, |\, \mathrm {Re\,}(\lambda _j x)>0\ \mathrm{{for\ all\,}}j\ \mathrm{{with\ }}C_j\ne 0\}. \end{aligned}$$

(69)

1.2 Correspondence Between Transseries and Actual Solutions: Generalized Borel Summation

Consider the linear equation (59). Since its formal solution (60) is factorially divergent and ${\mathcal {L}}(p^{n-1})=(n-1)!x^{-n}$, heuristically, it is natural to attempt to write $\tilde{y}_0$ as a Laplace transform: this is central to Borel summation.

Recall that the Borel transform is defined as the formal inverse Laplace transform, $\mathcal {B}(x^{-\alpha })=\tfrac{p^{\alpha -1}}{\Gamma (\alpha )}$ for $\alpha >0$ (where ${\mathcal {L}}$ is the Laplace transform), and, more generally, we have (9).

Taking the inverse Laplace transform, equation (59) becomes $(1-p)Y(p)=p$; therefore,

$$\begin{aligned} y={\mathcal {L}}_\phi Y \quad \mathrm{where}\,Y(p)=\frac{p}{1-p}, \end{aligned}$$

(70)

and ${\mathcal {L}}_\phi$ is the Laplace transform along a direction of argument $\phi$, see (8).

Note that we cannot take the Laplace transform along ${\mathbb {R}}_+$ in (70)(except in the sense of distributions [13]), but we can integrate on any half-lines above or below ${\mathbb {R}}_+$, obtaining

$$\begin{aligned} y_0^+(x)={\mathcal {L}}_\phi Y(x) \quad \mathrm{for }-\phi =\arg x\in \left( 0,\frac{\pi }{2}\right) \end{aligned}$$

and

$$\begin{aligned} y_0^-(x)={\mathcal {L}}_\phi Y(x)\quad \mathrm{for }-\phi =\arg x\in \left( -\frac{\pi }{2},0\right) . \end{aligned}$$

The values of $y_0^\pm (x)$ do not depend on the value of $\phi$ in its specific quadrant; they can be both analytically continued in the right half-plane (and beyond) $y_0^+(x)\ne y_0^-(x)$ and in fact the difference $\tfrac{1}{2\pi i}(y_0^+(x)- y_0^-(x))= \mathrm{e}^{-x}$ recovers the exponentially small term in (61).

These facts generalize to nonlinear equations. Consider the example (63); taking the inverse Laplace transform, one obtains the convolution equation

$$\begin{aligned} (1-p)Y(p)=p+Y^{*4}(p), \end{aligned}$$

(71)

which has a unique solution $Y=Y_0(p)$ analytic at $p=0$. In fact, $Y_0(p)$ is analytic for $|p|<1$, and it is singular at $p=1$. Due to the convolution term in (71), the singularity at $p=1$ gives rise to an equally spaced array of singularities in the Borel plane at $p=2,3,4,\ldots$. $Y_0(p)$ is analytic along any direction $p=|p|\mathrm{e}^{i\phi }$ with $0<|\phi |<\tfrac{\pi }{2}$, is Laplace transformable, and $y_0={\mathcal {L}}_\phi Y_0={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_0$ is an actual solution of (63) for $x$ large with $\arg x=-\phi$: $y_0(x)$ is the Borel summation, along the direction of $x$, of $\tilde{y}_0(x)$.

The Borel sum $y_0^+={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_0$ of $\tilde{y_0}$ is the same for all $-\phi =\arg x\in (0,\tfrac{\pi }{2})$, and $y_0^-={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_0$ is the same for all $-\phi =\arg x\in (-\tfrac{\pi }{2},0)$; both $y_0^\pm$ can be analytically continued in the right half-plane; and $y_0^+-y_0^-$ is exponentially small.

The other series $\tilde{y}_k$ in (64) are Borel summed similarly (using convolutions equations which are found for $Y_k=\mathcal {B}\tilde{y}_k$ in [10, 13]), yielding functions $y_k={\mathcal {L}}_\phi Y_k={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_k$ analytic for large $x$; the series

$$\begin{aligned} y_0(x)+\sum _{k=1}^\infty C^k\mathrm{e}^{-kx}y_k(x) \end{aligned}$$

converges for $x$ large with $\arg x=-\phi \in (-\tfrac{\pi }{2},0)\cup (0,\tfrac{\pi }{2})$ to a solution of (63)³ [10, 13].

The general one-dimensional case (65) is similar; $Y_0(p)$ will have equally spaced arrays of singularities along $\arg p=\arg \lambda$. Along any other direction $p=|p|\mathrm{e}^{i\phi }$ with $0<|\phi -\arg \lambda |<\tfrac{\pi }{2}$ the continuation of $Y_0(p)$ is analytic (generally on a Riemann surface) and Laplace transformable, and ${\mathcal {L}}_\phi Y_0={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_0$ is an actual solution of (65) for $x$ large with $\arg x=-\phi$.

Remark 18

To Borel sum the series in (66) for $k\geqslant 1$, we may consider $y_k={\mathcal {L}}_\phi \mathcal {B}\tilde{y}_k$ (if $\alpha <0$), or we can choose any $m$ (large enough so that $\alpha -m<0$) and find solutions as Borel summed transseries in the form

$$\begin{aligned} y_0(x)+\sum _{k=1}^\infty C^k\mathrm{e}^{-kx}x^{mk}{\mathcal {L}}_\phi \mathcal {B}(x^{-mk}\tilde{y}_k). \end{aligned}$$

(72)

The final result does not depend on $m$, since

$$\begin{aligned} x^N{\mathcal {L}}_\phi \mathcal {B}(x^{-N}x^{-n}) ={\mathcal {L}}_\phi \mathcal {B}(x^{-n}). \end{aligned}$$

(73)

Finally, the series (72) converges to actual solutions for $x\in S_\mathrm{an}$, where

$$\begin{aligned} S_\mathrm{an}=\{x\, \big |\, -\frac{\pi }{2}+\epsilon <\arg (x)<\frac{\pi }{2}-\epsilon ,\ |x|>R\}. \end{aligned}$$

Generic nonlinear equations (67) have their transseries solutions (68) summed similarly along directions $d$ in ${\mathbb {C}}$. Furthermore, there exists a one-to-one correspondence between solutions s.t. ${\mathbf {y}}(x)\rightarrow 0$ ($x\in d,\,x\rightarrow \infty$) and (generalized) Borel sums of $\tilde{{\mathbf {y}}}(x;C)$ transseries solutions along $d$. These solutions ${\mathbf {y}}(x;C)$ are analytic in a sector containing $d$ for $|x|$ large. These results are stated and proved in [10, 13].

Theorem 16 is an application of these results for the system (44) associated with the Painlevé equation PI.

1.3 The Stokes Phenomenon

The directions $\pm i\overline{\lambda }_j{\mathbb {R}}_+$ are called antistokes lines; along these directions, some exponential $\mathrm{e}^{-\lambda _jx}$ in (68) is purely oscillatory. Antistokes directions border the sectors where transseries exist, (69). Directions with $\lambda _jx\in {\mathbb {R}}_+$ (for some $j$) are called Stokes lines; along these, some exponential $\mathrm{e}^{-\lambda _jx}$ has fastest decay. At Stokes directions, the constants beyond all order in the one-to-one association between small solutions and transseries may change: this is the Stokes phenomenon.

To illustrate this, consider (59). As noted above, solutions can be written using $Y_0$, the Borel sum of $\tilde{y}_0$, as

$$\begin{aligned} y(x)=\left\{ \begin{array}{l} {\mathcal {L}}_\phi Y_0\,(x)+ C_+ \mathrm{e}^{-x} \quad \mathrm{for } - \phi = \arg x\in (0,\tfrac{\pi }{2}),\\ {\mathcal {L}}_\phi Y_0\,(x)+ C_- \mathrm{e}^{-x} \quad \mathrm{for } - \phi = \arg x\in (-\tfrac{\pi }{2},0). \end{array}\right. \end{aligned}$$

(74)

The value of the jump in the constant beyond all orders, $C_+ -C_-$, is called the Stokes constant.

More generally, a fixed solution of (65) can be written as Borel summed transseries (72) for some fixed $C$ for all $\phi$ with $\arg \phi \in \arg \lambda +(0,\tfrac{\pi }{2})$, and with a different $C$ for all $\phi$ with $\arg \phi \in \arg \lambda +(-\tfrac{\pi }{2},0)$.

For general equations, the situation is similar: the vector parameter ${\mathbf {C}}$ in a transseries (68) associated via Borel summation along a direction to a true solution does not change when this direction varies between two consecutive Stokes or antistokes lines, but it generally changes across a Stokes line.

Consider systems (67), with $\lambda _1=1$, $|\lambda _j|\geqslant 1$, and $\beta :=\beta _1<1$ (which can be arranged by a suitable substitution) and solutions obtained by Borel summation of the transseries solution (68) along directions slightly above and below the Stokes line $\arg x=0$:

$$\begin{aligned} {\mathbf {y}}(x)=\left\{ \begin{array}{l} {\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {0}}\,(x)+ \sum _ {{\mathbf {k}}\in {\mathbb {N}}^d\setminus {\mathbf {0}}} \mathbf {C_+}^{\mathbf {k}}\mathrm{{e}}^{-\varvec{\lambda }\cdot {\mathbf {k}}x} {\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {k}}(x) \quad \mathrm{for } - \phi = \arg x\in (0,a_2),\\ {\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {0}}\,(x)+ \sum _ {{\mathbf {k}}\in {\mathbb {N}}^d\setminus {\mathbf {0}}} \mathbf {C_-}^{\mathbf {k}}\mathrm{{e}}^{-\varvec{\lambda }\cdot {\mathbf {k}}x} {\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {k}}(x) \quad \mathrm{for } - \phi = \arg x\in (a_1,0), \end{array}\right. \end{aligned}$$

(75)

where ${\mathbf {Y}}_{\mathbf {k}}=\mathcal {B}_\phi \tilde{{\mathbf {y}}}_{\mathbf {k}}$ (the analytic continuation of the Borel transform of $\tilde{{\mathbf {y}}}_{\mathbf {k}}$ along the direction of argument $\phi$), and the sector $a_1<\arg x<a_2$ does not contain another Stokes or antistokes line besides $\arg x=0$.

The first component $C_1$ of the constant beyond all orders in (75) changes when $\arg x$ crosses the Stokes line $\arg x=0$, corresponding to $\lambda _1=1$ [13].

Changes in the constant beyond all orders occur upon analytic continuation across* a Stokes line; the leading order change, which is exponentially small, is due to the continuation of ${\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {k}}$. The continuations of ${\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {k}}$ generally add further, but these are of order $\mathrm{e}^{-x}$ or smaller, and for $|{\mathbf {k}}|\geqslant 1$, the ${\mathcal {L}}_\phi {\mathbf {Y}}_{\mathbf {k}}$ already multiplies an exponential, so this change does not affect the coefficient of $\mathrm{e}^{-x}$. The fact that the changes in all ${\mathbf {C}}^{\mathbf {k}}$ with $|{\mathbf {k}}|\geqslant 1$ match to give an overall jump equivalent to ${\mathbf {C}}_+\rightarrow {\mathbf {C}}_-$ is due to the so-called resurgence, which links the singularities of all ${\mathbf {Y}}_{\mathbf {k}}$ in a precise manner.)

1.3.1 The Stokes Multiplier

A calculation analogous to the one in the proof of Proposition 3 gives the change in $C_1$, and the argument is as follows. To analytically continue ${\mathcal {L}}_{0^-}{\mathbf {Y}}_0(x)$ past $\arg x=0$, we write ${\mathcal {L}}_{0^-}{\mathbf {Y}}_0={\mathcal {L}}_{0^+}{\mathbf {Y}}_0+\delta$, where $\delta =({\mathcal {L}}_{0^-}-{\mathcal {L}}_{0^+}){\mathbf {Y}}_0$. Since ${\mathbf {Y}}_0(p)$ is analytic for $|p|<1$ (by [10] Proposition 1), the path of integration in $\delta$ can be deformed to the path from $\infty$ to $1$ below $[1,+\infty )$, going around 1, and then going to $+\infty$ above $[1,+\infty )$.

Using the fact that ${\mathbf {Y}}_0(p)=S_{\beta }(1-p)^{\beta -1} (\mathbf {e}_1+o(1))$ (by [10] Proposition 1), and that $(1-p)^{\beta -1}=\mathrm{e}^{\mp i\pi (\beta -1)}|1-p|^{\beta -1}$ for $|p|>1, \arg (p)=\pm 0$, we obtain, using Watson’s lemma,

$$\begin{aligned} \delta= & {} -2i S_{\beta } \sin (\pi \beta ) \int _1^\infty |1-p|^{\beta -1}\mathrm{e}^{-px}dp (\mathbf {e}_1+o(1))\\= & {} -2iS_{\beta } \sin (\pi \beta )\Gamma (\beta ) \mathrm{e}^{-x}x^{-\beta } (\mathbf {e}_1+o(1)), \end{aligned}$$

so that the jump in the constant $C_1$ across the Stokes line $\arg x=0$ is⁴

$$\begin{aligned} C_{1;+}-C_{1;-}=-S=-\mu \ \mathrm{with}\ S=2iS_{\beta } \sin (\pi \beta )\Gamma (\beta ), \quad \beta =\beta _1; \quad \mathrm {Re\,}(\beta )\in (0,1). \end{aligned}$$

For general equations, the values of Stokes constants are transcendental.

Note 19

The five special directions of PI are Stokes or antistokes lines of its normalized form (5).

1.4 Further References

Double expansions of solutions of linear equations as power series multiplying exponentials have been studied starting with Fabry [19] (1885), and then Cope [8] (1936). Iwano (1957–’59) analyzed solutions of nonlinear systems as a convergent series of functions analytic in sectors, multiplying exponentials [27]. The subject has been developed and expanded substantially after the fundamental work of Ecalle (1981), with results in multisummability of power series of linear ODEs [1], nonlinear ones [4], transseries of nonlinear ODEs [10, 13], similar results for discrete equations [5, 6], and for PDEs [16]; singularity formation near antistokes lines was studied in a general setting in [11].

1.5 Rewriting (5) as a Normalized System

We write (5) as usual,

$$\begin{aligned} \begin{pmatrix} h\\ h' \end{pmatrix}'= \begin{pmatrix} 0\\ \tfrac{392}{625}x^{-4} \end{pmatrix}+ \begin{pmatrix} 0 &{} 1\\ 1 &{}0 \end{pmatrix} \begin{pmatrix} h\\ h' \end{pmatrix}+ \begin{pmatrix} 0 &{} 0 \\ 0 &{} -\tfrac{1}{x} \end{pmatrix}\begin{pmatrix} h\\ h' \end{pmatrix}+ \begin{pmatrix} 0\\ \tfrac{1}{2} h^2 \end{pmatrix}. \end{aligned}$$

(76)

The transformation

$$\begin{aligned} \begin{pmatrix} h\\ h' \end{pmatrix}= \frac{1}{2}\begin{pmatrix} 1-\tfrac{1}{4x} &{} 1+\tfrac{1}{4x}\\ -1-\tfrac{1}{4x} &{} 1-\tfrac{1}{4x} \end{pmatrix} \begin{pmatrix} y_1\\ y_2 \end{pmatrix} \end{aligned}$$

brings (76) to (44), (45), which is in the normal form (67).

More precisely,

$$\begin{aligned} g_1(x,{\mathbf {y}})= & {} -{\frac{1568}{625}}\,{\frac{4\,x+1}{ \left( 16\,{x}^{2}+1 \right) {x }^{3}}}-\frac{1}{16}\,{\frac{ \left( 4\,x-1 \right) \left( 4\,x+1 \right) ^{ 2}{y_1} { y_2} }{ \left( 16\,{x }^{2}+1 \right) x}}\nonumber \\&-\frac{1}{32}\,{\frac{ \left( 4\,x+1 \right) \left( 4\,x- 1 \right) ^{2} { y_1}\, ^{2}}{ \left( 16\,{x}^{2}+1 \right) x}}-\frac{1}{32}\,{\frac{ \left( 4\,x+1 \right) ^{3} { y_2}\, ^{2}}{ \left( 16\,{x}^{2}+1 \right) x}}\nonumber \\&-{\frac{ \left( 2\,x-1 \right) { y_1} }{ \left( 16\,{x}^{2}+1 \right) x}}+\frac{1}{2}\,{\frac{ \left( 8\,x-1 \right) { y_2} }{ \left( 16\,{x}^{2} +1 \right) x}},\nonumber \\ g_2(x,{\mathbf {y}})= & {} {\frac{1568}{625}}\,{\frac{4\,x-1}{ \left( 16\,{x}^{2}+1 \right) {x} ^{3}}}+\frac{1}{16}\,{\frac{ \left( 4\,x+1 \right) \left( 4\,x-1 \right) ^{2 }{ y_1} { y_2} }{ \left( 16\,{x} ^{2}+1 \right) x}}\nonumber \\&+\frac{1}{32}\,{\frac{ \left( 4\,x-1 \right) ^{3} { y_1} \, ^{2}}{ \left( 16\,{x}^{2}+1 \right) x}}+\frac{1}{32}\,{\frac{ \left( 4\,x-1 \right) \left( 4\,x+1 \right) ^{2} { y_2} \,^{2}}{ \left( 16\,{x}^{2}+1 \right) x}}\nonumber \\&-\frac{1}{2}\,{\frac{ \left( 8\,x+1 \right) { y_1} }{ \left( 16\,{x}^{2}+1 \right) x}}+{\frac{ \left( 2\,x+1 \right) { y_2} }{ \left( 16\,{x}^{2}+1 \right) x}}. \end{aligned}$$

1.6 Calculation of the Functions $F_n(\xi )$

Substituting the two scale expansion (16) in (5), we obtain an asymptotic series, for $1\ll x\ll \xi$ and $F_0(\xi )\ll x$, in integer powers of $x^{-1}$, with coefficients functions of $\xi$; the first term is

$$\begin{aligned} {\xi }^{2}\, {\frac{\hbox {d}^{2}}{\hbox {d}{\xi }^{2}}}F_{{0}} \left( \xi \right) + \xi \, {\frac{\hbox {d}}{\hbox {d}\xi }}F_{{0}} \left( \xi \right) -\frac{1}{2}\, F_{{0}}(\xi ) ^{2} -F_{{0 }} \left( \xi \right) =O \left( {x}^{-1} \right) , \end{aligned}$$

and we look for $F_0$ analytic at $\xi =0$ and $F_0(0)=0,\, F_0'(0)=1$.

Substituting $F_0(\xi )=G_0(s),\,s=\ln \xi$, we get $G_0''-\tfrac{1}{2}G_0^2-G_0=0$, an equation having, as expected, Weierstrass elliptic functions as general solution, a one parameter family of rational solutions, as well as two constant solutions: multiplying the equation by $2G_0'$, we obtain $G_0'^2=\tfrac{1}{3}G_0^3+G_0^2+Const.$, whose solution contains a term $s=\ln \xi$ unless $Const.=0$, in which case we obtain $F_0(\xi )= 12\xi /[c(1-\xi /c)^{2}]$ (degenerate elliptic) and $F_0'(0)=1$ implies the formula in (18).

The coefficient of $x^{-1}$ gives the equation for $F_1(\xi )$:

$$\begin{aligned} \xi ^2F_1''+\xi F_1'-(1+F_0)F_1=-\xi ^2 F_0'', \end{aligned}$$

which shows that the only possible singularities for $F_1$ are at $\xi =0$ and $\xi =12$. Similarly, the differential equation for all $F_n$ are linear, with coefficients depending on $F_0,\ldots ,F_{n-1}$, and by induction, the only possible singularities for $F_1$ are at $\xi =0$ and $\xi =12$.

To determine $F_1$, we need two constants; one is determined from the condition that $F_1$ be analytic at $0$ (thus the coefficient multiplying $\ln \xi$ must vanish), and the other constant is determined at the next step, when solving for $F_2$ (from the condition that $F_2$ does not contain $\ln \xi$ terms). This pattern continues for all $F_n$, and is typical for generic equations.

An additional potential obstruction to $F_n$ rational occurs at $n=6$: $F_6$ also contains, in principle, a term $\ln (\xi -12)$ multiplied by a constant. This term vanishes precisely when the coefficient of $x^{-4}$ in (5) equals $-\tfrac{392}{625}$. Any other value of this coefficient produces an equation with movable branch points, hence not having the Painlevé property! This is the special feature of integrability of PI.

For practical calculation of $F_n$, for $n\geqslant 3$ it is better to substitute $F_n(\xi )={\frac{\xi \, \left( \xi +12 \right) }{ \left( \xi -12 \right) ^{3}}}V_n(\xi )$; the functions $V_n(\xi )$ can be calculated recursively using only two successive integrations of rational functions.