Explicit error term of the elliptic asymptotics for the first Painlevé transcendents

The first Painlevé equation

$$y^{\prime\prime}=6y^{2}+x$$

(1.1)

governs the isomonodromy deformation of the two-dimensional linear system

	$\displaystyle\frac{d\Psi}{d\xi}=$	$\displaystyle\Bigl{(}(4\xi^{4}+x+2y^{2})\sigma_{3}-i(4y\xi^{2}+x+2y^{2})\sigma_{2}-(2y^{\prime}\xi+\tfrac{1}{2}\xi^{-1})\sigma_{1}\Bigr{)}\Psi,$		(1.2)
		$\displaystyle\sigma_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$

([8], [10], [3]). System (1.2) admits the canonical solutions $\Psi_{k}(\xi)$ $(k\in\mathbb{Z})$ represented by $\Psi_{k}(\xi)=(I+O(\xi^{-1}))\exp((\tfrac{4}{5}\xi^{5}+x\xi)\sigma_{3})$ as $\xi\to\infty$ through the sector $-3\pi/{10}+\pi k/5<\arg\xi<\pi/{10}+\pi k/5$, and the Stokes matrices

$$S_{2l+1}=\begin{pmatrix}1&s_{2l+1}\\ 0&1\end{pmatrix},\quad S_{2l}=\begin{pmatrix}1&0\\ s_{2l}&1\end{pmatrix},\quad l=0,1,2,\ldots$$

are defined by $\Psi_{k+1}(\xi)=\Psi_{k}(\xi)S_{k}$. Each solution of (1.1) is parametrised by $(s_{j})_{1\leq j\leq 5}$ on the manifold of monodromy data for (1.2), which is a two-dimensional complex manifold in $\mathbb{C}^{5}$ given by $S_{1}S_{2}S_{3}S_{4}S_{5}=i\sigma_{2}$ (see [10], [12], [14], [15]). For a general solution thus parametrised, Kapaev [10] obtained asymptotic representations as $x\to\infty$ along the Stokes rays $\arg x=\pi+2\pi k/5$ $(k=0,\pm 1,\pm 2)$, and subsequent studies on related connection formulas and the Stokes phenomenon are found in [21], [22], [11], [20], [1], and [16], [7] for the $\tau$-function. Along generic directions the Boutroux ansatz [2] suggests the asymptotic behaviour of a general solution of (1.1) expressed by the Weierstrass $\wp$-function. An approach to such an expression was made by Joshi and Kruskal [9] by the method of multiple-scale expansions, the labelling of the solution by $(s_{j})$ not being considered. Based on the isomonodromy deformation of (1.2), using WKB analysis, Kitaev [14], [15], and Kapaev and Kitaev [12] presented the elliptic asymptotic representation of a general solution of (1.1) as $x\to\infty$ through a cheese-like strip along any direction in the sector $3\pi/5<\phi-2\pi k/5<\pi,$ $k\in\mathbb{Z}.$ For $k=-2$ the asymptotic result is described as follows [14, p. 593, Section 8], [15, Theorem 2], [12, Theorem 2], in which $\omega_{\mathbf{a}},$ $\omega_{\mathbf{b}}$ and $A_{\phi}$ are, respectively, the periods of an elliptic curve and a unique solution of the Boutroux equations explained in (1) at the end of this section.

The leading term of (1.3) depends on the integration constant $t_{0}=t_{0}((s_{j})_{1\leq j\leq 5})$ only and the other one, which may be called the second integration constant, is hidden in the error term $O(t^{-\delta})$. Thus, in treating $y(x)$ as a general solution, it is preferable to know the explicit error term. This high-order part is also related, say, to the $\tau$-function [15, p. 121] or to degeneration to trigonometric asymptotics [15, Section 4]. For the $\tau$-function associated with (1.1), by the method of topological recursion, Iwaki [6] obtained a conjectural full-order formal expansion yielding the elliptic expression (see also [7]).

The justification procedure for the asymptotics of $y(x)$ [15, pp. 105–106, pp. 120–121], [13] is based on the leading terms of (3.1) and of the correction function $B(\phi,t)$, the pair of which should depend on both integration constants labelling the solution $y(x)$. The second integration constant is also related to $B(\phi,t)$ given in the following [15, Theorem 4], where $J_{\mathbf{a}}$ and $\vartheta(z,\tau)$ with $\vartheta^{\prime}(z,\tau)=(d/dz)\vartheta(z,\tau)$ are as in (1) and (2).

It is natural to seek explicitly the error term of (1.3). In this paper we present an explicit asymptotic representation of the error term, and this formula leads to the error bound $O(t^{-1})=O(x^{-5/4})$ in (1.3). The main results are stated in Theorems 2.1 and 2.2. Corollary 2.3 describes the dependence on the other integration constant $\beta_{0}$ in this explicit error term. In Section 4 these results are derived from the $t^{-\delta}$-asymptotics of $y(x)$ in Theorem A and of $B(\phi,t)$ in Theorem B combined with a system of integral equations equivalent to (1.1), which is constructed in Section 3. Necessary lemmas in our argument are shown in Section 5. The final section is devoted to discussions on our method in deriving the explicit error with the bound $O(t^{-1})$, which is quite different from that in [5], [9] and [17].

Throughout this paper we use the symbols summed up below (cf. [14], [15]).

(1) For each $\phi\in\mathbb{R}$, the condition

$$\mathrm{Re\,}\int_{\mathbf{c}}wd\lambda=0,$$

where $\mathbf{c}$ is any cycle on the elliptic curve $w^{2}=\lambda^{3}+\tfrac{1}{2}e^{i\phi}\lambda+\tfrac{1}{4}A,$ uniquely determines $A=A_{\phi}\in\mathbb{C}$ [15, Lemma 3], [14, Section 7] (The conditions for basic cycles $\mathbf{c}=\mathbf{c}_{1},$ $\mathbf{c}_{2}$ are the Boutroux equations). Then $A_{\phi}$ is continuous in $\phi$ and smooth for $\phi\not=\pi+2\pi k/5,$ $(k\in\mathbb{Z})$, and fulfils the relations $A_{-\phi}=\overline{A_{\phi}},$ $A_{\phi+2\pi}=A_{\phi},$ $A_{\phi-2\pi k/5}=e^{2\pi ik/5}A_{\phi},$ and $A_{\pi}=-(\tfrac{2}{3})^{3/2},$ $0.336<A_{0}<0.380$ [15, Lemma 4], [14, Section 7]. Suppose that $|\phi|<\pi/5$, and set $w=w(A_{\phi},\lambda)=\sqrt{\lambda^{3}+\tfrac{1}{2}e^{i\phi}\lambda+\tfrac{1}{4}A_{\phi}}=\sqrt{\lambda-\lambda_{1}}\,\sqrt{\lambda-\lambda_{2}}\,\sqrt{\lambda-\lambda_{3}}.$ Let $\mathbf{a}$ and $\mathbf{b}$ be basic cycles as in Figure 1.1 on the elliptic curve $\Gamma=\Gamma_{+}\cup\Gamma_{-}$ defined by $w(A_{\phi},\lambda)$ such that the upper and lower sheets $\Gamma_{+}$ and $\Gamma_{-}$ are glued along the cuts $[\lambda_{1},\lambda_{2}]\cup[-\infty,\lambda_{3}]$; where the branch points $\lambda_{j}=\lambda_{j}(\phi)$ for $\phi$ around $\phi=0$ are specified in such a way that $\mathrm{Re\,}\lambda_{1}(0)>0,$ $\mathrm{Im\,}\lambda_{1}(0)>0,$ $\lambda_{2}(0)=\overline{\lambda_{1}(0)},$ $\lambda_{3}(0)<0,$ and the branches of $\sqrt{\lambda-\lambda_{j}}$ are determined by $\sqrt{\lambda-\lambda_{j}}\to+\infty$ as $\lambda\to+\infty$ along the positive real axis on $\Gamma_{+}.$

For the cycles $\mathbf{a}$ and $\mathbf{b}$ on $\Gamma$ write

	$\displaystyle\omega_{\mathbf{a}}=\omega_{\mathbf{a}}(\phi)=\frac{1}{2}\int_{\mathbf{a}}\frac{d\lambda}{w},\quad\omega_{\mathbf{b}}=\omega_{\mathbf{b}}(\phi)=\frac{1}{2}\int_{\mathbf{b}}\frac{d\lambda}{w},\quad$
	$\displaystyle J_{\mathbf{a}}=J_{\mathbf{a}}(\phi)=2\int_{\mathbf{a}}wd\lambda,\quad J_{\mathbf{b}}=J_{\mathbf{b}}(\phi)=2\int_{\mathbf{b}}wd\lambda,$

where $\omega_{\mathbf{a}},$ $\omega_{\mathbf{b}}$ fulfil $\mathrm{Im\,}\omega_{\mathbf{b}}/{\omega_{\mathbf{a}}}>0.$ For $\phi\not=\pi+2\pi k/5$ the elliptic curve $\Gamma$ with the cycles $\mathbf{a},$ $\mathbf{b}$ and the integrals $\omega_{\mathbf{a},\,\mathbf{b}}$ and $J_{\mathbf{a},\,\mathbf{b}}$ may be extended by the use of the substitution $(\phi,\lambda,w)\mapsto(\phi+2\pi/5,e^{-4\pi i/5}\lambda,e^{-6\pi i/5}w)$ [15, pp. 95–96].

(2) For $\tau=\omega_{\mathbf{b}}/\omega_{\mathbf{a}}$ such that $\mathrm{Im\,}\tau>0$,

$$\vartheta(z,\tau)=\sum_{n\in\mathbb{Z}}e^{\pi i\tau n^{2}+2\pi izn}$$

is the theta-function [4], [23] with $\nu=(1+\tau)/2,$ which satisfies $\vartheta(z\pm 1,\tau)=\vartheta(z,\tau),$ $\vartheta(z\pm\tau,\tau)=e^{-\pi i(\tau\pm 2z)}\vartheta(z,\tau).$

(3) We write $f\ll g$ or $g\gg f$ if $f=O(g)$, and $f\asymp g$ if $f\ll g$ and $g\ll f.$

To state our results let us define some strips similar to $\mathcal{D}_{(s_{j})}(\phi,c,\varepsilon)$. Recall the phase shift $t_{0}$ given by (1.4), and the zeros $\lambda_{j}$ $(1\leq j\leq 3)$ of $w(A_{\phi},\lambda)^{2}$ specified as in (1).

Let $t_{j}^{m,n}\in\mathcal{S}(t_{\infty},c):=\{t\,|\,\mathrm{Re\,}t>t_{\infty},|\mathrm{Im\,}t|<c\}$ $(j=1,2,3,\infty)$ be such that $t_{j}^{m,n}=t_{j}+e^{-i\phi}(t_{0}+m\omega_{\mathbf{a}}+n\omega_{\mathbf{b}})$ if $1\leq j\leq 3$, and that $t_{\infty}^{m,n}=e^{-i\phi}(t_{0}+m\omega_{\mathbf{a}}+n\omega_{\mathbf{b}})$ if $j=\infty$. We may suppose $\varepsilon$ so small that all the circles $|t-t_{j}^{m,n}|=\varepsilon$ $(j=1,2,3,\infty;m,n\in\mathbf{Z})$ are disjoint. Let $l(t^{m,n}_{j})\in\mathcal{S}(t_{\infty},c)$ $(j=1,2,3,\infty)$ be the lines defined by $t=\mathrm{Re\,}t^{m,n}_{j}+i\eta$, $\eta\geq\mathrm{Im\,}t^{m,n}_{j}$ if $\mathrm{Im\,}t^{m,n}_{j}\geq 0$ (respectively, $\eta\leq\mathrm{Im\,}t^{m,n}_{j}$ if $\mathrm{Im\,}t^{m,n}_{j}<0$), and let $l^{x}(t^{m,n}_{j})$ be the images of $l(t^{m,n}_{j})$ under the mapping $x=x(t)=e^{i\phi}(\tfrac{5}{4}t)^{4/5}$.

These domains are strips lying in the $x$-plane. For a function $f(x)$ we may deal with $\hat{f}(t)=f(x(t))$ with $t=\tfrac{4}{5}(e^{-i\phi}x)^{5/4}$ or $\hat{f}(z)=f(x(z))$ with $z=e^{i\phi}t=\tfrac{4}{5}e^{i\phi}(e^{-i\phi}x)^{5/4}$ as $x\to\infty$, say, through $\check{\mathcal{D}}_{\mathrm{cut}}(\phi,t_{\infty},c,\varepsilon)$, and then we write, in short, $f(t)$ or $f(z)$ in $\check{\mathcal{D}}_{\mathrm{cut}}(\phi,t_{\infty},c,\varepsilon)$.

Suppose that $|\phi|<\pi/5.$ Let $y(x)$ be a solution of (1.1) corresponding to $(s_{j})_{1\leq j\leq 5}$ such that $s_{1}s_{4}\not=0.$ Then we have the following.

To deal with the error term explicitly, let us write (1.3) in the form

$$y(x)=(e^{-i\phi}x)^{1/2}\wp(e^{i\phi}t-t_{0}+h(e^{i\phi}t);g_{2}(\phi),g_{3}(\phi)).$$

Set $z=e^{i\phi}t$ and write

	$\displaystyle\mathfrak{p}=\mathfrak{p}(z)=\wp(z-t_{0};g_{2}(\phi),g_{3}(\phi)),$
	$\displaystyle\beta=\beta(z)=\frac{8e^{i\phi}}{5\omega_{\mathbf{a}}}\Bigl{(}\beta_{0}-\frac{5}{4}e^{-i\phi}J_{\mathbf{a}}(z-t_{0})+\frac{\vartheta^{\prime}}{\vartheta}\Bigl{(}\frac{z-t_{0}}{\omega_{\mathbf{a}}}+\nu,\tau\Bigr{)}\Bigr{)},$
	$\displaystyle\beta_{0}=\ln\frac{s_{1}}{s_{4}}-\frac{5}{4}e^{-i\phi}J_{\mathbf{a}}t_{0}+\pi i.$

The quantity $\beta(z)=e^{i\phi}B_{\mathrm{as}}(\phi,t)$ is the leading term of the correction function $e^{i\phi}B(\phi,t)$ in Theorem B, and $\beta(z)$ and $\mathfrak{p}(z)$ are bounded uniformly in $\mathcal{D}(\phi,t_{\infty},c,\varepsilon)$. Then we have the following explicit expression of $h(z).$

The integration constant $\beta_{0}$ appears in $h(z)$ as described in the following.

Let $c$, $\varepsilon$ and $\delta$ be as in Theorem A. The number $t_{\infty}$ is retaken if necessary, being denoted by the same letter $t_{\infty}$ in each appearance in our argument. In the proofs of our theorems we may suppose that $0<\delta<1.$

By the change of variables $t=\tfrac{4}{5}(e^{-i\phi}x)^{5/4},$ $y=(e^{-i\phi}x)^{1/2}v$, equation (1.1) or $(d/dx)((y^{\prime})^{2}-4y^{3}-2xy)=-2y$ is taken to

$$t\frac{d}{dt}a_{\phi}+\frac{6}{5}a_{\phi}=-\frac{8}{5}e^{i\phi}v$$

with

$$a_{\phi}=e^{-2i\phi}\bigl{(}v_{t}+(2/5)t^{-1}v\bigr{)}^{2}-4v^{3}-2e^{i\phi}v,$$

which is the Lagrangian or the Hamiltonian $2(e^{-i\phi}x)^{-3/2}\mathrm{H}(x,y(x),y^{\prime}(x))$ appearing in Theorem B, [15, Theorem 4]. Setting $a_{\phi}=A_{\phi}+t^{-1}B(\phi,t)$ with $e^{i\phi}t=z,$ $e^{i\phi}B(\phi,t)=b,$ we have the system of equations

$$\begin{split}&\bigl{(}v_{z}+(2/5)z^{-1}v\bigr{)}^{2}=4v^{3}+2e^{i\phi}v+A_{\phi}+z^{-1}b,\\ &b_{z}=-\frac{8}{5}e^{i\phi}v-\frac{6}{5}A_{\phi}-\frac{1}{5}z^{-1}b.\end{split}$$

(3.1)

Note that (1.5) is written as $B(\phi,t)=B_{\mathrm{as}}(\phi,t)+O(t^{-\delta})$. The system

$$\begin{split}&\mathfrak{p}_{z}^{2}=4\mathfrak{p}^{3}+2e^{i\phi}\mathfrak{p}+A_{\phi},\\ &\beta_{z}=-\frac{8}{5}e^{i\phi}\mathfrak{p}-\frac{6}{5}A_{\phi}\end{split}$$

(3.2)

admits a solution $(\mathfrak{p},\beta)=(\mathfrak{p}(z),\beta(z))=(\wp(z-t_{0};g_{2}(\phi),g_{3}(\phi)),e^{i\phi}B_{\mathrm{as}}(\phi,e^{-i\phi}z))$, where validity of the second equation is due to [15, Proposition 6, (3.13)]. System (3.2) is, at least formally, an approximation to (3.1).