Algebraic Solutions of the Painlevé-III $({\rm D}_{7})$ Equation

Differential Equations for Approximate Solutions
of Painlevé Equations: Application to the Algebraic
Solutions of the Painlevé-III $\boldsymbol{({\rm D}_{7})}$ Equation††This
paper is a contribution to the Special Issue on Evolution Equations, Exactly
Solvable Models and Random Matrices in honor of Alexander Its’ 70th birthday.
The full collection is available at
https://www.emis.de/journals/SIGMA/Its.html

Robert J. BUCKINGHAM a and Peter D. MILLER b

R.J. Buckingham and P.D. Miller

a) Department of Mathematical Sciences, University of Cincinnati,
a) P.O. Box 210025, Cincinnati, OH 45221, USA<EMAIL_ADDRESS>
b) Department of Mathematics, University of Michigan,
b) East Hall, 530 Church St., Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>
Received August 31, 2023, in final form January 05, 2024; Published online
January 20, 2024

It is well known that the Painlevé equations can formally degenerate to
autonomous differential equations with elliptic function solutions in suitable
scaling limits. A way to make this degeneration rigorous is to apply
Deift–Zhou steepest-descent techniques to a Riemann–Hilbert representation of
a family of solutions. This method leads to an explicit approximation formula
in terms of theta functions and related algebro-geometric ingredients that is
difficult to directly link to the expected limiting differential equation.
However, the approximation arises from an outer parametrix that satisfies
relatively simple conditions. By applying a method that we learned from
Alexander Its, it is possible to use these simple conditions to directly
obtain the limiting differential equation, bypassing the details of the
algebro-geometric solution of the outer parametrix problem. In this paper, we
illustrate the use of this method to relate an approximation of the algebraic
solutions of the Painlevé-III (D7) equation valid in the part of the complex
plane where the poles and zeros of the solutions asymptotically reside to a
form of the Weierstraß equation.

Painlevé-III (D7) equation; isomonodromy method; algebraic solutions;
Weierstraß equation

34E05; 34M55; 37K10

##  1 Introduction

###  1.1 Algebraic solutions of the Painlevé-III ($\boldsymbol{{\rm D}_{7}}$)
equation

The Painlevé-III (D6) equation for a function
$u\colon\mathbb{C}\to\mathbb{C}$, $x\mapsto u(x)$ is

$u^{\prime\prime}=\frac{(u^{\prime})^{2}}{u}-\frac{u^{\prime}}{x}+\frac{\alpha
u^{2}+\beta}{x}+\gamma u^{3}+\frac{\delta}{u},$

where $\alpha,\beta,\gamma,\delta\in\mathbb{C}$ are parameters. The Painlevé-
III (D7) equation

$u^{\prime\prime}=\frac{(u^{\prime})^{2}}{u}-\frac{u^{\prime}}{x}+\frac{\alpha
u^{2}+\beta}{x}+\frac{\delta}{u}$

is a special degenerate case in which $\gamma=0$ and $\alpha\delta\neq 0$. For
more information on this equation see, for example, Kitaev and Vartanian [11].
With the choice of parameters $\alpha=8$, $\beta=2n$, and $\delta=-1$, namely

$u^{\prime\prime}=\frac{(u^{\prime})^{2}}{u}-\frac{u^{\prime}}{x}+\frac{8u^{2}+2n}{x}-\frac{1}{u},$
(1.1)

the Painlevé-III (D7) equation admits an algebraic solution for each
$n\in\mathbb{Z}$. Specifically, define functions $R_{n}(\zeta)$ for
$n\in\mathbb{Z}$ via the recurrence relation

$\displaystyle 2\zeta
R_{n+1}(\zeta)R_{n-1}(\zeta)=-R_{n}(\zeta)R_{n}^{\prime\prime}(\zeta)+R_{n}^{\prime}(\zeta)^{2}-\frac{1}{\zeta}R_{n}(\zeta)R_{n}^{\prime}(\zeta)+2\bigl{(}\zeta^{2}-n\bigr{)}R_{n}(\zeta)^{2},$
$\displaystyle R_{0}(\zeta):=1,\qquad R_{1}(\zeta):=\zeta^{2}.$

Examples of these functions are

$\displaystyle R_{-3}(\zeta)=1+\frac{4}{\zeta^{2}}+\frac{5}{\zeta^{4}},\qquad$
$\displaystyle R_{-2}(\zeta)=\frac{1}{\zeta}+\frac{1}{\zeta^{3}},\qquad$
$\displaystyle R_{-1}(\zeta)=\frac{1}{\zeta},$ $\displaystyle
R_{2}(\zeta)=\zeta^{5}-\zeta^{3},\qquad$ $\displaystyle
R_{3}(\zeta)=\zeta^{9}-4\zeta^{7}+5\zeta^{5}.\qquad$

If $n\geq 0$, then $R_{n}(\zeta)$ is a polynomial in $\zeta$ known as an
_Ohyama polynomial_ [12]. The unique (on the Riemann surface of $x^{1/3}$)
algebraic solution to (1.1) is $u(x)=u_{n}(x)$, where

$u_{n}(x):=\frac{R_{n+1}\bigl{(}\sqrt{3}x^{1/3}\bigr{)}R_{n-1}\bigl{(}\sqrt{3}x^{1/3}\bigr{)}}{2\sqrt{3}R_{n}\bigl{(}\sqrt{3}x^{1/3}\bigr{)}^{2}},\qquad
n\in\mathbb{Z}.$

The $u_{n}(x)$ are rational functions of $x^{1/3}$. If one selects the
principal branch for $x^{1/3}$, then each of these produces three distinct
algebraic solutions on the complex plane: $u_{n}(x)$ and
$u_{n}\bigl{(}\mathrm{e}^{\pm 2\pi\mathrm{i}}x\bigr{)}$. Some examples are

$\displaystyle
u_{-2}(x)=\frac{9x^{5/3}+12x+5x^{1/3}}{2\bigl{(}3x^{2/3}+1\bigr{)}^{2}},\qquad$
$\displaystyle u_{-1}(x)=\frac{3x^{2/3}+1}{6x^{1/3}},\qquad$ $\displaystyle
u_{0}(x)=\frac{1}{2}x^{1/3},$ $\displaystyle
u_{1}(x)=\frac{3x^{2/3}-1}{6x^{1/3}},\qquad$ $\displaystyle
u_{2}(x)=\frac{9x^{5/3}-12x+5x^{1/3}}{2\bigl{(}3x^{2/3}-1\bigr{)}^{2}}.\qquad$

See Clarkson [5] for additional background on these functions. The Painlevé-
III (D7) equation (1.1) is invariant under the symmetries
$u(x)\mapsto\pm\mathrm{i}u(\pm\mathrm{i}x)$, $n\mapsto-n$, and it is easily
seen that $\pm\mathrm{i}u_{n}(\pm\mathrm{i}x)=u_{-n}\bigl{(}\mathrm{e}^{\pm
2\pi\mathrm{i}}x\bigr{)}$. In this paper, we will assume that $n\geq 0$ and
also restrict attention to the principal sheet $-\pi<\arg(x)<\pi$.

It is natural to introduce a scaled independent variable $y$ via

$x=n^{3/2}y.$ (1.2)

Under this scaling, plots show that the zeros and poles of
$u_{n}\bigl{(}n^{3/2}y\bigr{)}$ appear to be confined for $n$ large to a “bow-
tie” shaped bounded region in the $Y$-plane with $Y:=y^{1/3}$ that is
asymptotically independent of $n$. See Figure 1.

Figure 1: Left panel: a density plot of
$\big{|}U_{10}\bigl{(}Y^{3}\bigr{)}\big{|}$ and the boundary of the “bow-tie”
region $\mathcal{B}$. Right panel: a similar plot of $|U_{10}(y)|$ on the
principal sheet of the $y$-plane with $-\pi<\arg(y)<\pi$ and the sheet
boundary (branch cut) shown with a red line. In both plots, lighter/darker
color indicates larger/smaller modulus.

In [4], the limiting region was characterized precisely and it was proved that
on the unbounded exterior of this region, the related function
$U_{n}(y):=n^{-1/2}u_{n}\bigl{(}n^{3/2}y\bigr{)}$ converges as $n\to\infty$ to
the solution $\breve{U}(y)$ of the cubic equation

$8\breve{U}^{3}+2\breve{U}-y=0$ (1.3)

that behaves as $\breve{U}\approx\frac{1}{2}y^{1/3}$ for large $y$ with
$|\arg(y)|\leq 3\pi$.

###  1.2 Formal degeneration of Painlevé-III (D7)

Motivated by this convergence result, we may refine the scaling (1.2) by
introducing a second parameter $z\in\mathbb{C}$ and considering

$U(z)=U_{n}\bigl{(}y+n^{-1}z\bigr{)}=n^{-1/2}u_{n}\bigl{(}n^{3/2}\bigl{(}y+n^{-1}z\bigr{)}\bigr{)}$
(1.4)

as a function of $z$ for fixed $y\in\mathbb{C}$. Then, the Painlevé-III (D7)
equation (1.1) with parameter $n$ for $u_{n}(x)$ implies

$U^{\prime\prime}(z)=\frac{U^{\prime}(z)^{2}}{U(z)}+\frac{8}{y}U(z)^{2}+\frac{2}{y}-\frac{1}{U(z)}+\mathcal{O}\bigl{(}n^{-1}\bigr{)},\qquad
n\to\infty.$

Thus, neglecting the $\mathcal{O}\bigl{(}n^{-1}\bigr{)}$ error term, the
Painlevé-III (D7) equation formally degenerates to a one-parameter family,
parametrized by $y\in\mathbb{C}$, of autonomous differential equations that we
write for an unknown $\breve{U}(z)=\breve{U}(z;y)$:

$\breve{U}^{\prime\prime}(z)=\frac{\breve{U}^{\prime}(z)^{2}}{\breve{U}(z)}+\frac{8}{y}\breve{U}(z)^{2}+\frac{2}{y}-\frac{1}{\breve{U}(z)}.$
(1.5)

The cubic equation (1.3) corresponds to the solutions of (1.5) that are
independent of $z$. For non-constant solutions, (1.5) can be multiplied by the
integrating factor $\breve{U}^{\prime}(z)/\breve{U}(z)^{2}$ and then
integrated once to obtain

$\breve{U}^{\prime}(z)^{2}=\frac{16}{y}\breve{U}(z)^{3}+2E\breve{U}(z)^{2}-\frac{4}{y}\breve{U}(z)+1,$
(1.6)

wherein $E\in\mathbb{C}$ is an integration constant. Setting
$\breve{U}(z)=\frac{1}{4}y\wp(z-z_{0})-\frac{1}{24}yE$ for arbitrary $z_{0}$
one finds that $\wp(z)$ solves the Weierstraß equation [13, Chapter 23]

$\wp^{\prime}(z)^{2}=4\wp(z)^{3}-g_{2}\wp(z)-g_{3}$ (1.7)

with invariants

$g_{2}=\frac{16}{y^{2}}+\frac{E^{2}}{3}\qquad\text{and}\qquad
g_{3}=-\frac{16}{y^{2}}-\frac{8E}{3y^{2}}-\frac{E^{3}}{27}.$

Thus, one might expect that the algebraic solutions might be locally
approximated near a point $y$ with $y^{1/3}$ in the bounded “bow-tie” region
by a Weierstraß elliptic function of $z$ with invariants depending on $y$ and
$E$. However, this formalism does not explain how $E$ should be chosen given
$y$, nor does it determine the offset $z_{0}$, and it is not a rigorous
argument. For that one can use a Riemann–Hilbert characterization of
$U_{n}(y)$ that was also found in [4]. We describe this characterization next.

###  1.3 Riemann–Hilbert representation of
$\boldsymbol{U_{n}(y):=n^{-1/2}u_{n}(n^{3/2}y)}$

Given $y\in\mathbb{C}$ with $|\arg(y)|<\pi$, let $\Sigma_{0}^{\pm}$,
$\Sigma_{\infty}^{\pm}$, and $C^{\pm}$ be smooth, pairwise disjoint, oriented
open arcs in the $\eta$-plane as shown in Figure 2.

Figure 2: The jump contour for Riemann–Hilbert Problem 1.1.

The important properties of these arcs are the following.

  * •

$\Sigma_{0}^{+}$ terminates at the origin and $\Sigma_{0}^{-}$ originates from
the origin tangent to the line making the angle $2\arg(y)$ with the vertical.

  * •

$\Sigma_{\infty}^{+}$ terminates vertically at $\eta=\mathrm{i}\infty$ and
$\Sigma_{\infty}^{-}$ originates vertically from $\eta=-\mathrm{i}\infty$.

  * •

$\Sigma_{\infty}^{+}$, $\Sigma_{0}^{+}$, $C^{+}$, and $C^{-}$ share a common
initial point.

  * •

$\Sigma_{\infty}^{-}$, $\Sigma_{0}^{-}$, $C^{+}$, and $C^{-}$ share a common
terminal point.

  * •

$C^{+}$ (resp. $C^{-}$) lies in the component of
$\mathbb{C}\setminus\overline{\Sigma_{\infty}^{-}\cup\Sigma_{0}^{-}\cup\Sigma_{0}^{+}\cup\Sigma_{\infty}^{+}}$
containing large positive (resp. negative) real values of $\eta$.

Denote

$\Phi(\eta,y):=\mathrm{i}\eta-y(-\mathrm{i}\eta)^{-1/2}$ (1.8)

with the branch cut of the function $\eta\mapsto\Phi(\eta,y)$ taken to
coincide with $\overline{\Sigma_{0}^{-}\cup\Sigma_{\infty}^{-}}$ and
$(-\mathrm{i}\eta)^{-1/2}$ taken to be real and positive for positive
imaginary $\eta$ sufficiently large, and set

$\widetilde{\mathbf{E}}:=\frac{1}{\sqrt{2}}\begin{bmatrix}\mathrm{e}^{5\pi\mathrm{i}/6}&\mathrm{e}^{-5\pi\mathrm{i}/6}\\\
\mathrm{e}^{-\mathrm{i}\pi/6}&\mathrm{e}^{-5\pi\mathrm{i}/6}\end{bmatrix}.$
(1.9)

Consider the following problem, in which $\sigma_{3}:=\mathrm{diag}(1,-1)$
denotes the third Pauli matrix.

######  Riemann–Hilbert Problem 1.1 (scaled algebraic Painlevé-III (D7)
solutions, [4, Section 4.1]).

Let $y\in\mathbb{C}$ with $|\arg(y)|\leq\pi$ and $n\in\mathbb{Z}_{>0}$ be
given. Seek a $2\times 2$ matrix function
$\eta\mapsto\mathbf{Z}^{(n)}(\eta,y)$ with the following properties:

  * Analyticity: $\eta\mapsto\mathbf{Z}^{(n)}(\eta,y)$ is analytic for $\eta\in\mathbb{C}\setminus\overline{\Sigma_{\infty}^{+}\cup\Sigma_{\infty}^{-}\cup\Sigma_{0}^{-}\cup\Sigma_{0}^{+}\cup C^{+}\cup C^{-}}$.

  * Jump conditions: $\eta\mapsto\mathbf{Z}^{(n)}(\eta,y)$ takes continuous boundary values on the jump contour except at $\eta=0$, and these boundary values are related by the jump conditions

$\displaystyle\mathbf{Z}_{+}^{(n)}(\eta,y)=\mathbf{Z}^{(n)}_{-}(\eta,y)\mathrm{e}^{-n\Phi(\eta,y)\sigma_{3}}\begin{bmatrix}1&0\\\
\mathrm{i}&1\end{bmatrix}\mathrm{e}^{n\Phi(\eta,y)\sigma_{3}},\qquad\eta\in\Sigma_{\infty}^{+}\cup\Sigma_{0}^{+},$
$\displaystyle\mathbf{Z}_{+}^{(n)}(\eta,y)=\mathbf{Z}^{(n)}_{-}(\eta,y)\mathrm{e}^{-n\Phi(\eta,y)\sigma_{3}}\begin{bmatrix}1&0\\\
-\mathrm{i}&1\end{bmatrix}\mathrm{e}^{n\Phi(\eta,y)\sigma_{3}},\qquad\eta\in
C^{+}\cup C^{-},$
$\displaystyle\mathbf{Z}_{+}^{(n)}(\eta,y)=\mathbf{Z}^{(n)}_{-}(\eta,y)\mathrm{e}^{-n\Phi_{-}(\eta,y)\sigma_{3}}(-1)^{n}\begin{bmatrix}1&-\mathrm{i}\\\
0&1\end{bmatrix}\mathrm{e}^{n\Phi_{+}(\eta,y)\sigma_{3}},\qquad\eta\in\Sigma_{\infty}^{-},$
$\displaystyle\mathbf{Z}_{+}^{(n)}(\eta,y)=\mathbf{Z}^{(n)}_{-}(\eta,y)\mathrm{e}^{-n\Phi_{-}(\eta,y)\sigma_{3}}\begin{bmatrix}0&(-1)^{n}\mathrm{i}\\\
(-1)^{n}\mathrm{i}&0\end{bmatrix}\mathrm{e}^{n\Phi_{+}(\eta,y)\sigma_{3}},\qquad\eta\in\Sigma_{0}^{-}.$

Here a subscript $+$ $($resp. $-)$ refers to a boundary value taken on an
oriented arc from its left $($resp. right$)$ side.

  * Normalization: $\mathbf{Z}^{(n)}(\eta,y)(-\mathrm{i}\eta)^{n\sigma_{3}/2}\to\mathbb{I}$ as $\eta\to\infty$.

  * Behavior as $\eta\to 0$: The limit of $\mathbf{Z}^{(n)}(\eta,y)\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$ as $\eta\to 0$ exists.

In the last two conditions, powers of $-\mathrm{i}\eta$ are taken to be cut on
$\Sigma_{0}^{-}\cup\Sigma_{\infty}^{-}$ and agree with principal branches for
large positive imaginary $\eta$.

The last property can be used to define a matrix $\mathbf{B}^{(n)}_{0}(y)$ by

$\mathbf{B}^{(n)}_{0}(y):=(-\mathrm{i}ny)^{n\sigma_{3}}\Big{[}\lim_{\eta\to
0}\mathbf{Z}^{(n)}(\eta,y)\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}\Big{]}(-\mathrm{i}ny)^{(-1)^{n}\sigma_{3}/2},$
(1.10)

with fractional powers of $-\mathrm{i}ny$ defined by continuation of the
principal branch for $y>0$. Then, a rescaling of the algebraic solution of
Painlevé-III (D7) is encoded in the solution of Riemann–Hilbert Problem 1.1 by
the formula

$U_{n}(y):=n^{-1/2}u_{n}\bigl{(}n^{3/2}y\bigr{)}=ny\begin{cases}\mathrm{e}^{-5\pi\mathrm{i}/6}B_{0,12}^{(n)}(y)B_{0,22}^{(n)}(y),&\text{$n$
even},\\\
\mathrm{e}^{5\pi\mathrm{i}/6}B_{0,11}^{(n)}(y)B_{0,21}^{(n)}(y),&\text{$n$
odd.}\end{cases}$ (1.11)

###  1.4 Main aims of the paper

The conditions of Riemann–Hilbert Problem 1.1 involve the large parameter $n$
in an explicit way and there are well-known techniques originating in the
Deift–Zhou steepest-descent method [6] for analyzing such problems. One needs
to firstly control the large exponential factors in the jump matrices by
introducing an appropriate scalar $g$-function. The difference between $g$ and
$\Phi$ is a function $h$ whose derivative satisfies an algebraic equation
defining the _spectral curve_. We show below that when $y$ corresponds to a
point in the “bow-tie”, the spectral curve has genus $1$ and that the
landscape of $\operatorname{Re}(h)$ has the properties necessary to continue
the analysis. The next step involves exploiting analytic factorizations of
jump matrices to “open lenses” by moving certain factors off the jump contour
onto nearby arcs. After this step, all jump matrices decay rapidly to the
identity as $n\to\infty$ except along certain arcs where in the same limit a
nontrivial limiting jump matrix emerges instead. In the third step one uses
the limiting jump matrix to define a Riemann–Hilbert problem for an
approximation called an _outer parametrix_ ; in addition one or more _inner
parametrices_ are needed near certain points where the convergence of the jump
matrix is not uniform. One pieces together a _global parametrix_ from the
outer and inner parametrices to define an unjustified (at this point)
approximation of the solution of the “opened lenses” problem. Finally, one
proves a convergence theorem by showing that the matrix ratio of the unknown
and its global parametrix solves a special kind of Riemann–Hilbert problem (a
_small-norm problem_) for which the solution is uniformly close to the
identity.

In this scheme, the approximate formula for $U_{n}(y)$ comes from the outer
parametrix. In the situation we discuss in this paper, that $y$ corresponds to
a point in the “bow-tie”, this outer parametrix can be written explicitly in
terms of theta functions of genus $1$ and elliptic integrals. Actually, we
first replace $y$ with $y+n^{-1}z$ as in (1.4) but use a $g$-function
depending on $y$ only, and then one obtains an approximate formula for
$U_{n}\bigl{(}y+n^{-1}z\bigr{)}$ explicitly involving the independent variable
$z$ that should be related to the Weierstraß equation if the formal reasoning
described in Section 1.2 above is correct. However, it is very difficult to
prove such a connection directly from the approximation formula for
$U_{n}\bigl{(}y+n^{-1}z\bigr{)}$; at the very least it is a calculation that
is a complicated diversion from what should be a relatively simple path from
(1.1) to (1.6) or (1.7).

Our aim in this paper is not to give all details of the convergence proof but
rather we focus on explaining a reasonably effective way to make the
connection between the outer parametrix Riemann–Hilbert problem – whose
conditions are far simpler than the elliptic solution they generate – and the
limiting differential equation (1.6). Our approach also determines the value
of the integration constant $E$ in (1.6) as a function of $y$ (equivalently
the value of both invariants in the Weierstraß equation (1.7) are so-
determined). For those who would like to see the basic idea of this method
illustrated in a simple setting, in Appendix A, we show directly (without
reference to the known exact solution) that certain quantities derived from a
toy Riemann–Hilbert problem satisfy simple differential equations. We
originally learned this method from Alexander Its (see, for example, [8,
Chapter 8] and [9]), and it is a pleasure to write this article in his honor.

##  2 Spectral curves of genus 1

Motivated by [4, Section 4.3], for given complex parameters $y$ and $c$, we
introduce a function $\eta\mapsto h_{\eta}(\eta,y,c)$ determined up to a sign
by the equation

$\left(\frac{\partial
h}{\partial\eta}(\eta,y,c)\right)^{2}=f(\eta,y,c):=\frac{P(-\mathrm{i}\eta,y,c)}{(-\mathrm{i}\eta)^{3}},\qquad
P(\mu,y,c):=-\mu^{3}+\mu^{2}+c\mu-\frac{y^{2}}{4}.$ (2.1)

Considered as an algebraic relation between $h_{\eta}$ and $\eta$, this
defines the _spectral curve_ , which will have genus 1 provided $c$ and $y$
are chosen so that the three roots of the cubic $\mu\mapsto P(\mu,y,c)$ are
distinct and nonzero. Note that if $y\neq 0$, $P(0,y,c)\neq 0$ so $\mu=0$
cannot be a root. Let us label the three distinct nonzero (for $y\neq 0$ and
generic $c$) roots by $\mu=s_{j}$, $j=1,2,3$, so that $P(\mu,y,c)=-(\mu-
s_{1})(\mu-s_{2})(\mu-s_{3})$. Let $\Sigma_{0,1}$ be an arc in the
$\eta$-plane joining $\eta=0$ to $\eta=\mathrm{i}s_{1}$, let $\Sigma_{2,3}$ be
an arc in the $\eta$-plane joining $\eta=\mathrm{i}s_{2}$ to
$\eta=\mathrm{i}s_{3}$, and assume that
$\Sigma_{0,1}\cap\Sigma_{2,3}=\varnothing$. Then we may define
$h_{\eta}(\eta,y,c)$ unambiguously using (2.1) by assuming that $\eta\mapsto
h_{\eta}(\eta,y,c)$ is analytic for
$\eta\in\mathbb{C}\setminus\overline{\Sigma_{0,1}\cup\Sigma_{2,3}}$ and that
$h_{\eta}(\eta,y,c)=-\mathrm{i}-\frac{1}{2}\eta^{-1}+\mathcal{O}\bigl{(}\eta^{-2}\bigr{)}$
as $\eta\to\infty$. Then also,

$h_{\eta}(\eta,y,c)=\frac{1}{2}\mathrm{i}y(-\mathrm{i}\eta)^{-3/2}+O\bigl{(}\eta^{-1/2}\bigr{)},\qquad\eta\to
0,$

with the power function being cut on $\Sigma_{\infty}^{-}\cup\Sigma_{0,1}$ and
coinciding with the principal branch for large positive imaginary $\eta$.

Next we attempt to determine $c$ given $y\neq 0$ by imposing two real
_Boutroux conditions_ :

$\displaystyle
I_{1,2}(y,c):=\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\partial
h}{\partial\eta}(\eta,y,c)\,\mathrm{d}\eta\bigg{)}=0,$ $\displaystyle
I_{2,3}(y,c):=\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{2}}^{\mathrm{i}s_{3}}\frac{\partial
h}{\partial\eta}(\eta,y,c)\,\mathrm{d}\eta\bigg{)}=0,$ (2.2)

where the path of integration in each case lies in the domain of analyticity
of $\eta\mapsto h_{\eta}(\eta,y,c)$. Although the latter domain is multiply
connected, and hence $I_{1,2}$ and $I_{2,3}$ are only well-defined modulo a
finitely-generated symmetry group, the conditions (2.2) are independent of the
specific choice of paths due to the fact that $h_{\eta}(\eta,y,c)$ changes
sign across its branch cuts and the fact that the residue of
$h_{\eta}(\eta,y,c)$ at $\eta=\infty$ is real. If we introduce the real and
imaginary parts of $c$ by $c_{\mathrm{R}}:=\operatorname{Re}(c)$ and
$c_{\mathrm{I}}:=\operatorname{Im}(c)$ so that
$c=c_{\mathrm{R}}+\mathrm{i}c_{\mathrm{I}}$, then we have

$\displaystyle\frac{\partial}{\partial
c_{\mathrm{R}}}\bigg{[}\bigg{(}\frac{\partial
h}{\partial\eta}(\eta,y,c)\bigg{)}^{2}\bigg{]}=-\eta^{-2},$
$\displaystyle\frac{\partial}{\partial
c_{\mathrm{I}}}\bigg{[}\bigg{(}\frac{\partial
h}{\partial\eta}(\eta,y,c)\bigg{)}^{2}\bigg{]}=-\mathrm{i}\eta^{-2},$

from which it follows that if paths of integration are selected so that
$I_{1,2}(y,c)$ and $I_{2,3}(y,c)$ depend smoothly on
$(c_{\mathrm{R}},c_{\mathrm{I}})$,

$\begin{bmatrix}\dfrac{\partial I_{1,2}}{\partial
c_{\mathrm{R}}}&\dfrac{\partial I_{1,2}}{\partial
c_{\mathrm{I}}}\vspace{1mm}\\\ \dfrac{\partial I_{2,3}}{\partial
c_{\mathrm{R}}}&\dfrac{\partial I_{2,3}}{\partial
c_{\mathrm{I}}}\end{bmatrix}=\begin{bmatrix}\displaystyle-2\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)}&\displaystyle
2\operatorname{Im}\bigg{(}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)}\vspace{1mm}\\\
\displaystyle-2\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{2}}^{\mathrm{i}s_{3}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)}&\displaystyle
2\operatorname{Im}\bigg{(}\int_{\mathrm{i}s_{2}}^{\mathrm{i}s_{3}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)}\end{bmatrix}.$

Therefore, the Jacobian determinant is

$\frac{\partial(I_{1,2},I_{2,3})}{\partial(c_{\mathrm{R}},c_{\mathrm{I}})}=-4\operatorname{Im}\bigg{(}\bigg{[}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{]}^{*}\int_{\mathrm{i}s_{2}}^{\mathrm{i}s_{3}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)}.$

Since $\bigl{(}\eta^{2}h_{\eta}(\eta,y,c)\bigr{)}^{2}=-\mathrm{i}\eta
P(-\mathrm{i}\eta,y,c)$ is a quartic polynomial with roots at $\eta=0$,
$\mathrm{i}s_{1}$, $\mathrm{i}s_{2}$, $\mathrm{i}s_{3}$, the two integral
factors on the right-hand side are complete elliptic integrals of the first
kind over paths that form a basis for homology on the corresponding elliptic
curve. Hence under the assumption that all four roots are distinct, the
Jacobian is nonzero [7, Corollary 1]. By the implicit function theorem,
whenever a pair $y\neq 0$ and $c\in\mathbb{C}$ are such that both equations
(2.2) hold and that $P(\cdot,y,c)$ has distinct roots, the solution
$(c_{\mathrm{R}},c_{\mathrm{I}})$ of (2.2) can therefore be continued smoothly
to nearby values of $y\in\mathbb{C}$.

We now assume that $y$ and $c$ are related so that the conditions (2.2) hold.
The function $\eta\mapsto h_{\eta}(\eta,y,c)$ extends to a single-valued
function on the two-sheeted Riemann surface $\mathcal{R}$ over the
$\eta$-plane defined by the spectral curve (2.1). The differential
$h_{\eta}(\eta,y,c)\,\mathrm{d}\eta$ is meromorphic on $\mathcal{R}$ with
double poles (in suitable local coordinates) at the two points over
$\eta=\infty$ and at the branch point $\eta=0$ and no other singularities. The
residues of $h_{\eta}(\eta,y,c)\,\mathrm{d}\eta$ at the two points over
$\eta=\infty$ are opposite real values $\pm\frac{1}{2}$ and then the residue
at $\eta=0$ necessarily vanishes. It follows that if $c$ is such that the
conditions (2.2) hold, then the multi-valued function $h(\eta,y,c)$ defined on
$\mathcal{R}$ up to an integration constant by contour integration of
$h_{\eta}(\eta,y,c)\,\mathrm{d}\eta$ has a real part that is single-valued on
$\mathcal{R}$. Selecting the integration constant such that
$\operatorname{Re}(h(\eta,y,c))$ vanishes at any one of the points
$\eta=\mathrm{i}s_{j}$, $j=1,2,3$ (and hence at all three of them by (2.2)),
the projection of the zero level set of $\operatorname{Re}(h(\eta,y,c))$ to
either sheet of $\mathcal{R}$ is the same set in the $\eta$-plane, which we
denote by $K$.

It is known [4, Theorem 3] that for large $n$,
$u_{n}\bigl{(}n^{3/2}Y^{3}\bigr{)}$ is pole- and zero-free for $Y$ on an
unbounded domain $\mathcal{E}$ whose complement is a “bow-tie” shaped region
$\mathcal{B}:=\mathbb{C}\setminus\mathcal{E}$ in the $Y$-plane that is
symmetric with respect to reflection in the real and imaginary axes. The
interior of $\mathcal{B}$ is the disjoint union of two “wings”, one on either
side of the imaginary axis. The wings are joined at the origin only, and they
are bounded in part by the straight-line segments joining the pairs
$Y=\pm\bigl{(}2^{1/3}/3^{1/2}\bigr{)}\mathrm{e}^{\mathrm{i}\pi/6}$ and
$\pm\bigl{(}2^{1/3}/3^{1/2}\bigr{)}\mathrm{e}^{5\pi\mathrm{i}/6}$. The set
$\mathcal{B}\cap\mathbb{R}$ consists of the interval
$\bigl{[}-y_{\mathrm{c}}^{1/3},y_{\mathrm{c}}^{1/3}\bigr{]}$, where
$y_{\mathrm{c}}\approx 0.29177$. See the left-hand panel of Figure 1.

######  Lemma 2.1.

Assume that $\operatorname{Re}(y)>0$ and that $Y=y^{\frac{1}{3}}$ lies in the
open interior of $\mathcal{B}$. Then, there is a well-defined value
$c=c_{1}(y)\in\mathbb{C}$, a smooth function of real variables
$\operatorname{Re}(y)$ and $\operatorname{Im}(y)$ but not analytic in $y$,
such that the following hold.

  * •

$\eta\mapsto P(-\mathrm{i}\eta,y,c)$ has three distinct complex roots denoted
$\eta=\mathrm{i}s_{j}$, $j=1,2,3$.

  * •

The Boutroux conditions (2.2) are satisfied.

  * •

There is a simple arc $\Sigma_{h}$ originating at $\eta=-\mathrm{i}\infty$ and
terminating at one of the roots denoted $\eta=\mathrm{i}s_{3}$ that passes
through in order the intermediate points $\eta=\mathrm{i}s_{1}$, $\eta=0$, and
$\eta=\mathrm{i}s_{2}$, and an integration constant, such that $\eta\mapsto
h(\eta,y,c)$ is analytic in $\mathbb{C}\setminus\Sigma_{h}$ and continuous up
to $\Sigma_{h}\setminus\\{0\\}$ with
$\operatorname{Re}(h(\mathrm{i}s_{j},y,c))=0$ for $j=1,2,3$, and such that
$\eta\mapsto\operatorname{Re}(h(\eta,y,c))$ is continuous on
$\mathbb{C}\setminus\\{0\\}$ and harmonic on
$\mathbb{C}\setminus(\Sigma_{0,1}\cup\Sigma_{2,3})$, where $\Sigma_{0,1}$
denotes the arc of $\Sigma_{h}$ between $\eta=\mathrm{i}s_{1}$ and $\eta=0$
while $\Sigma_{2,3}$ denotes the arc of $\Sigma_{h}$ between
$\eta=\mathrm{i}s_{2}$ and $\eta=\mathrm{i}s_{3}$, and the latter are taken to
be the branch cuts of $\eta\mapsto h_{\eta}(\eta,y,c)$.

  * •

The zero level set $K$ of $\operatorname{Re}(h(\eta,y,c))$ consists of the
arcs $\Sigma_{0,1}$ and $\Sigma_{2,3}$, two arcs joining
$\eta=\mathrm{i}s_{1}$ to $\eta=\mathrm{i}s_{2}$ that bound a region
containing $\eta=0$, and two unbounded arcs emanating from
$\eta=\mathrm{i}s_{3}$, one tending to $\eta=\infty$ in the left half-plane
and one tending to $\eta=\infty$ in the right half-plane.
$\operatorname{Re}(h(\eta,y,c))$ changes sign across each of these arcs except
for $\Sigma_{0,1}$ and $\Sigma_{2,3}$.

  * •

If $y>0$ (and also $y<y_{\mathrm{c}}\approx 0.29177$ so that $Y=y^{1/3}$ lies
in $\mathcal{B}$) then $s_{1}<0<s_{2}<s_{3}$, $\Sigma_{h}$ consists of the
part of the imaginary axis in the $\eta$-plane below the point
$\eta=\mathrm{i}s_{3}$, and $K^{*}=-K$.

We give the proof in the appendix. The structure of the set $K$ allows the
arcs of the jump contour for Riemann–Hilbert Problem 1.1 to be chosen in a
useful way, as illustrated in the left-hand panel of Figure 3 for
$0<y<y_{\mathrm{c}}$. The picture is topologically equivalent provided that
$\operatorname{Re}(y)>0$ and $Y=y^{1/3}$ lies in the interior of $\mathcal{B}$
as in the conditions of Lemma 2.1.

Figure 3: Jump contours and sign chart of $\operatorname{Re}(h)$ in the
$\eta$-plane for $0<y<y_{\mathrm{c}}$. Left panel: the zero level set $K$ of
$\operatorname{Re}(h(\eta,y,c))$ shown in gray and orange (orange indicates
the branch cuts $\Sigma_{0,1}$ and $\Sigma_{2,3}$ of $h_{\eta}(\eta,y,c)$),
and the relative placement of the jump contour for Riemann–Hilbert Problem
1.1. The sign of $\operatorname{Re}(h)$ is as indicated and
$\operatorname{Re}(h)$ only changes sign across the gray arcs of $K$. Note
that the contour $\Sigma_{\infty}^{+}$ actually extends from
$\eta=\mathrm{i}s_{2}$, taken as the junction point of $C^{-}$, $C^{+}$, and
$\Sigma_{0}^{+}$, all the way up the positive imaginary axis, passing through
$\Sigma_{2,3}$. Likewise $\Sigma_{\infty}^{-}$ extends from
$-\mathrm{i}\infty$ up to $\mathrm{i}s_{1}$. The arc $\Sigma_{0}^{-}$
coincides with the branch cut $\Sigma_{0,1}$. Center panel: the jump contour
for $\mathbf{N}^{(n)}(\eta,y,z)$ has two additional arcs on the left and right
of the branch cut $\Sigma_{2,3}$ after opening a lens. Right panel: the jump
contour for $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ consists of the
arcs $\Sigma_{\infty}^{-}$, $\Sigma_{0,1}$, $\Sigma_{0}^{+}$, and
$\Sigma_{2,3}$ (shown with solid curves; the dashed arcs in the jump contour
for $\mathbf{N}^{(n)}(\eta,y,z)$ have been neglected).

##  3 Introduction of $\boldsymbol{g}$-function and lens opening

We assume from now on that $\operatorname{Re}(y)>0$ and that $Y=y^{1/3}$ lies
in the interior of $\mathcal{B}$. Also, since $c=c_{1}(y)$ is determined from
$y$ according to Lemma 2.1, we will write $h(\eta,y)=h(\eta,y,c_{1}(y))$ going
forward. Under these assumptions, in this section we will implement the first
two steps of the asymptotic analysis of Riemann–Hilbert Problem 1.1 with $y$
replaced by $y+n^{-1}z$.

###  3.1 First step: introduction of $\boldsymbol{g}$-function

From $h(\eta,y)$, we define a related function by

$g=g(\eta,y)=\Phi(\eta,y)+h(\eta,y),\qquad\eta\in\mathbb{C}\setminus\Sigma_{h},$

with $\Phi(\eta,y)$ defined by (1.8). In particular, there is a function
$g_{0}(y)$ such that

$g(\eta,y)=-\frac{1}{2}\log(-\mathrm{i}\eta)+g_{0}(y)+\mathcal{O}\bigl{(}\eta^{-1/2}\bigr{)},\qquad\eta\to\infty.$
(3.1)

We then use $g(\eta,y)$ to modify the matrix
$\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}$ by setting

$\mathbf{M}^{(n)}(\eta,y,z):=\mathrm{e}^{ng_{0}(y)\sigma_{3}}\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}\mathrm{e}^{-ng(\eta,y)\sigma_{3}}.$
(3.2)

Note that while $\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}$ depends on
$(y,z)$ only through the combination $y+n^{-1}z$, the function
$\mathbf{M}^{(n)}(\eta,y,z)$ involves these variables in a more complicated
fashion. However, as a function of $\eta$, $\mathbf{M}^{(n)}(\eta,y,z)$ is
analytic where $\mathbf{Z}^{(n)}(\eta,y,z)$ is, and according to (3.1), is
normalized to the identity as $\eta\to\infty$:
$\mathbf{M}^{(n)}(\infty,y,z)=\mathbb{I}$.

###  3.2 Second step: opening a lens

We next open a lens about the branch cut $\Sigma_{2,3}$ of $h_{\eta}(\eta,y)$.
Let $\Lambda^{+}$ (resp. $\Lambda^{-}$) denote a lens-shaped region abutting
$\Sigma_{2,3}$ (oriented from $\mathrm{i}s_{2}(y)$ to $\mathrm{i}s_{3}(y)$) on
its left (resp. right) side and lying in the domain where
$\operatorname{Re}(h)<0$. We then define $\mathbf{N}^{(n)}(\eta,y,z)$ by
setting

$\displaystyle\mathbf{N}^{(n)}(\eta,y,z):=\mathbf{M}^{(n)}(\eta,y,z)\begin{bmatrix}1&\mathrm{i}\mathrm{e}^{2nh(\eta,y)}\mathrm{e}^{2z(-\mathrm{i}\eta)^{-1/2}}\\\
0&1\end{bmatrix},\qquad\eta\in\Lambda^{+},$
$\displaystyle\mathbf{N}^{(n)}(\eta,y,z):=\mathbf{M}^{(n)}(\eta,y,z)\begin{bmatrix}1&-\mathrm{i}\mathrm{e}^{2nh(\eta,y)}\mathrm{e}^{2z(-\mathrm{i}\eta)^{-1/2}}\\\
0&1\end{bmatrix},\qquad\eta\in\Lambda^{-},$

and elsewhere we take
$\mathbf{N}^{(n)}(\eta,y,z):=\mathbf{M}^{(n)}(\eta,y,z)$. The jumps of
$\mathbf{N}^{(n)}(\eta,y,z)$ on the outer lens boundaries, oriented toward the
endpoint $\eta=\mathrm{i}s_{3}$ as shown in the center panel of Figure 3, both
read

$\mathbf{N}^{(n)}_{+}(\eta,y,z)=\mathbf{N}^{(n)}_{-}(\eta,y,z)\begin{bmatrix}1&-\mathrm{i}\mathrm{e}^{2nh(\eta,y)}\mathrm{e}^{2z(-\mathrm{i}\eta)^{-1/2}}\\\
0&1\end{bmatrix}.$

On the part of $\Sigma_{\infty}^{+}$ coinciding with the branch cut
$\Sigma_{2,3}$ in between the lenses, we get the modified jump condition

$\displaystyle\mathbf{N}^{(n)}_{+}(\eta,y,z)$
$\displaystyle\qquad=\mathbf{N}^{(n)}_{-}(\eta,y,z)\begin{bmatrix}0&\mathrm{i}\mathrm{e}^{n(h_{+}(\eta,y)+h_{-}(\eta,y))}\mathrm{e}^{2z(-\mathrm{i}\eta)^{-1/2}}\\\
\mathrm{i}\mathrm{e}^{-n(h_{+}(\eta,y)+h_{-}(\eta,y))}\mathrm{e}^{-2z(-\mathrm{i}\eta)^{-1/2}}&0\end{bmatrix}.$

By the Boutroux conditions (2.2) asserted in Lemma 2.1, it holds that
$h_{+}(\eta,y)+h_{-}(\eta,y)=\mathrm{i}\psi(y)$ for $\eta\in\Sigma_{2,3}$,
where $\psi(y)$ is a real quantity. By similar arguments as in [4], it also
holds that $h_{+}(\eta,y)+h_{-}(\eta,y)=\mathrm{i}\xi(y)$ for
$\eta\in\Sigma_{0}^{-}$, where $\xi(y)$ is a real quantity. Finally,
$h_{+}(\eta,y)-h_{-}(\eta,y)=\mathrm{i}\kappa(y)$ for
$\eta\in\Sigma_{0}^{+}=\Sigma_{0,1}$ where $\kappa(y)$ is a real quantity.

##  4 Outer parametrix Riemann–Hilbert problem

In addition to the conditions placed so far on $y$, we now suppose that
$z\in\mathbb{C}$ is bounded. Then the jump matrices for
$\mathbf{N}^{(n)}(\eta,y,z)$ all decay exponentially rapidly to the identity
except near the two branch cuts $\Sigma_{0,1}$ and $\Sigma_{2,3}$ of
$h_{\eta}(\eta,y)$, near the arc $\Sigma_{\infty}^{-}$, and near the arc
$\Sigma_{0}^{+}$. In the branch-cut arcs, the jump conditions read as follows:

$\displaystyle\mathbf{N}^{(n)}_{+}(\eta,y,z)=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\\!\\!\begin{bmatrix}0&\mathrm{i}\mathrm{e}^{\mathrm{i}n\psi(y)}\\\
\mathrm{i}\mathrm{e}^{-\mathrm{i}n\psi(y)}&0\end{bmatrix}\\!\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\\!$
(4.1)

for $\eta\in\Sigma_{2.3}$, where the boundary values are determined by
orientation of $\Sigma_{2,3}$ toward $\eta=\mathrm{i}s_{3}(y)$, and

$\displaystyle\mathbf{N}^{(n)}_{+}(\eta,y,z)=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}_{-}\sigma_{3}}\begin{bmatrix}0&(-1)^{n}\mathrm{i}\mathrm{e}^{\mathrm{i}n\xi(y)}\\\
(-1)^{n}\mathrm{i}\mathrm{e}^{-\mathrm{i}n\xi(y)}&0\end{bmatrix}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}_{+}\sigma_{3}}\\!\\!\\!$
(4.2)

for $\eta\in\Sigma_{0,1}$, where $\Sigma_{0,1}$ is oriented toward
$\eta=\mathrm{i}s_{1}(y)$. Note that the sum of the boundary values of
$(-\mathrm{i}\eta)^{-1/2}$ vanishes on this contour, so as the jump matrix is
off-diagonal, the outer exponential factors in (4.2) could have been omitted,
but it is convenient to write them here anyway. The jump matrices in (4.1) and
(4.2) are rapidly oscillatory in the parameter $y$, but the only dependence on
$\eta$ enters via the diagonal conjugating factors with exponents proportional
to $z$. Next, there is a residual jump across the contour
$\Sigma_{\infty}^{-}$ with orientation toward $\eta=\mathrm{i}s_{1}(y)$ in the
limit $n\to\infty$:

$\displaystyle\mathbf{N}^{(n)}_{+}(\eta,y,z)$
$\displaystyle=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}_{-}\sigma_{3}}\begin{bmatrix}1&-\mathrm{i}\mathrm{e}^{n(h_{+}(\eta,y)+h_{-}(\eta,y))}\\\
0&1\end{bmatrix}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}_{+}\sigma_{3}}$
$\displaystyle=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}_{-}\sigma_{3}}\left(\mathbb{I}+\text{exponentially
small}\right)\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}_{+}\sigma_{3}}$

for $\eta\in\Sigma_{\infty}^{-}$, with the estimate of the error arising in
the limit $n\to\infty$ because $\operatorname{Re}(h)<0$ on
$\Sigma_{\infty}^{-}$ (see Figure 3). Finally, there is also a residual jump
across the contour $\Sigma_{0}^{+}$ with orientation toward $\eta=0$ in the
limit $n\to\infty$:

$\displaystyle\mathbf{N}^{(n)}_{+}(\eta,y,z)$
$\displaystyle=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\begin{bmatrix}\mathrm{e}^{-\mathrm{i}n\kappa(y)}&0\\\
\mathrm{i}\mathrm{e}^{-n(h_{+}(\eta,y)+h_{-}(\eta,y))}&\mathrm{e}^{\mathrm{i}n\kappa(y)}\end{bmatrix}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}$
(4.3)
$\displaystyle=\mathbf{N}^{(n)}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\bigl{(}\mathrm{e}^{-\mathrm{i}n\kappa(y)\sigma_{3}}+\text{exponentially
small}\bigr{)}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}$

for $\eta\in\Sigma_{0}^{+}$.

Neglecting the exponentially small terms, (4.1)–(4.3) define the limiting jump
conditions to be satisfied by an _outer parametrix_. The convergence of the
jump matrices overall to these three limits is not uniform near the three
points $\eta=\mathrm{i}s_{j}(y)$, $j=1,2,3$, and one can install standard
_inner parametrices_ near each of these points constructed from Airy functions
to correctly approximate $\mathbf{N}^{(n)}(\eta,y,z)$ nearby; see [4, Section
4.4.2] for some details in a very similar setting. However, no inner
parametrix is needed near $\eta=0$ if one specifies suitable behavior for the
outer parametrix at this point matching that inherited from
$\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}$.

By definition, the outer parametrix
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ is then the solution of the
following Riemann–Hilbert problem, which retains all of the most important
properties of $\mathbf{N}^{(n)}(\eta,y,z)$ when $\eta$ is bounded away from
$\eta=\mathrm{i}s_{j}(y)$, $j=1,2,3$ and builds in the key property near these
points needed to facilitate a good match between the outer and inner (Airy)
parametrices near those points, namely a negative one-fourth power
singularity.

######  Riemann–Hilbert Problem 4.1.

Given $n\in\mathbb{Z}_{>0}$, $y\in\mathbb{C}$ with $\operatorname{Re}(y)>0$
and $Y=y^{1/3}$ in the interior of $\mathcal{B}$, and $z\in\mathbb{C}$, seek a
$2\times 2$ matrix-valued function
$\bigl{(}\mathbb{C}\setminus\overline{\Sigma_{0,1}\cup\Sigma_{0}^{+}\cup\Sigma_{2,3}\cup\Sigma_{\infty}^{-}}\bigr{)}\ni\eta\mapsto\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$
with the following properties:

  * Analyticity: $\eta\mapsto\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ is analytic in its domain of definition.

  * Jump conditions: $\eta\mapsto\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ takes continuous boundary values from the left $(+$ subscript$)$ and right $(-$ subscript$)$ on the arc $\Sigma_{0,1}$ oriented toward $\eta=\mathrm{i}s_{1}(y)$, the arc $\Sigma_{0}^{+}$ oriented toward $\eta=0$, the arc $\Sigma_{2,3}$ oriented toward $\eta=\mathrm{i}s_{3}(y)$, and the unbounded arc $\Sigma_{\infty}^{-}$ oriented toward $\eta=\mathrm{i}s_{1}(y)$, except near the finite endpoints $\eta=\mathrm{i}s_{j}(y)$, $j=1,2,3$. The jump conditions on these arcs are exactly the $n\to\infty$ limits of those satisfied by $\mathbf{N}^{(n)}(\eta,y,z)$ $($see (4.1)–(4.3)$)$:

$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}_{+}(\eta,y,z)=\breve{\mathbf{N}}^{(n),\mathrm{out}}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}_{-}\sigma_{3}}$
$\displaystyle\hphantom{\breve{\mathbf{N}}^{(n),\mathrm{out}}_{+}(\eta,y,z)=}{}\times\begin{bmatrix}0&(-1)^{n}\mathrm{i}\mathrm{e}^{\mathrm{i}n\xi(y)}\\\
(-1)^{n}\mathrm{i}\mathrm{e}^{-\mathrm{i}n\xi(y)}&0\end{bmatrix}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}_{+}\sigma_{3}},\qquad\eta\in\Sigma_{0,1},$
$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}_{+}(\eta,y,z)=\breve{\mathbf{N}}^{(n),\mathrm{out}}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\mathrm{e}^{-\mathrm{i}n\kappa(y)\sigma_{3}}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}},\qquad\eta\in\Sigma_{0}^{+},$
$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}_{+}(\eta,y,z)$
$\displaystyle\qquad=\breve{\mathbf{N}}^{(n),\mathrm{out}}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}\begin{bmatrix}0&\mathrm{i}\mathrm{e}^{\mathrm{i}n\psi(y)}\\\
\mathrm{i}\mathrm{e}^{-\mathrm{i}n\psi(y)}&0\end{bmatrix}\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}},\qquad\eta\in\Sigma_{2,3},$
$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}_{+}(\eta,y,z)=\breve{\mathbf{N}}^{(n),\mathrm{out}}_{-}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)_{-}^{-1/2}\sigma_{3}}\mathrm{e}^{-z(-\mathrm{i}\eta)_{+}^{-1/2}\sigma_{3}},\qquad\eta\in\Sigma_{\infty}^{-}.$
(4.4)

  * Normalization: $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\to\mathbb{I}$ as $\eta\to\infty$.

  * Endpoint behavior: $\eta\mapsto\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ is allowed to blow up like a negative one-fourth power near each of the four finite endpoints. In particular, there are matrices $\mathbf{A}^{(n)}_{j}=\mathbf{A}^{(n)}_{j}(y,z)$, $j=0,1,2,\dots$ independent of $\eta$ such that

$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sim\\!\Bigg{(}\sum_{j=0}^{\infty}\mathbf{A}_{j}(-\mathrm{i}\eta)^{j}\Bigg{)}\\!(-\mathrm{i}\eta)^{(-1)^{n}\sigma_{3}/4}\widetilde{\mathbf{E}}^{-1}\mathrm{e}^{\mathrm{i}n\eta\sigma_{3}}\mathrm{e}^{-ng(\eta,y)\sigma_{3}},\quad\eta\to
0,\\!$ (4.5)

where $\tilde{\mathbf{E}}$ is defined by (1.9).

The jump contour for $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ is
illustrated in the right-hand panel of Figure 3. This Riemann–Hilbert problem
can be solved explicitly, but the construction is not as simple as the above
conditions suggest. It involves elliptic integrals on the genus-1 spectral
curve and corresponding Jacobi theta functions. Full details of the solution
of a similar problem can be found in [1, Section 4.4.2], for instance. The
solution formula shows that, given $n$ and $y$,
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ exists for all
$z\in\mathbb{C}$ except for a doubly periodic lattice of isolated points.
However, we will have no need of the resulting complicated formulæ in this
paper.

Replacing $y$ with $y+n^{-1}z$ in (1.10) and (1.11), the rescaled algebraic
solution $U_{n}\bigl{(}y+n^{-1}z\bigr{)}$ is expressed in terms of
$\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}$ by

$U_{n}\bigl{(}y+n^{-1}z\bigr{)}=(ny+z)\begin{cases}\mathrm{e}^{-5\pi\mathrm{i}/6}B^{(n)}_{0,12}\bigl{(}y+n^{-1}z\bigr{)}B^{(n)}_{0,22}\bigl{(}y+n^{-1}z\bigr{)},&\text{$n$
even},\\\
\mathrm{e}^{5\pi\mathrm{i}/6}B^{(n)}_{0,11}\bigl{(}y+n^{-1}z\bigr{)}B^{(n)}_{0,21}\bigl{(}y+n^{-1}z\bigr{)},&\text{$n$
odd},\end{cases}$

wherein

$\displaystyle\mathbf{B}_{0}^{(n)}\bigl{(}y+n^{-1}z\bigr{)}=(-\mathrm{i}(ny+z))^{n\sigma_{3}}$
$\displaystyle\qquad{}\times\Big{[}\lim_{\eta\to
0}\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}\Big{]}(-\mathrm{i}(ny+z))^{(-1)^{n}\sigma_{3}/2}.$

Also, writing $\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}$ in terms of
$\mathbf{M}^{(n)}(\eta,y,z)$ by (3.2) and using the fact that
$\mathbf{M}^{(n)}(\eta,y,z)=\mathbf{N}^{(n)}(\eta,y,z)$ identically for $\eta$
in a neighborhood of the origin,

$\displaystyle\lim_{\eta\to
0}\mathbf{Z}^{(n)}\bigl{(}\eta,y+n^{-1}z\bigr{)}\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad=\mathrm{e}^{-ng_{0}(y)\sigma_{3}}\lim_{\eta\to
0}\mathbf{N}^{(n)}(\eta,y,z)\mathrm{e}^{ng(\eta,y)\sigma_{3}}\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}.$

It then follows that

$U_{n}\bigl{(}y+n^{-1}z\bigr{)}=\begin{cases}\mathrm{i}\mathrm{e}^{-5\pi\mathrm{i}/6}\widetilde{B}^{(n)}_{0,12}(y,z)\widetilde{B}^{(n)}_{0,22}(y,z),&\text{$n$
even},\\\
\mathrm{i}\mathrm{e}^{5\pi\mathrm{i}/6}\widetilde{B}^{(n)}_{0,11}(y,z)\widetilde{B}^{(n)}_{0,21}(y,z),&\text{$n$
odd},\end{cases}$

wherein

$\widetilde{\mathbf{B}}^{(n)}_{0}(y,z):=\lim_{\eta\to
0}\mathbf{N}^{(n)}(\eta,y,z)\mathrm{e}^{ng(\eta,y)\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}.$

To obtain an approximation $\breve{U}_{n}(z;y)$ of
$U_{n}\bigl{(}y+n^{-1}z\bigr{)}$, we replace the expression
$\mathbf{N}^{(n)}(\eta,y,z)$ with
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ in this formula and, then
using (4.5) we get

$\breve{U}_{n}(z;y):=\begin{cases}\mathrm{i}\mathrm{e}^{-5\pi\mathrm{i}/6}A^{(n)}_{0,12}(y,z)A^{(n)}_{0,22}(y,z),&\text{$n$
even},\\\
\mathrm{i}\mathrm{e}^{5\pi\mathrm{i}/6}A^{(n)}_{0,11}(y,z)A^{(n)}_{0,21}(y,z),&\text{$n$
odd}.\end{cases}$ (4.6)

Note that by taking the limit $\eta\to 0$ from the left and right sides of the
jump contour through the origin, one obtains two equivalent formulæ for the
matrix coefficient $\mathbf{A}^{(n)}_{0}(y,z)$:

$\mathbf{A}^{(n)}_{0}(y,z)=\lim_{\eta\to
0}\breve{\mathbf{N}}^{(n),\mathrm{out}}_{\pm}(\eta,y,z)\mathrm{e}^{ng_{\pm}(0,y)\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4},$

where the $\pm$ signs correspond in the two instances.

Accuracy of the approximation of $U_{n}\bigl{(}y+n^{-1}z\bigr{)}$ by
$\breve{U}_{n}(z;y)$ in the limit of large $n$ hinges on the details of the
analysis of a small-norm Riemann–Hilbert problem for the matrix ratio between
$\mathbf{N}^{(n)}(\eta,y,z)$ and its global parametrix. This is important, but
it takes us far from our main goal in this work, which is to explain how one
can prove, relatively easily and directly from the conditions of
Riemann–Hilbert Problem 4.1, that $\breve{U}(z)=\breve{U}_{n}(z;y)$ as defined
in (4.6) is an exact solution of the elliptic differential equation (1.6) for
a specific choice of the integration constant $E$ as a function of $y$.

##  5 Derivation of the Weierstraß differential equation for
$\boldsymbol{\breve{U}_{n}(z;y)}$

It is a familiar outcome that various coefficients in the expansion of the
solution of a Riemann–Hilbert problem depending on a parameter $z$ satisfy
important differential equations. Indeed, this is exactly how one can be sure
that Riemann–Hilbert Problem 1.1 generates a rescaled solution of the
Painlevé-III (D7) equation (1.1) by formula (1.11) for each $n$. Such a
computation is done in [4, Section 3.2] for a Riemann–Hilbert problem
equivalent to Riemann–Hilbert Problem 1.1 but with an unknown denoted
$\mathbf{W}^{(n)}(\lambda,x)$. The steps are as follows:

  * •

One first introduces a diagonal exponential transformation by setting

$\boldsymbol{\Psi}^{(n)}(\lambda,x):=\mathbf{W}^{(n)}(\lambda,x)\mathrm{e}^{-\mathrm{i}(x\lambda-x(\mathrm{i}x\lambda)^{-1/2})\sigma_{3}}.$

This has the effect of making the induced jump matrices for
$\boldsymbol{\Psi}^{(n)}(\lambda,x)$ arcwise independent of both $\lambda$
(the complex variable of the Riemann–Hilbert problem) and $x$ (the independent
variable of the Painlevé-III (D7) equation in the form (1.1)).

  * •

It then follows by differentiation of the jump conditions that the matrices

$\displaystyle\boldsymbol{\Lambda}^{(n)}(\lambda,x):=\frac{\partial\boldsymbol{\Psi}^{(n)}}{\partial\lambda}(\lambda,x)\boldsymbol{\Psi}^{(n)}(\lambda,x)^{-1},$
$\displaystyle\mathbf{X}^{(n)}(\lambda,x):=\frac{\partial\boldsymbol{\Psi}^{(n)}}{\partial
x}(\lambda,x)\boldsymbol{\Psi}^{(n)}(\lambda,x)^{-1}$ (5.1)

are analytic in $\lambda$ except at isolated singular points which in this
case are $\lambda=\infty$ and $\lambda=0$.

  * •

By expanding $\boldsymbol{\Psi}^{(n)}(\lambda,x)$ and its derivatives near the
singular points using information from the Riemann–Hilbert problem for
$\mathbf{W}^{(n)}(\lambda,x)$, one deduces that both
$\boldsymbol{\Lambda}^{(n)}(\lambda,x)$ and $\mathbf{X}^{(n)}(\lambda,x)$ are
rational functions of $\lambda$ with principal parts expressed in terms of
expansion coefficients of $\mathbf{W}^{(n)}(\lambda,x)$.

  * •

Re-arranging the equations (5.1) with this new knowledge, one sees that
$\boldsymbol{\Psi}^{(n)}(\lambda,x)$ satisfies an overdetermined system
consisting of two first-order $2\times 2$ linear systems, one with respect to
$\lambda$ and another with respect to $x$.

  * •

Expressing the compatibility condition between the two systems in terms of the
elements of the matrices $\boldsymbol{\Lambda}^{(n)}(\lambda,x)$ and
$\mathbf{X}^{(n)}(\lambda,x)$, one separates out from the various powers of
$\lambda$ a closed system of nonlinear differential equations on the
coefficients with respect to $x$ alone. This system implies the Painlevé-III
(D7) equation (1.1).

Analogues of these steps are frequently called the _dressing method_ in many
papers.

It is a natural expectation that a similar approach might apply to
Riemann–Hilbert Problem 4.1 to allow one to deduce a differential equation
with respect to $z$ satisfied by $\breve{U}_{n}(z;y)$. Indeed, the matrix
function

$\eta\mapsto\mathbf{F}^{(n)}(\eta,y,z):=\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\mathrm{e}^{z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}$
(5.2)

satisfies modified jump conditions that simply omit the factors
$\mathrm{e}^{\pm z(-\mathrm{i}\eta)_{\mp}^{-1/2}\sigma_{3}}$ from the jump
matrix. Hence the jump matrices for $\mathbf{F}^{(n)}(\eta,y,z)$ are arcwise
independent of both $\eta$ and $z$. One can then derive a linear first-order
differential equation for $\mathbf{F}^{(n)}(\eta,y,z)$ with respect to $z$
(see Section 5.2 below). However, derivation of a linear first-order
differential equation for $\mathbf{F}^{(n)}(\eta,y,z)$ with respect to $\eta$
is more challenging. One can deduce that
$\mathbf{F}^{(n)}_{\eta}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}$ is rational
in $\eta$ with simple poles at $\eta=0$ and $\eta=\mathrm{i}s_{j}(y)$,
$j=1,2,3$, but it turns out that there is not enough information available to
deduce fully the residue matrices. Without the first-order system with respect
to $\eta$ one cannot obtain the desired nonlinear differential equation from
any compatibility condition.

About a decade ago, we approached Alexander Its with a similar conundrum in
the setting of a project to study elliptic function approximations of rational
solutions of the second Painlevé equation [2]. His advice was to eschew the
undetermined Fuchsian linear system with respect to the Riemann–Hilbert
complex variable (spectral parameter) in favor of a remarkable algebraic
identity satisfied by the matrix solutions of Riemann–Hilbert problems whose
jump matrices have a certain structure. Expanding this identity with respect
to the spectral parameter produces numerous identities among functions of the
independent variable alone that serve to close the system of differential
equations; squaring it produces a scalar identity that links the spectral
curve and the target differential equation.

The jump matrices of Riemann–Hilbert Problem 4.1 have the necessary structure
for this method to apply. In the rest of this section, we implement the method
and show how it yields the expected differential equation (1.6). Specifically,
we prove the following.

######  Theorem 5.1.

Fix $n\in\mathbb{Z}_{>0}$ and $y$ with $\operatorname{Re}(y)>0$ and
$Y=y^{1/3}$ in the interior of $\mathcal{B}$. Then the function
$z\mapsto\breve{U}_{n}(z;y)$ defined from the solution of Riemann–Hilbert
Problem 4.1 by (4.6) is a solution of the differential equation (1.6), in
which the integration constant $E$ is given by

$E=-\frac{8c}{y^{2}},$ (5.3)

where $c=c_{1}(y)$ is the smooth but non-analytic function of $y$ defined in
Lemma 2.1.

######  Remark 5.2.

This result shows that the first-order autonomous differential equation (1.6),
which is now well-defined given $y$ as in the theorem statement, is solved by
the approximation $\breve{U}_{n}(z;y)$, which also depends on the index
$n\in\mathbb{Z}_{>0}$. However, the space of solutions of the differential
equation is mapped out by translations in $z$, and the particular translate
needed to identify $\breve{U}_{n}(z;y)$ will generally depend on $n$ and is
not specified by Theorem 5.1.

######  Remark 5.3.

Theorem 5.1 shows that the (scaled) algebraic function
$n^{-1/2}u_{n}\bigl{(}n^{3/2}\bigl{(}y+n^{-1}z\bigr{)}\bigr{)}$, which has a
finite number of poles, is well approximated in its pole region as
$n\to\infty$ by a solution of the Weierstraß equation in the form (1.6) having
an infinite number of poles. Interestingly, the same Weierstraß equation has
recently been shown to govern large-$x$ asymptotic behavior of general (non-
algebraic) solutions of the Painlevé-III (D7) equation (1.1) by Shimomura
[14].

Now we continue with the proof of Theorem 5.1. An elementary example
illustrating the basic steps in the method we use can be found in Appendix A.

###  5.1 Expansion of
$\boldsymbol{\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)}$ near
$\boldsymbol{\eta=\infty}$

It is easy to see from the jump condition (4.4) that if $y$, $z$, and $n$ are
such that $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ exists, the
product $\mathbf{F}(\eta)=\mathbf{F}^{(n)}(\eta,y,z)$ defined by (5.2) is
analytic for large $\eta$ and decays to $\mathbb{I}$ as $\eta\to\infty$.
Therefore, there are matrix coefficients
$\mathbf{F}_{j}=\mathbf{F}_{j}^{(n)}(y,z)$ such that

$\displaystyle\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)=\mathbf{F}(\eta)\mathrm{e}^{-z(-\mathrm{i}\eta)^{-1/2}\sigma_{3}}=\Bigg{(}\\!\mathbb{I}+\sum_{j=1}^{\infty}\mathbf{F}_{j}\cdot(-\mathrm{i}\eta)^{-j}\\!\Bigg{)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!}(-\mathrm{i}\eta)^{-k/2}\sigma_{3}^{k}\\!\\!\\!$
(5.4)
$\displaystyle\qquad=\Bigg{(}\mathbb{I}+\sum_{j=1}^{\infty}\mathbf{F}_{j}\cdot(-\mathrm{i}\eta)^{-j}\Bigg{)}\Bigg{(}\mathbb{I}\sum_{k=0}^{\infty}\frac{z^{2k}}{(2k)!}(-\mathrm{i}\eta)^{-k}-(-\mathrm{i}\eta)^{-1/2}\sigma_{3}\sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)!}(-\mathrm{i}\eta)^{-k}\Bigg{)}.$

This immediately implies that
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ has an expansion in
nonnegative integer powers of $(-\mathrm{i}\eta)^{-1/2}$

$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)=\sum_{m=0}^{\infty}\mathbf{C}_{m/2}\cdot(-\mathrm{i}\eta)^{-m/2},\qquad\mathbf{C}_{0}=\mathbb{I}$
(5.5)

that is convergent for $|\eta|$ large enough.

By matching the coefficients of like powers of $-\mathrm{i}\eta$ in the
expansions (5.4)–(5.5), one can express the coefficients
$\mathbf{C}_{m/2}=\mathbf{C}_{m/2}^{(n)}(y,z)$ in terms of the
$\mathbf{F}_{j}$. Therefore, we find

$\displaystyle\mathbf{C}_{1/2}=-z\sigma_{3},\qquad\mathbf{C}_{1}=\mathbf{F}_{1}+\frac{z^{2}}{2!}\mathbb{I},\qquad\mathbf{C}_{3/2}=-z\mathbf{F}_{1}\sigma_{3}-\frac{z^{3}}{3!}\sigma_{3},$
(5.6)

and so on. Eliminating $\mathbf{F}_{1}$ in favor of $\mathbf{C}_{1}$, we can
also write

$\mathbf{C}_{3/2}=-z\mathbf{C}_{1}\sigma_{3}+\frac{z^{3}}{3}\sigma_{3}.$ (5.7)

######  Remark 5.4.

Similar analysis of the growth and jump conditions of Riemann–Hilbert Problem
4.1 near $\eta=0$ shows that the product
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\mathrm{e}^{ng(\eta,y)\sigma_{3}}\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
is analytic at $\eta=0$. This implies that the series in (4.5) is convergent,
and that “$\sim$” can be written instead as “$=$”.

###  5.2 Differential equations in $\boldsymbol{z}$

####  5.2.1 Lax equation satisfied by
$\boldsymbol{\mathbf{F}^{(n)}(\eta,y,z)}$

Assuming it exists, the matrix function
$\eta\mapsto\mathbf{F}^{(n)}(\eta,y,z)$ defined in (5.2) above satisfies
modified jump conditions with jump matrices that are independent of $z$ (and
also of $\eta$, but we will not use that). Then by standard arguments,
$\mathbf{F}^{(n)}_{z}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}$ is analytic in
$\eta$ except possibly at $\eta=0,\infty$. In terms of the outer parametrix,
we have from (5.2)

$\displaystyle\frac{\partial\mathbf{F}^{(n)}}{\partial
z}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}=\frac{\partial\breve{\mathbf{N}}^{(n),\mathrm{out}}}{\partial
z}(\eta,y,z)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}$
$\displaystyle\hphantom{\frac{\partial\mathbf{F}^{(n)}}{\partial
z}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}=}{}+(-\mathrm{i}\eta)^{-1/2}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}.$

The expansion (5.5) is differentiable term-by-term with respect to $z$, and
therefore

$\displaystyle\frac{\partial\breve{\mathbf{N}}^{(n),\mathrm{out}}}{\partial
z}(\eta,y,z)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}$
$\displaystyle\qquad{}=\bigg{(}\frac{\partial\mathbf{C}_{1/2}}{\partial
z}\cdot(-\mathrm{i}\eta)^{-1/2}+\frac{\partial\mathbf{C}_{1}}{\partial
z}\cdot(-\mathrm{i}\eta)^{-1}+\cdots\bigg{)}$
$\displaystyle\qquad\quad{}\times\bigl{(}\mathbb{I}-\mathbf{C}_{1/2}(-\mathrm{i}\eta)^{-1/2}+\big{[}\mathbf{C}_{1/2}^{2}-\mathbf{C}_{1}\big{]}(-\mathrm{i}\eta)^{-1}+\cdots\bigr{)}$
$\displaystyle\qquad{}=\frac{\partial\mathbf{C}_{1/2}}{\partial
z}\cdot(-\mathrm{i}\eta)^{-1/2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-1}\bigr{)}=-\sigma_{3}(-\mathrm{i}\eta)^{-1/2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-1}\bigr{)},\qquad\eta\to\infty,$

where we used $\mathbf{C}_{1/2}=-z\sigma_{3}$. Also, directly from (5.5),

$\displaystyle(-\mathrm{i}\eta)^{-1/2}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}=\sigma_{3}(-\mathrm{i}\eta)^{-1/2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-1}\bigr{)},\quad\eta\to\infty.$

Therefore,
$\mathbf{F}^{(n)}_{z}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}=\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-1}\bigr{)}$
as $\eta\to\infty$. Likewise, using (4.5) we get

$\displaystyle\frac{\partial\breve{\mathbf{N}}^{(n),\mathrm{out}}}{\partial
z}(\eta,y,z)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}$
$\displaystyle\qquad{}=\bigg{(}\frac{\partial\mathbf{A}^{(n)}_{0}}{\partial
z}(y,z)+\frac{\partial\mathbf{A}^{(n)}_{1}}{\partial
z}(y,z)(-\mathrm{i}\eta)+\cdots\bigg{)}$
$\displaystyle\qquad\quad{}\times\bigl{(}\mathbf{A}^{(n)}_{0}(y,z)^{-1}-\mathbf{A}^{(n)}_{0}(y,z)^{-1}\mathbf{A}^{(n)}_{1}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}(-\mathrm{i}\eta)+\cdots\bigr{)}=\mathcal{O}(1),$

as $\eta\to 0$, and in the same limit

$\displaystyle(-\mathrm{i}\eta)^{-1/2}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}$
$\displaystyle\qquad=\bigl{(}\mathbf{A}_{0}^{(n)}(y,z)+\mathbf{A}_{1}^{(n)}(y,z)(-\mathrm{i}\eta)+\cdots\bigr{)}$
$\displaystyle\qquad\quad{}\times(-\mathrm{i}\eta)^{-1/2}(-\mathrm{i}\eta)^{(-1)^{n}\sigma_{3}/4}\widetilde{\mathbf{E}}^{-1}\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad\quad{}\times\bigl{(}\mathbf{A}^{(n)}_{0}(y,z)^{-1}-\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}(-\mathrm{i}\eta)+\cdots\bigr{)}.$

But, using the identity

$\widetilde{\mathbf{E}}^{-1}\sigma_{3}\widetilde{\mathbf{E}}=\begin{bmatrix}0&\mathrm{e}^{-5\pi\mathrm{i}/3}\\\
\mathrm{e}^{5\pi\mathrm{i}/3}&0\end{bmatrix},$

the central factor becomes

$\displaystyle(-\mathrm{i}\eta)^{-1/2}(-\mathrm{i}\eta)^{(-1)^{n}\sigma_{3}/4}\widetilde{\mathbf{E}}^{-1}\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad=(-\mathrm{i}\eta)^{[(-1)^{n}\sigma_{3}-\mathbb{I}]/2}\begin{bmatrix}0&\mathrm{e}^{-5\pi\mathrm{i}/3}\\\
\mathrm{e}^{5\pi\mathrm{i}/3}&0\end{bmatrix}=\begin{cases}\mathbf{U}+\mathbf{L}\cdot(-\mathrm{i}\eta)^{-1},&\text{$n$
even},\\\ \mathbf{U}\cdot(-\mathrm{i}\eta)^{-1}+\mathbf{L},&\text{$n$
odd},\end{cases}$ (5.8)

where

$\mathbf{U}:=\begin{bmatrix}0&\mathrm{e}^{-5\pi\mathrm{i}/3}\\\
0&0\end{bmatrix}\qquad\text{and}\qquad\mathbf{L}:=\begin{bmatrix}0&0\\\
\mathrm{e}^{5\pi\mathrm{i}/3}&0\end{bmatrix}.$

Therefore,
$\mathbf{F}^{(n)}_{z}(\eta,y,z)\mathbf{F}^{(n)}(\eta,y,z)^{-1}=\mathbf{T}\cdot(-\mathrm{i}\eta)^{-1}+\mathcal{O}(1)$
as $\eta\to 0$, wherein

$\mathbf{T}=\mathbf{T}^{(n)}(y,z):=\begin{cases}\mathbf{A}^{(n)}_{0}(y,z)\mathbf{L}\mathbf{A}_{0}^{(n)}(y,z)^{-1},&\text{$n$
even,}\\\
\mathbf{A}_{0}^{(n)}(y,z)\mathbf{U}\mathbf{A}_{0}^{(n)}(y,z)^{-1},&\text{$n$
odd}.\end{cases}$ (5.9)

It will also be convenient later to define the following related matrix

$\widetilde{\mathbf{T}}=\widetilde{\mathbf{T}}^{(n)}(y,z):=\begin{cases}\mathbf{A}_{0}^{(n)}(y,z)\mathbf{U}\mathbf{A}_{0}^{(n)}(y,z)^{-1},&\text{$n$
even,}\\\
\mathbf{A}_{0}^{(n)}(y,z)\mathbf{L}\mathbf{A}_{0}^{(n)}(y,z)^{-1},&\text{$n$
odd}.\end{cases}$ (5.10)

These definitions imply that

$\mathbf{T}^{2}=\mathbf{0}\qquad\text{and}\qquad\widetilde{\mathbf{T}}^{2}=\mathbf{0}$
(5.11)

and that

$\mathbf{T}\widetilde{\mathbf{T}}+\widetilde{\mathbf{T}}\mathbf{T}=\mathbb{I}.$
(5.12)

Note that, using (4.6) and $\det(\mathbf{A}_{0})=1$, whether $n$ is even or
odd one obtains the same formula for $\breve{U}_{n}$ in terms of $\mathbf{T}$:

$\breve{U}_{n}=T_{11}=-T_{22}.$ (5.13)

Comparing the expansions as $\eta\to\infty$ and as $\eta\to 0$ and multiplying
on the right by $\mathbf{F}^{(n)}(\eta,y,z)$, we obtain the differential
equation

$\frac{\partial\mathbf{F}^{(n)}}{\partial
z}(\eta,y,z)=\mathbf{T}\cdot(-\mathrm{i}\eta)^{-1}\mathbf{F}^{(n)}(\eta,y,z).$
(5.14)

####  5.2.2 Implied differential equations for coefficients

Combining (5.2) and (5.14), one obtains a differential equation for
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$, namely

$\frac{\partial\breve{\mathbf{N}}^{(n),\mathrm{out}}}{\partial
z}(\eta,y,z)=\mathbf{T}\cdot(-\mathrm{i}\eta)^{-1}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)-(-\mathrm{i}\eta)^{-1/2}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}.$
(5.15)

Multiplying (5.15) on the right by
$\mathrm{e}^{ng(\eta,y)\sigma_{3}}\mathrm{e}^{-\mathrm{i}n\eta\sigma_{3}}\widetilde{\mathbf{E}}(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
and using (4.5) gives the convergent (by Remark 5.4) series

$\displaystyle\sum_{j=0}^{\infty}\frac{\partial\mathbf{A}^{(n)}_{j}}{\partial
z}(y,z)(-\mathrm{i}\eta)^{j}=\sum_{j=0}^{\infty}\mathbf{T}\mathbf{A}^{(n)}_{j}(y,z)(-\mathrm{i}\eta)^{j-1}$
$\displaystyle\qquad\quad{}-(-\mathrm{i}\eta)^{-1/2}\Bigg{(}\sum_{j=0}^{\infty}\mathbf{A}_{j}^{(n)}(y,z)(-\mathrm{i}\eta)^{j}\Bigg{)}(-\mathrm{i}\eta)^{(-1)^{n}\sigma_{3}/4}\widetilde{\mathbf{E}}^{-1}\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad{}=\sum_{j=0}^{\infty}\mathbf{T}\mathbf{A}_{j}^{(n)}(y,z)(-\mathrm{i}\eta)^{j-1}-\sum_{j=0}^{\infty}\mathbf{A}_{j}^{(n)}(y,z)(-\mathrm{i}\eta)^{j}(-\mathrm{i}\eta)^{[(-1)^{n}\sigma_{3}-\mathbb{I}]/2}(\mathbf{U}+\mathbf{L}),$

where we used (5.8) in the second equality. Matching the coefficient of
$(-\mathrm{i}\eta)^{-1}$ on both sides gives an identity according to the
definition (5.9) of $\mathbf{T}$. Matching the coefficient of
$(-\mathrm{i}\eta)^{0}$ then gives

$\frac{\partial\mathbf{A}_{0}^{(n)}}{\partial
z}(y,z)=\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)-\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)-\widetilde{\mathbf{T}}\mathbf{A}_{0}^{(n)}(y,z).$

Again using (5.9) gives

$\displaystyle\frac{\partial\mathbf{T}}{\partial
z}=\frac{\partial\mathbf{A}_{0}^{(n)}}{\partial
z}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}-\mathbf{T}\frac{\partial\mathbf{A}_{0}^{(n)}}{\partial
z}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}$
$\displaystyle\hphantom{\frac{\partial\mathbf{T}}{\partial
z}}{}=\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}-\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}^{2}-\widetilde{\mathbf{T}}\mathbf{T}$
$\displaystyle\hphantom{\frac{\partial\mathbf{T}}{\partial
z}=}{}-\mathbf{T}^{2}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}+\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}+\mathbf{T}\widetilde{\mathbf{T}}$
$\displaystyle\hphantom{\frac{\partial\mathbf{T}}{\partial
z}}{}=2\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}+\big{[}\mathbf{T},\widetilde{\mathbf{T}}\big{]}.$
(5.16)

Here $[\mathbf{A},\mathbf{B}]:=\mathbf{AB}-\mathbf{BA}$ denotes the matrix
commutator. Note that (5.16) is not a closed system of differential equations;
in particular elements of $\mathbf{A}_{1}^{(n)}(y,z)$ appear on the right-hand
side. To close the system, we next follow the suggestion of Its and
investigate algebraic identities.

###  5.3 Algebraic matrix identity

Denoting $R(\eta,y):=-h_{\eta}(\eta,y,c_{1}(y))$, let

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z):={}$ $\displaystyle
R(\eta,y)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)^{-1}$
$\displaystyle={}$ $\displaystyle
R(\eta,y)\mathbf{F}^{(n)}(\eta,y,z)\sigma_{3}\mathbf{F}^{(n)}(\eta,y,z)^{-1},$
(5.17)

with $c=c_{1}(y)$ being the function defined in Lemma 2.1. Recalling (2.1), we
have

$\displaystyle
R(\eta,y)^{2}=f(\eta,y,c)=\frac{P(-\mathrm{i}\eta,y,c)}{(-\mathrm{i}\eta)^{3}}$
$\displaystyle=-1+(-\mathrm{i}\eta)^{-1}+c(-\mathrm{i}\eta)^{-2}-\tfrac{1}{4}y^{2}(-\mathrm{i}\eta)^{-3}$
$\displaystyle=-\frac{(-\mathrm{i}\eta-s_{1}(y))(-\mathrm{i}\eta-
s_{2}(y))(-\mathrm{i}\eta-s_{3}(y))}{(-\mathrm{i}\eta)^{3}}.$

Note that $\eta\mapsto R(\eta,y)$ changes sign across branch cuts where the
jump matrix for $\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)$ is off-
diagonal, while $R(\eta,y)$ is analytic on $\Sigma_{\infty}^{-}$ and
$\Sigma_{0}^{+}$ where the jump matrix is diagonal (this is the required
special structure of the jump matrices for the method to apply). It follows
easily that $\eta\mapsto\mathbf{G}^{(n)}(\eta,y,z)$ is analytic except at
$\eta=0,\infty$.

Note also that

$\displaystyle
R(\eta,y)=\begin{cases}\mathrm{i}\Bigl{(}1-\tfrac{1}{2}(-\mathrm{i}\eta)^{-1}-\tfrac{1}{2}\bigl{(}c+\tfrac{1}{4}\bigr{)}(-\mathrm{i}\eta)^{-2}&\\\
\qquad{}+\tfrac{1}{16}\bigl{(}2y^{2}-4c-1\bigr{)}(-\mathrm{i}\eta)^{-3}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-4}\bigr{)}\Bigr{)},&\eta\to\infty,\\\
-\tfrac{1}{2}\mathrm{i}y(-\mathrm{i}\eta)^{-3/2}\Bigl{(}1-\frac{2c}{y^{2}}(-\mathrm{i}\eta)-\frac{2(c^{2}+y^{2})}{y^{4}}(-\mathrm{i}\eta)^{2}&\\\
\qquad{}+\frac{2(y^{4}-2c^{3}-2cy^{2})}{y^{6}}(-\mathrm{i}\eta)^{3}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{4}\bigr{)}\Bigr{)},&\eta\to
0.\end{cases}$ (5.18)

From the fact that
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\to\mathbb{I}$ as
$\eta\to\infty$, analyticity of $\mathbf{G}^{(n)}(\eta,y,z)$ in
$\mathbb{C}\setminus\\{0\\}$ implies that it has a Laurent expansion of the
form

$\mathbf{G}^{(n)}(\eta,y,z)=\mathbf{G}^{\infty}_{0}+\mathbf{G}^{\infty}_{1}\cdot(-\mathrm{i}\eta)^{-1}+\mathbf{G}^{\infty}_{2}\cdot(-\mathrm{i}\eta)^{-2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{-3}\bigr{)},\qquad\eta\to\infty.$

Multiplying the definition (5.17) of $\mathbf{G}^{(n)}(\eta,y,z)$ on the right
by $\breve{\mathbf{N}}^{(n)}(\eta,y,z)$, we insert the expansions (5.4) and
(5.18) and equate the coefficients of like powers of $(-\mathrm{i}\eta)^{-1}$
to obtain a hierarchy of equations:

$\displaystyle\mathbf{G}^{\infty}_{0}=\mathrm{i}\sigma_{3},$
$\displaystyle\mathbf{G}^{\infty}_{0}\mathbf{C}_{1/2}=\mathrm{i}\mathbf{C}_{1/2}\sigma_{3},$
$\displaystyle\mathbf{G}^{\infty}_{1}+\mathbf{G}^{\infty}_{0}\mathbf{C}_{1}=\mathrm{i}\mathbf{C}_{1}\sigma_{3}-\tfrac{1}{2}\mathrm{i}\sigma_{3},$
$\displaystyle\mathbf{G}^{\infty}_{1}\mathbf{C}_{1/2}+\mathbf{G}^{\infty}_{0}\mathbf{C}_{3/2}=\mathrm{i}\mathbf{C}_{3/2}\sigma_{3}-\tfrac{1}{2}\mathrm{i}\mathbf{C}_{1/2}\sigma_{3},$
$\displaystyle\mathbf{G}^{\infty}_{2}+\mathbf{G}^{\infty}_{1}\mathbf{C}_{1}+\mathbf{G}^{\infty}_{0}\mathbf{C}_{2}=\mathrm{i}\mathbf{C}_{2}\sigma_{3}-\tfrac{1}{2}\mathrm{i}\mathbf{C}_{1}\sigma_{3}-\tfrac{1}{2}\mathrm{i}\bigl{(}c+\tfrac{1}{4}\bigr{)}\sigma_{3},$

and so on. The first, third, and fifth equations give, in order,

$\displaystyle\mathbf{G}_{0}^{\infty}=\mathrm{i}\sigma_{3},$
$\displaystyle\mathbf{G}_{1}^{\infty}=\mathrm{i}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\mathrm{i}\sigma_{3},$
$\displaystyle\mathbf{G}_{2}^{\infty}=\mathrm{i}[\mathbf{C}_{2},\sigma_{3}]-\tfrac{1}{2}\mathrm{i}[\mathbf{C}_{1},\sigma_{3}]-\mathrm{i}[\mathbf{C}_{1},\sigma_{3}]\mathbf{C}_{1}-\tfrac{1}{2}\mathrm{i}\bigl{(}c+\tfrac{1}{4}\bigr{)}\sigma_{3}.$

According to (5.6) and (5.7), the second and fourth equations are trivial
identities. Likewise, $\mathbf{G}^{(n)}(\eta,y,z)$ has a Laurent expansion
about $\eta=0$ of the form

$\mathbf{G}^{(n)}(\eta,y,z)=\mathbf{G}^{0}_{0}\cdot(-\mathrm{i}\eta)^{-2}+\mathbf{G}^{0}_{1}\cdot(-\mathrm{i}\eta)^{-1}+\mathbf{G}^{0}_{2}+\mathcal{O}((-\mathrm{i}\eta)),\qquad\eta\to
0.$

Multiplying the definition of $\mathbf{G}^{(n)}(\eta,y,z)$ on the right by
$\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
and using the expansions (4.5) and (5.18) again gives a hierarchy of
equations. To see them, first we expand the left-hand side:

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad{}=\bigl{(}\mathbf{G}^{0}_{0}\cdot(-\mathrm{i}\eta)^{-2}+\mathbf{G}^{0}_{1}\cdot(-\mathrm{i}\eta)^{-1}+\mathbf{G}^{0}_{2}+\mathcal{O}((-\mathrm{i}\eta))\bigr{)}$
$\displaystyle\qquad\quad{}\times\bigl{(}\mathbf{A}_{0}^{(n)}(y,z)+\mathbf{A}_{1}^{(n)}(y,z)(-\mathrm{i}\eta)+\mathbf{A}_{2}^{(n)}(y,z)(-\mathrm{i}\eta)^{2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{3}\bigr{)}\bigr{)}$
$\displaystyle\qquad{}=\mathbf{G}_{0}^{0}\mathbf{A}_{0}^{(n)}(y,z)(-\mathrm{i}\eta)^{-2}+\bigl{(}\mathbf{G}_{1}^{0}\mathbf{A}_{0}^{(n)}(y,z)+\mathbf{G}_{0}^{0}\mathbf{A}_{1}^{(n)}(y,z)\bigr{)}(-\mathrm{i}\eta)^{-1}$
$\displaystyle\qquad\quad{}+\bigl{(}\mathbf{G}_{2}^{0}\mathbf{A}_{0}^{(n)}(y,z)+\mathbf{G}_{1}^{0}\mathbf{A}_{1}^{(n)}(y,z)+\mathbf{G}_{0}^{0}\mathbf{A}_{2}^{(n)}(y,z)\bigr{)}+\mathcal{O}((-\mathrm{i}\eta)),\qquad\eta\to
0.$

Then we expand the right-hand side:

$\displaystyle
R(\eta,y)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad{}=R(\eta,y)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad\quad{}\times(-\mathrm{i}\eta)^{(-1)^{n}\sigma_{3}/4}\widetilde{\mathbf{E}}^{-1}\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\qquad{}=-\tfrac{1}{2}\mathrm{i}y\Big{(}1-\tfrac{2c}{y^{2}}(-\mathrm{i}\eta)-\tfrac{2(c^{2}+y^{2})}{y^{4}}(-\mathrm{i}\eta)^{2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{3}\bigr{)}\Big{)}$
$\displaystyle\qquad\quad{}\times\bigl{(}\mathbf{A}_{0}^{(n)}(y,z)+\mathbf{A}_{1}^{(n)}(y,z)(-\mathrm{i}\eta)+\mathbf{A}_{2}^{(n)}(y,z)(-\mathrm{i}\eta)^{2}+\mathcal{O}\bigl{(}(-\mathrm{i}\eta)^{3}\bigr{)}\bigr{)}$
$\displaystyle\qquad\quad{}\times(-\mathrm{i}\eta)^{[(-1)^{n}\sigma_{3}-3\mathbb{I}]/2}(\mathbf{U}+\mathbf{L}).$

Recalling the definitions (5.9) and (5.10), one sees that

$(-\mathrm{i}\eta)^{[(-1)^{n}\sigma_{3}-3\mathbb{I}]/2}(\mathbf{U}+\mathbf{L})=\mathbf{A}_{0}^{(n)}(y,z)^{-1}\bigl{(}\mathbf{T}\cdot(-\mathrm{i}\eta)^{-2}+\widetilde{\mathbf{T}}\cdot(-\mathrm{i}\eta)^{-1}\bigr{)}\mathbf{A}_{0}^{(n)}(y,z).$

This shows that

$\displaystyle
R(\eta,y)\breve{\mathbf{N}}^{(n),\mathrm{out}}(\eta,y,z)\sigma_{3}\widetilde{\mathbf{E}}\cdot(-\mathrm{i}\eta)^{-(-1)^{n}\sigma_{3}/4}$
$\displaystyle\quad{}=-\frac{\mathrm{i}y}{2}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)(-\mathrm{i}\eta)^{-2}$
$\displaystyle\qquad{}+\biggl{(}-\frac{\mathrm{i}y}{2}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)+\frac{\mathrm{i}c}{y}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)-\frac{\mathrm{i}y}{2}\widetilde{\mathbf{T}}\mathbf{A}_{0}^{(n)}(y,z)\biggr{)}(-\mathrm{i}\eta)^{-1}$
$\displaystyle\qquad{}-\frac{\mathrm{i}y}{2}\mathbf{A}_{2}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)+\frac{\mathrm{i}c}{y}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)$
$\displaystyle\qquad{}+\frac{\mathrm{i}(c^{2}+y^{2})}{y^{4}}\mathbf{T}\mathbf{A}_{0}^{(n)}(y,z)-\frac{\mathrm{i}y}{2}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\widetilde{\mathbf{T}}\mathbf{A}_{0}^{(n)}(y,z)+\frac{\mathrm{i}c}{y}\widetilde{\mathbf{T}}\mathbf{A}_{0}^{(n)}(y,z)$
$\displaystyle\qquad{}+\mathcal{O}(-\mathrm{i}\eta),\qquad\eta\to 0.$

Hence, matching the left and right-hand sides,

$\displaystyle\mathbf{G}_{0}^{0}=-\tfrac{1}{2}\mathrm{i}y\mathbf{T},$
$\displaystyle\mathbf{G}_{1}^{0}=\tfrac{1}{2}\mathrm{i}y\big{[}\mathbf{T},\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\big{]}+\frac{\mathrm{i}c}{y}\mathbf{T}-\tfrac{1}{2}\mathrm{i}y\widetilde{\mathbf{T}},$

and so on. Since Liouville’s theorem implies that $\mathbf{G}^{(n)}(\eta,y,z)$
is a Laurent polynomial in $-\mathrm{i}\eta$ of degree $(-2,0)$, we have the
identities

$\mathbf{G}^{\infty}_{2}=\mathbf{G}_{0}^{0},\qquad\mathbf{G}^{\infty}_{1}=\mathbf{G}_{1}^{0},\qquad\mathbf{G}^{\infty}_{0}=\mathbf{G}_{2}^{0}.$

When used with other facts obtained from (5.9) and (5.10) such as (5.11) and
(5.12) these imply several interesting relations. For instance,
$\mathbf{T}^{2}=\mathbf{0}$ implies that

$\mathbf{G}_{1}^{0}\mathbf{T}=\tfrac{1}{2}\mathrm{i}y\bigl{(}\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}-\widetilde{\mathbf{T}}\mathbf{T}\bigr{)}$

and

$\mathbf{T}\mathbf{G}_{1}^{0}=\tfrac{1}{2}\mathrm{i}y\bigl{(}-\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}-\mathbf{T}\widetilde{\mathbf{T}}\bigr{)},$

so using (5.12) the sum of these is
$\mathbf{G}_{1}^{0}\mathbf{T}+\mathbf{T}\mathbf{G}_{1}^{0}=-\tfrac{1}{2}\mathrm{i}y\mathbb{I}$.
However, by $\mathbf{G}_{1}^{0}=\mathbf{G}_{1}^{\infty}$, we get

$\mathrm{i}\mathbf{T}\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}+\mathrm{i}\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}\mathbf{T}+\tfrac{1}{2}\mathrm{i}y\mathbb{I}=\mathbf{0}.$
(5.19)

The difference is
$\big{[}\mathbf{G}_{1}^{0},\mathbf{T}\big{]}=\tfrac{1}{2}\mathrm{i}y\bigl{(}2\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}+\big{[}\mathbf{T},\widetilde{\mathbf{T}}\big{]}\bigr{)}$.
Again using $\mathbf{G}_{1}^{0}=\mathbf{G}_{1}^{\infty}$ gives

$\mathrm{i}\big{[}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3},\mathbf{T}\big{]}=\tfrac{1}{2}\mathrm{i}y\bigl{(}2\mathbf{T}\mathbf{A}_{1}^{(n)}(y,z)\mathbf{A}_{0}^{(n)}(y,z)^{-1}\mathbf{T}+\big{[}\mathbf{T},\widetilde{\mathbf{T}}\big{]}\bigr{)}.$

This implies that the differential equation (5.16) can be written as a Lax
commutator equation:

$\frac{\partial\mathbf{T}}{\partial
z}=\frac{2}{y}\big{[}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3},\mathbf{T}\big{]}.$

In particular,

$\frac{\partial T_{11}}{\partial
z}=-\frac{4}{y}(C_{1,12}T_{21}+T_{12}C_{1,21}),$

which further implies that

$\displaystyle\bigg{(}\frac{\partial T_{11}}{\partial z}\bigg{)}^{2}$
$\displaystyle=\frac{16}{y^{2}}\bigl{(}C_{1,12}^{2}T_{21}^{2}+T_{12}^{2}C_{1,21}^{2}+2C_{1,12}C_{1,21}T_{12}T_{21}\bigr{)}$
$\displaystyle=\frac{16}{y^{2}}\bigl{(}C_{1,12}^{2}T_{21}^{2}+T_{12}^{2}C_{1,21}^{2}-2C_{1,12}C_{1,21}T_{11}^{2}\bigr{)},$
(5.20)

where on the second line we used $\det(\mathbf{T})=\mathrm{tr}(\mathbf{T})=0$.

###  5.4 Scalar identity and completion of the proof of Theorem 5.1

The most remarkable identity stemming from the definition of
$\mathbf{G}^{(n)}(\eta,y,z)$ comes from $\sigma_{3}^{2}=\mathbb{I}$ which
implies that $\mathbf{G}^{(n)}(\eta,y,z)^{2}$ is the scalar (independent of
both $n$ and $z$, and rational in $\eta$) $R(\eta,y)^{2}\mathbb{I}$. We can
write $\mathbf{G}^{(n)}(\eta,y,z)$ in the form

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z)$
$\displaystyle=\mathbf{G}_{0}^{0}(-\mathrm{i}\eta)^{-2}+\mathbf{G}_{1}^{\infty}(-\mathrm{i}\eta)^{-1}+\mathbf{G}_{0}^{\infty}$
$\displaystyle=-\tfrac{1}{2}\mathrm{i}y\mathbf{T}\cdot(-\mathrm{i}\eta)^{-2}+\mathrm{i}\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}(-\mathrm{i}\eta)^{-1}+\mathrm{i}\sigma_{3}.$

Therefore, using $\mathbf{T}^{2}=\mathbf{0}$,

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z)^{2}=\tfrac{1}{2}y\bigl{(}\mathbf{T}\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}+\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}\mathbf{T}\bigr{)}(-\mathrm{i}\eta)^{-3}$
$\displaystyle\hphantom{\mathbf{G}^{(n)}(\eta,y,z)^{2}=}{}+\bigl{(}\tfrac{1}{2}y(\mathbf{T}\sigma_{3}+\sigma_{3}\mathbf{T})-\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}^{2}\bigr{)}(-\mathrm{i}\eta)^{-2}$
$\displaystyle\hphantom{\mathbf{G}^{(n)}(\eta,y,z)^{2}=}{}-\bigl{(}\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}\sigma_{3}+\sigma_{3}([\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3})\bigr{)}(-\mathrm{i}\eta)^{-1}-\mathbb{I}.$

Using (5.19) and the fact that $[\mathbf{C}_{1},\sigma_{3}]$ is off-diagonal,
this becomes

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z)^{2}=-\tfrac{1}{4}y^{2}\mathbb{I}(-\mathrm{i}\eta)^{-3}+\bigl{(}\tfrac{1}{2}y(\mathbf{T}\sigma_{3}+\sigma_{3}\mathbf{T})-\bigl{(}[\mathbf{C}_{1},\sigma_{3}]-\tfrac{1}{2}\sigma_{3}\bigr{)}^{2}\bigr{)}(-\mathrm{i}\eta)^{-2}$
$\displaystyle\hphantom{\mathbf{G}^{(n)}(\eta,y,z)^{2}=}{}+\mathbb{I}(-\mathrm{i}\eta)^{-1}-\mathbb{I}.$

Using $\mathrm{tr}(\mathbf{T})=0$, we verify that the coefficient of
$(-\mathrm{i}\eta)^{-2}$ is also a multiple of the identity, and therefore

$\displaystyle\mathbf{G}^{(n)}(\eta,y,z)^{2}$
$\displaystyle{}=\bigl{(}-\tfrac{1}{4}y^{2}(-\mathrm{i}\eta)^{-3}+\bigl{(}yT_{11}+4C_{1,12}C_{1,21}-\tfrac{1}{4}\bigr{)}(-\mathrm{i}\eta)^{-2}+(-\mathrm{i}\eta)^{-1}-1\bigr{)}\mathbb{I}$
$\displaystyle{}=R(\eta,y)^{2}\mathbb{I}.$

Then using this with $\eta=\mathrm{i}y/(4T_{11})$ and (5.20), we have

$\displaystyle\bigg{(}\frac{\partial T_{11}}{\partial
z}\bigg{)}^{2}+R\bigg{(}\frac{\mathrm{i}y}{4T_{11}},y\bigg{)}^{2}$
$\displaystyle\qquad{}=\bigg{(}\frac{16}{y^{2}}\bigl{(}C_{1,12}^{2}T_{21}^{2}+T_{12}^{2}C_{1,21}^{2}\bigr{)}-\frac{32}{y^{2}}C_{1,12}C_{1,21}T_{11}^{2}\bigg{)}$
$\displaystyle\qquad\quad{}+\bigg{(}\frac{64}{y^{2}}C_{1,12}C_{1,21}T_{11}^{2}-\frac{4T_{11}^{2}}{y^{2}}+\frac{4T_{11}}{y}-1\bigg{)}$
$\displaystyle\qquad{}=\frac{16}{y^{2}}\bigl{(}C_{1,12}^{2}T_{21}^{2}+T_{12}^{2}C_{1,21}^{2}\bigr{)}+\frac{32}{y^{2}}C_{1,12}C_{1,21}T_{11}^{2}-\frac{4T_{11}^{2}}{y^{2}}+\frac{4T_{11}}{y}-1$
$\displaystyle\qquad{}=\frac{16}{y^{2}}\bigl{(}C_{1,12}^{2}T_{21}^{2}+T_{12}^{2}C_{1,21}^{2}\bigr{)}-\frac{32}{y^{2}}C_{1,12}C_{1,21}T_{12}T_{21}-\frac{4T_{11}^{2}}{y^{2}}+\frac{4T_{11}}{y}-1$
$\displaystyle\qquad{}=\frac{16}{y^{2}}\bigl{(}C_{1,12}T_{21}-C_{1,21}T_{12}\bigr{)}^{2}-\frac{4T_{11}^{2}}{y^{2}}+\frac{4T_{11}}{y}-1,$

where we used $\det(\mathbf{T})=\mathrm{tr}(\mathbf{T})=0$. But now, using the
$(1,1)$-entry of the identity (5.19) shows that

$\left(\frac{\partial T_{11}}{\partial
z}\right)^{2}+R\left(\frac{\mathrm{i}y}{4T_{11}},y\right)^{2}=0.$

In other words, recalling from (5.13) that
$T_{11}=\breve{U}=\breve{U}_{n}(z;y)$, we have shown that

$\left(\frac{\partial\breve{U}}{\partial
z}\right)^{2}=-R\left(\frac{\mathrm{i}y}{4\breve{U}},y\right)^{2}=\frac{16}{y}\breve{U}^{3}-\frac{16c}{y^{2}}\breve{U}^{2}-\frac{4}{y}\breve{U}+1.$
(5.21)

Comparing (1.6) and (5.21) shows that $\breve{U}(z)=\breve{U}_{n}(z;y)$
satisfies the expected differential equation equivalent to the Weierstraß
equation (1.7) with constant of integration $E$ connected to $y$ via (5.3),
which completes the proof of Theorem 5.1.

######  Remark 5.5.

It is an interesting coincidence that the cubic polynomial in $\breve{U}$
appearing in the differential equation (5.21) is related to the rational
function $R(\eta,y)^{2}=f(\eta,y,c_{1}(y))$ defining the underlying spectral
curve (see (2.1)) by $\eta=\mathrm{i}y/(4\breve{U})$. Similar correspondences
have been noted with each application of this method; see [2] for the original
application to Painlevé-II, [1] for an application to Painlevé-III (D6), and
[3] for an application to Painlevé-IV.

##  Appendix A Elementary illustration of the method

In this appendix we illustrate the method of proof of Theorem 5.1 with a
toy111In fact, Riemann–Hilbert Problem A.1 arises in the description of unit-
amplitude plane-wave solutions for the defocusing nonlinear Schrödinger
equation $\mathrm{i}q_{t}+\frac{1}{2}q_{zz}-|q|^{2}q=0$ at time $t=0$ via
$q(z,0)=2\mathrm{i}\lim_{\eta\to\infty}\eta\breve{N}_{12}(\eta,z)$.  example.
Suppose the function $\breve{\mathbf{N}}(\eta,z)$ satisfies the following
Riemann–Hilbert problem:

######  Riemann–Hilbert Problem A.1 (toy outer parametrix).

Given $z\in\mathbb{C}$, seek a $2\times 2$ matrix-valued function
$(\mathbb{C}\setminus[-1,1])\ni\eta\mapsto\breve{\mathbf{N}}(\eta,z)$ with the
following properties:

  * Analyticity: $\eta\mapsto\breve{\mathbf{N}}(\eta,z)$ is analytic in its domain of definition.

  * Jump condition: $\eta\mapsto\breve{\mathbf{N}}(\eta,z)$ takes continuous boundary values from the left $(+$ subscript$)$ and right $(-$ subscript$)$ on $(-1,1)$ oriented toward $\eta=1$ except at the endpoints $\eta=\pm 1$. The jump condition relating the boundary values is

$\breve{\mathbf{N}}_{+}(\eta,z)=\breve{\mathbf{N}}_{-}(\eta,z)\begin{bmatrix}0&\mathrm{e}^{2\mathrm{i}z\eta}\\\
-\mathrm{e}^{-2\mathrm{i}z\eta}&0\end{bmatrix},\qquad z\in(-1,1).$

  * Normalization: $\breve{\mathbf{N}}(\eta,z)\to\mathbb{I}$ as $\eta\to\infty$.

  * Endpoint behavior: $\eta\mapsto\breve{\mathbf{N}}(\eta,z)$ is allowed to blow up like a negative one-fourth power near each endpoint $\eta=\pm 1$.

It is easy to check that this problem has a unique solution for every
$z\in\mathbb{C}$ given explicitly by

$\breve{\mathbf{N}}(\eta,z)=\frac{1}{\sqrt{2}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{bmatrix}\left(\frac{\eta-1}{\eta+1}\right)^{\sigma_{3}/4}\left(\frac{1}{\sqrt{2}}\begin{bmatrix}1&\mathrm{i}\\\
\mathrm{i}&1\end{bmatrix}\right)^{-1}\mathrm{e}^{-\mathrm{i}z(\eta-R(\eta))\sigma_{3}},$
(A.1)

where the diagonal matrix power is defined as the principal branch,
$R(\eta)^{2}=\eta^{2}-1$, and $R(\eta)$ is analytic for $\eta\not\in[-1,1]$
with $R(\eta)=\eta-\frac{1}{2}\eta^{-1}+\mathcal{O}\bigl{(}\eta^{-3}\bigr{)}$
as $\eta\to\infty$. If we let $\breve{\mathbf{N}}^{(k)}(z)$ denote the
coefficients in the convergent Laurent expansion

$\breve{\mathbf{N}}(\eta,z)=\mathbb{I}+\sum_{k=1}^{\infty}\breve{\mathbf{N}}^{(k)}(z)\eta^{-k},\qquad|\eta|>1,$
(A.2)

then it is straightforward to obtain from (A.1) that

$\breve{N}^{(1)}_{11}(z)=-\frac{1}{2}\mathrm{i}z,\qquad\breve{N}^{(1)}_{22}(z)=\frac{1}{2}\mathrm{i}z.$

In particular, this implies that the diagonal elements of
$\breve{\mathbf{N}}^{(1)}(z)$ satisfy simple differential equations:

$\frac{\mathrm{d}\breve{N}^{(1)}_{11}}{\mathrm{d}z}=-\frac{1}{2}\mathrm{i},\qquad\frac{\mathrm{d}\breve{N}^{(1)}_{22}}{\mathrm{d}z}=\frac{1}{2}\mathrm{i}.$
(A.3)

We will now derive the differential equations (A.3) without using the explicit
solution formula (A.1). (Analogously, in the proof of Theorem 5.1, we directly
derive (1.6).) First set
$\mathbf{F}(\eta,z):=\breve{\mathbf{N}}(\eta,z)\mathrm{e}^{\mathrm{i}z\eta\sigma_{3}}$,
and observe that

$\mathbf{Z}(\eta,z):=\frac{\partial\mathbf{F}}{\partial
z}(\eta,z)\mathbf{F}(\eta,z)^{-1}=\frac{\partial\breve{\mathbf{N}}}{\partial
z}(\eta,z)\breve{\mathbf{N}}(\eta,z)^{-1}+\mathrm{i}\eta\breve{\mathbf{N}}(\eta,z)\sigma_{3}\breve{\mathbf{N}}(\eta,z)^{-1}$

is an entire function. Since
$\breve{\mathbf{N}}_{z}(\eta,z)=\mathcal{O}\bigl{(}\eta^{-1}\bigr{)}$ as
$\eta\to\infty$ because $\breve{\mathbf{N}}(\eta,z)$ itself tends to a
constant matrix $\mathbb{I}$ as $\eta\to\infty$, we get the expansion

$\displaystyle\mathbf{Z}(\eta,z)=\mathrm{i}\eta\breve{\mathbf{N}}(\eta,z)\sigma_{3}\breve{\mathbf{N}}(\eta,z)^{-1}+\mathcal{O}\bigl{(}\eta^{-1}\bigr{)}=\mathrm{i}\eta\sigma_{3}+\mathrm{i}\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}+\mathcal{O}\bigl{(}\eta^{-1}\bigr{)}$

as $\eta\to\infty$, so by Liouville’s theorem,

$\mathbf{Z}(\eta,z)=\mathrm{i}\eta\sigma_{3}+\mathrm{i}\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}.$

Therefore, $\mathbf{F}(\eta,z)$ satisfies the differential equation

$\frac{\partial\mathbf{F}}{\partial
z}(\eta,z)=\mathbf{Z}(\eta,z)\mathbf{F}(\eta,z)$

or, in terms of $\breve{\mathbf{N}}(\eta,z)$ itself,

$\frac{\partial\breve{\mathbf{N}}}{\partial
z}(\eta,z)=\mathrm{i}\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}\breve{\mathbf{N}}(\eta,z)-\mathrm{i}\eta\big{[}\breve{\mathbf{N}}(\eta,z),\sigma_{3}\big{]}.$
(A.4)

Using the expansion (A.2) in (A.4), the diagonal terms proportional to
$\eta^{-1}$ give

$\frac{\mathrm{d}\breve{N}^{(1)}_{11}}{\mathrm{d}z}(z)=-2\mathrm{i}\breve{N}^{(1)}_{12}(z)\breve{N}^{(1)}_{21}(z),\qquad\frac{\mathrm{d}\breve{N}^{(1)}_{22}}{\mathrm{d}z}(z)=2\mathrm{i}\breve{N}^{(1)}_{12}(z)\breve{N}^{(1)}_{21}(z).$
(A.5)

To close the system, we define a matrix function $\mathbf{G}(\eta,z)$ by

$\mathbf{G}(\eta,z):=R(\eta)\breve{\mathbf{N}}(\eta,z)\sigma_{3}\breve{\mathbf{N}}(\eta,z)^{-1}.$
(A.6)

Using the jump condition satisfied by $\eta\mapsto\breve{\mathbf{N}}(\eta,z)$,
one checks easily that $\mathbf{G}_{+}(\eta,z)=\mathbf{G}_{-}(\eta,z)$ holds
for $z\in(-1,1)$, and from the behavior of
$\eta\mapsto\breve{\mathbf{N}}(\eta,z)$ at the endpoints of $[-1,1]$ one sees
that $\mathbf{G}(\eta,z)$ is bounded at $z=\pm 1$. It follows that for each
$z\in\mathbb{C}$, $\eta\mapsto\mathbf{G}(\eta,z)$ is an entire function. The
expansion (A.2) then implies corresponding asymptotic behavior of
$\mathbf{G}(\eta,z)$:

$\mathbf{G}(\eta,z)=\eta\sigma_{3}+\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}+\mathcal{O}\bigl{(}\eta^{-1}\bigr{)},\qquad\eta\to\infty$

and hence by Liouville’s theorem we obtain the exact identity

$\mathbf{G}(\eta,z)=\eta\sigma_{3}+\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}.$
(A.7)

Moreover, since $\sigma_{3}^{2}=\mathbb{I}$ and $R(\eta)^{2}=z^{2}-1$, it
follows from (A.6) that

$\mathbf{G}(\eta,z)^{2}=R(\eta)^{2}\mathbb{I}=\eta^{2}\mathbb{I}-\mathbb{I},$

while on the other hand, by squaring the identity (A.7) and using the fact
that for any $2\times 2$ matrix $\mathbf{M}$ one has
$\sigma_{3}[\mathbf{M},\sigma_{3}]+[\mathbf{M},\sigma_{3}]\sigma_{3}=\mathbf{0}$,

$\mathbf{G}(\eta,z)^{2}=\eta^{2}\mathbb{I}+\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}^{2}.$

Comparing these two representations of $\mathbf{G}(\eta,z)^{2}$, we therefore
obtain

$\big{[}\breve{\mathbf{N}}^{(1)}(z),\sigma_{3}\big{]}^{2}=-\mathbb{I}\implies
4\breve{N}^{(1)}_{12}(z)\breve{N}^{(1)}_{21}(z)=1.$

Using this identity in (A.5) closes the system and yields (A.3).

##  Appendix B Proof of Lemma 2.1

###### Proof.

Suppose that the roots $\eta=\mathrm{i}s_{j}$, $j=1,2,3$ are distinct, and
that the Boutroux conditions (2.2) hold (this will be justified later via a
continuation argument). With an integration constant selected so that
$\operatorname{Re}(h(\eta,y,c))=0$ at any one of the roots, the zero level set
$K$ of $\operatorname{Re}(h(\eta,y,c))$ is well defined and it contains the
closure $K^{\prime}$ of the union of _critical trajectories_ emanating from
the points $\eta=\mathrm{i}s_{j}$, $j=1,2,3$, which are the curves along which
$f(\eta,y,c)\,\mathrm{d}\eta^{2}<0$, where $f$ is defined by (2.1). We claim
that $K^{\prime}$ has the following properties.

  * •

$K^{\prime}$ is a connected set consisting of six simple arcs pairwise
disjoint except for their endpoints:

    * –

One arc joining the origin $\eta=0$ to one of the three points that we label
as $\eta=\mathrm{i}s_{1}$. We take this arc to be the branch cut
$\Sigma_{0,1}$.

    * –

One arc joining the other two points, $\eta=\mathrm{i}s_{2}$ and
$\eta=\mathrm{i}s_{3}$. We take this arc to be the branch cut $\Sigma_{2,3}$.

    * –

Two arcs joining $\eta=\mathrm{i}s_{1}$ either to the same point that we label
as $\eta=\mathrm{i}s_{2}$ (case (i)) or one each to $\eta=\mathrm{i}s_{2}$ and
$\eta=\mathrm{i}s_{3}$ (case (ii)).

    * –

Two unbounded arcs tending to $\eta=\infty$ parallel to the real line, one in
the right half-plane and one in the left half-plane.

  * •

In case (i), the region bounded by the two arcs joining $\eta=\mathrm{i}s_{1}$
with $\eta=\mathrm{i}s_{2}$ contains the origin $\eta=0$ and both unbounded
arcs emanate from $\eta=\mathrm{i}s_{3}$. In case (ii), the region bounded by
the arc joining $\eta=\mathrm{i}s_{1}$ with $\eta=\mathrm{i}s_{2}$, the arc
joining $\eta=\mathrm{i}s_{1}$ with $\eta=\mathrm{i}s_{3}$, and the arc
$\Sigma_{2,3}$ contains the origin $\eta=0$ and one unbounded arc emanates
from each of $\eta=\mathrm{i}s_{2}$ and $\eta=\mathrm{i}s_{3}$.

Locally, near each of the simple roots $\eta=\mathrm{i}s_{j}$, $j=1,2,3$, of
$\eta\mapsto f(\eta,y,c)$, $K^{\prime}$ consists of a union of three
trajectories emanating from $\eta=\mathrm{i}s_{j}$ in directions separated by
equal angles of $2\pi/3$. Given an index $j=1,2,3$, each of the three
trajectories emanating from $\eta=\mathrm{i}s_{j}$ terminates in the other
direction at $\eta=\mathrm{i}s_{1}$, $\eta=\mathrm{i}s_{2}$,
$\eta=\mathrm{i}s_{3}$, $\eta=0$, or is unbounded in which case it tends to
$\eta=\infty$ asymptotically horizontally. This is because otherwise the
trajectory would be _divergent_ and hence _recurrent_ [15, Theorem 11.1]. But
the closure of a recurrent trajectory contains a nonempty domain in
$\mathbb{C}$ and since $\operatorname{Re}(h(\eta,y,c))=0$ on the trajectory,
this harmonic function would vanish identically on $\mathcal{R}$ (the Riemann
surface of $h_{\eta}(\eta,y,c)$, i.e., the spectral curve), which is a
contradiction because $h_{\eta}(\eta,y,c)$ is not identically zero. Taking
into account the Boutroux conditions (2.2) which imply that $K^{\prime}\subset
K$, similar local analysis shows that there can be at most one critical
trajectory terminating at $\eta=0$ and at most one unbounded critical
trajectory tending horizontally to $\eta=\infty$ in each of the left and right
half-planes.

To work out the global trajectory structure in order to prove the claim, it is
easiest to first assume that $y>0$ and $c\in\mathbb{R}$, in which case it is
easy to see that $h_{\eta}(-\eta^{*},y,c)=h_{\eta}(\eta,y,c)^{*}$ provided
that $\Sigma_{0,1}$ and $\Sigma_{2,3}$ are taken to be symmetric in the
imaginary $\eta$-axis, which we will also assume. Moreover we either have
(making a choice of labeling of the roots of $P$) $s_{1}<0<s_{2}<s_{3}$ or
$s_{1}<0$ and $s_{3}=s_{2}^{*}$. In either configuration the condition
$I_{2,3}=0$ holds automatically, and $c\in\mathbb{R}$ is presumed to be
determined from the remaining real condition $I_{1,2}=0$. We examine the two
configurations in turn.

If $s_{1}<0<s_{2}<s_{3}$, since $f(\eta,y,c)=h_{\eta}(\eta,y,c)^{2}>0$ holds
for $\eta\in\mathrm{i}\mathbb{R}$ between $\eta=\mathrm{i}s_{1}$ and $\eta=0$
as well as between $\eta=\mathrm{i}s_{2}$ and $\eta=\mathrm{i}s_{3}$, these
intervals of the imaginary axis are critical trajectories. Since elsewhere on
the imaginary axis we have ${f(\eta,y,c)=h_{\eta}(\eta,y,c)^{2}<0}$,
$\operatorname{Re}(h(\eta,y,c))$ is strictly monotone as $\eta$ varies in
these intervals of $\mathrm{i}\mathbb{R}$. Therefore, in this configuration
there are no points of either $K$ or $K^{\prime}\subset K$ on the imaginary
axis outside the two critical trajectories. Since $K^{\prime}$ is symmetric in
reflection through the imaginary axis, and since exactly one critical
trajectory goes to $\eta=\infty$ in each half-plane, there are only three
possibilities:

  * •

The two remaining trajectories emanating from $\eta=\mathrm{i}s_{1}$ tend to
infinity in opposite half-planes, and there is a symmetric pair of arcs in
each half-plane joining the points $\eta=\mathrm{i}s_{2}$ and
$\eta=\mathrm{i}s_{3}$. However, since $\operatorname{Re}(h(\eta,y,c))$ is
harmonic between the imaginary axis and each of these arcs and vanishes on
each critical trajectory, this would imply by the maximum principle that
$\operatorname{Re}(h(\eta,y,c))\equiv 0$ in each of these domains. This is a
contradiction since $h_{\eta}(\eta,y,c)$ does not vanish identically.

  * •

The two remaining trajectories emanating from $\eta=\mathrm{i}s_{2}$ tend to
infinity in opposite half-planes, and there is a symmetric pair of arcs in
each half-plane joining the points $\eta=\mathrm{i}s_{1}$ and
$\eta=\mathrm{i}s_{3}$. However, this would imply a crossing of two different
trajectories at a point in each half-plane where $f(\eta,y,c)$ is finite and
nonzero, which cannot occur.

  * •

Therefore, the remaining possibility must hold, namely that the two remaining
trajectories emanating from $\eta=\mathrm{i}s_{3}$ tend to infinity in
opposite half-planes, and there is a symmetric pair of arcs in each half-plane
joining the points $\eta=\mathrm{i}s_{1}$ and $\eta=\mathrm{i}s_{2}$.

This shows that the claimed structure holds in case (i) when
$s_{1}<0<s_{2}<s_{3}$.

If instead $s_{1}<0$ and $s_{3}=s_{2}^{*}$, since
$f(\eta,y,c)=h_{\eta}(\eta,y,c)^{2}>0$ holds for $\eta\in\mathrm{i}\mathbb{R}$
between $\eta=\mathrm{i}s_{1}$ and $\eta=0$, this interval of the imaginary
axis is a critical trajectory, while in the intervals between
$\eta=-\mathrm{i}\infty$ and $\eta=\mathrm{i}s_{1}$ and between $\eta=0$ and
$\eta=+\mathrm{i}\infty$ we have $f(\eta,y,c)=h_{\eta}(\eta,y,c)^{2}<0$ so
$\operatorname{Re}(h(\eta,y,c))$ is strictly monotone. This implies that there
are no points of either $K$ or $K^{\prime}\subset K$ on the imaginary axis
below $\eta=\mathrm{i}s_{1}$, but because $\operatorname{Re}(h(\eta,y,c))$
necessarily changes sign on the positive imaginary axis due to the singularity
at the origin and the linear growth at infinity there is exactly one point of
$K$ there, which may belong to $K^{\prime}$. In fact, this point does indeed
belong to $K^{\prime}$, because otherwise at least two of the critical
trajectories emanating from each of the points $\eta=\mathrm{i}s_{2}$ and
$\eta=\mathrm{i}s_{3}=-(\mathrm{i}s_{2})^{*}$ must tend to infinity in the
half-plane containing the point because only one of them can terminate at
$\eta=\mathrm{i}s_{1}$; this contradicts the fact that exactly one critical
trajectory tends to infinity in each half-plane. So the distinguished point in
the imaginary interval between $\eta=0$ and $\eta=+\mathrm{i}\infty$ belongs
to $K^{\prime}$ and lies on a critical trajectory crossing the imaginary axis
horizontally and connecting $\eta=\mathrm{i}s_{2}$ and
$\eta=\mathrm{i}s_{3}=-(\mathrm{i}s_{2})^{*}$. The remaining two trajectories
emanating from each of these points necessarily tend to $\eta=\mathrm{i}s_{1}$
and $\eta=\infty$ without crossing. This shows that the claimed structure of
$K^{\prime}$ holds in case (ii) when $s_{1}<0$ and $s_{3}=s_{2}^{*}$.

We next show that whenever $0<y<y_{\mathrm{c}}\approx 0.29177$, where
$y_{\mathrm{c}}$ is the critical value defined in [4, Section 4.6], there
exists a unique $c=c_{1}(y)\in\mathbb{R}$ for which the conditions (2.2)
(really just $I_{1,2}=0$ as $I_{2,3}=0$ is automatic) hold with root
configuration $s_{1}<0<s_{2}<s_{3}$ and hence $K^{\prime}$ has the claimed
structure in case (i). To do this, we first suppose that $y>0$ and choose
$c\in\mathbb{R}$ differently, so that the cubic $\mu\mapsto P(\mu,y,c)$
defined in (2.1) has a simple root $\mu=s$ and a double root $\mu=d$. Then by
setting $P(\mu,y,c)=-(\mu-d)^{2}(\mu-s)$ one sees that $d=(1-s)/2$ and that
$s$ satisfies the cubic equation $s(s-1)^{2}=-y^{2}$, while
$c=-d^{2}-2ds=\frac{1}{4}(3s^{2}-2s-1)$. The condition $y>0$ implies that the
equation $s(s-1)^{2}=-y^{2}$ has one real and two complex-conjugate solutions
for $s$. But if $s=u+\mathrm{i}v$ with $v\neq 0$, then
$\operatorname{Im}(c)=\frac{1}{2}(3u-1)v$ which vanishes for $v\neq 0$ only if
$u=\frac{1}{3}$. Then
$\operatorname{Im}\bigl{(}y^{2}\bigr{)}=-\operatorname{Im}\bigl{(}s(s-1)^{2}\bigr{)}=v^{3}\neq
0$ which contradicts $y>0$. Therefore, the conditions $y>0$ and
$c\in\mathbb{R}$ require that we select the real root $s=s(y)<0$ of
$s(s-1)^{2}=-y^{2}$ and then $d=d(y)>\frac{1}{2}$ is also real and
$c=c_{0}(y)$ is a corresponding well-defined real number. In this double-root
configuration, the function $\eta\mapsto h_{\eta}(\eta,y,c_{0}(y))$ is
analytic except on the imaginary segment between $\eta=\mathrm{i}s(y)$ and
$\eta=0$. Choosing an integration constant so that
$\operatorname{Re}(h(\eta,y,c_{0}(y)))=0$ for $\eta=\mathrm{i}s(y)$, the
function $\operatorname{Re}(h(\eta,y,c_{0}(y)))$ is well defined by contour
integration and it is harmonic except on the branch cut for
$h_{\eta}(\eta,y,c_{0}(y))$. According to [4, Section 4.6], if $y>0$ and
$y\neq y_{\mathrm{c}}$, then $y-y_{\mathrm{c}}$ and
$\operatorname{Re}(h(\eta,y,c_{0}(y)))$ always have the same sign for
$\eta=\mathrm{i}d(y)$. Now $c=c_{0}(y)$ is necessarily a real root of the
cubic discriminant of $\mu\mapsto P(\mu,y,c)$, which is proportional by a
positive numerical factor to $64c^{3}+16c^{2}+72y^{2}c+16y^{2}-27y^{4}$. The
discriminant of this latter polynomial with respect to $c$ is proportional by
a positive numerical factor to $-y^{2}\bigl{(}4+27y^{2}\bigr{)}^{3}$ which is
strictly negative for $y>0$, hence $c=c_{0}(y)$ is a simple root of the cubic
discriminant and it is the only real root thereof. Clearly the cubic
discriminant of $\mu\mapsto P(\mu,y,c)$ has the same sign as $c$ when $y>0$ is
fixed and $c\in\mathbb{R}$ is large. Hence $\mu\mapsto P(\mu,y,c)$ has three
real distinct roots whenever $c>c_{0}(y)$ and has only one real root (simple)
whenever ${c<c_{0}(y)}$.

Now returning to the general case of $y>0$ and $c\in\mathbb{R}$ arbitrary (so
that $P(\mu,y,c)$ need not have a double root), we wish to solve the equation
$I_{1,2}=0$ for $c$ using the above information. A simple calculation shows
that $\frac{\partial}{\partial c}h_{\eta}(\eta,y,c)^{2}=-\eta^{-2}$, which
implies that when $c\in\mathbb{R}$ and ${y>0}$,

$\frac{\partial I_{1,2}}{\partial
c}=\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\partial^{2}h}{\partial\eta\partial
c}(\eta,y,c)\,\mathrm{d}y\bigg{)}=-\frac{1}{2}\operatorname{Re}\bigg{(}\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}\frac{\mathrm{d}\eta}{\eta^{2}h_{\eta}(\eta,y,c)}\bigg{)},$

because $h_{\eta}(\eta,y,c)$ vanishes at $\eta=\mathrm{i}s_{1}$ and
$\eta=\mathrm{i}s_{2}$. Some contour deformations show that regardless of
whether $c<c_{0}(y)$ or $c>c_{0}(y)$, this derivative can be written in a
universal form:

$\frac{\partial I_{1,2}}{\partial
c}=\frac{1}{4}\int_{\mathbb{R}}\frac{\chi_{P>0}(\mu)\,\mathrm{d}\mu}{\sqrt{P(\mu,y,c)}},$

where $\chi_{P>0}(\mu)$ is the characteristic function of the union of
intervals of $\mathbb{R}$ on which $\mu\mapsto P(\mu,y,c)$ is positive, and
the square root is positive. Therefore, in both cases $c<c_{0}(y)$ and
$c>c_{0}(y)$ the partial derivative of $I_{1,2}$ with respect to
$c\in\mathbb{R}$ for $y>0$ is positive (the difference is that the support of
$\chi_{P>0}(\mu)$ consists of two intervals $(-\infty,s_{1})\cup(0,+\infty)$
for $c<c_{0}(y)$ and of three intervals
$(-\infty,s_{1})\cup(0,s_{2})\cup(s_{3},+\infty)$ for $c>c_{0}(y)$). Now, the
real-valued function $c\mapsto I(c):=I_{1,2}$ is certainly continuous as a
function of $c\in\mathbb{R}$ and is continuously differentiable for $c\neq
c_{0}(y)$ with positive derivative. We also know that
$I(c_{0}(y))=\operatorname{Re}(h(\mathrm{i}d(y),y,c_{0}(y)))<0$ for
$0<y<y_{\mathrm{c}}$. However, it is also true that eventually $I(c)>0$ as
$c\to+\infty$. To see this, first note that the roots of $P(\mu,y,c)$ for
large $c>0$ with fixed $y>0$ are $s_{1}=-c^{1/2}+O(1)$,
$s_{2}=\frac{1}{4}y^{2}c^{-1}+O\bigl{(}c^{-2}\bigr{)}$, and
$s_{3}=c^{1/2}+O(1)$. Then one rescales the integrand of $I_{1,2}$ by
$\eta=s_{2}w$ and notices that by contour deformations, the leading term of
$I_{1,2}$ as $c\to+\infty$ with $y>0$ fixed is computed as a residue. Indeed,
letting $C_{\mathrm{L}}$ (resp. $C_{\mathrm{R}}$) denote a contour in the left
(resp. right) half-plane beginning at a point on the imaginary axis between
$\eta=\mathrm{i}s_{1}$ and $\eta=0$ and terminating at a point on the
imaginary axis between $\eta=\mathrm{i}s_{2}$ and $\eta=\mathrm{i}s_{3}$,
letting $L$ denote a positively-oriented loop surrounding the imaginary
interval between $w=0$ and $w=\mathrm{i}$, and using principal branches for
all power functions,

$\displaystyle I_{1,2}$
$\displaystyle=\operatorname{Re}\left(\int_{\mathrm{i}s_{1}}^{\mathrm{i}s_{2}}h_{\eta}(\eta,y,c)\,\mathrm{d}\eta\right)$
$\displaystyle=\frac{1}{2}\operatorname{Re}\left(\int_{C_{\mathrm{L}}\cup
C_{\mathrm{R}}}\left[-\mathrm{i}(-\mathrm{i}\eta-s_{3})^{1/2}(-\mathrm{i}\eta-
s_{2})^{1/2}(-\mathrm{i}\eta)^{-3/2}(-\mathrm{i}\eta-
s_{1})^{1/2}\right]\,\mathrm{d}\eta\right)$
$\displaystyle=\frac{1}{2}\operatorname{Re}\left(\oint_{(-C_{\mathrm{L}})\cup
C_{\mathrm{R}}}\left[-\mathrm{i}(-\mathrm{i}\eta-
s_{1})^{1/2}\left(\frac{\eta-\mathrm{i}s_{2}}{\eta}\right)^{1/2}(\mathrm{i}\eta+s_{3})^{1/2}\right]\frac{\mathrm{d}\eta}{\eta}\right)$
$\displaystyle=\frac{1}{2}\sqrt{c}\operatorname{Re}\left(\oint_{L}\left[-\mathrm{i}\left(-\mathrm{i}\frac{s_{2}}{\sqrt{c}}w-\frac{s_{1}}{\sqrt{c}}\right)^{1/2}\left(\frac{w-\mathrm{i}}{w}\right)^{1/2}\left(\mathrm{i}\frac{s_{2}}{\sqrt{c}}+\frac{s_{3}}{\sqrt{c}}\right)^{1/2}\right]\,\frac{\mathrm{d}w}{w}\right)$
$\displaystyle=\frac{1}{2}\sqrt{c}\left[\operatorname{Re}\left(\oint_{L}\left[-\mathrm{i}\left(\frac{w-\mathrm{i}}{w}\right)^{1/2}\right]\frac{\mathrm{d}w}{w}\right)+o(1)\right]$
$\displaystyle=\frac{1}{2}\sqrt{c}\left[\operatorname{Re}\left(2\pi\mathrm{i}\mathop{\mathrm{Res}}_{w=\infty}\left[-\mathrm{i}\left(\frac{w-\mathrm{i}}{w}\right)^{1/2}\frac{1}{w}\right]\right)+o(1)\right]$
$\displaystyle=\pi\sqrt{c}+o(\sqrt{c}),\qquad c\to+\infty.$

Therefore if $0<y<y_{\mathrm{c}}$, $I(c)<0$ for $c=c_{0}(y)$ while $I(c)>0$ in
the limit $c\to+\infty$, and $c\mapsto I(c)$ is strictly increasing for
$c>c_{0}(y)$. It follows from the intermediate value theorem that there is a
unique solution $c=c_{1}(y)>c_{0}(y)$ of $I(c)=I_{1,2}=0$, and the inequality
$c_{1}(y)>c_{0}(y)$ implies that the roots of $\mu\mapsto P(\mu,y,c)$ are all
real and hence $K^{\prime}$ has the claimed structure in case (i) whenever
$0<y<y_{\mathrm{c}}$.

Now we continue this solution into the complex $y$-plane. Since for
$0<y<y_{\mathrm{c}}$ the roots of $\mu\mapsto P(\mu,y,c_{1}(y))$ are distinct,
the solution $c=c_{1}(y)$ of the system (2.2) can be continued uniquely by the
implicit function theorem to some maximal domain of the complex $y$-plane
containing the real interval $(0,y_{\mathrm{c}})$ (however note that
$c_{1}(y)$ is not an analytic function of $y$ in this domain). The boundary of
this domain consists of all points $y$ for which $c_{1}(y)=c_{0}(y)$, i.e.,
under continuation of the solution of the Boutroux conditions (2.2) one
arrives at a degenerate spectral curve with $\mu\mapsto P(\mu,y,c)$ having a
double root. This boundary was obtained in [4, Section 4.7], which shows that
the maximal domain under consideration is exactly that mapped by $Y=y^{1/3}$
onto the interior of the right “wing” of the “bow-tie” region $\mathcal{B}$ in
the $Y$-plane (see Figure 1, left-hand panel). The topological structure of
$K^{\prime}$ remains the same under continuation, and hence the claimed
structure of $K^{\prime}$ holds throughout this domain in case (i) as that is
the case for $0<y<y_{\mathrm{c}}$. The boundary of the domain consists of the
imaginary segment between $y=\pm 2\mathrm{i}/\sqrt{27}$ and a certain Schwarz-
symmetric curve in the right half-plane connecting those two imaginary
endpoints via the positive real value $y_{\mathrm{c}}\approx 0.29177$; see the
right-hand panel of Figure 1.

Given the proven structure of $K^{\prime}$ in case (i), it is clear that
$K^{\prime}$ divides the complex $\eta$-plane into three disjoint regions,
consistent with the _basic structure theorem_ [10, p. 37]. As each of these
regions has exactly one pole of $f(\eta,y,c)\,\mathrm{d}\eta^{2}$ on its
boundary (including the point at infinity in a suitable local coordinate),
they are all _end domains_ , which means that they are conformally mapped by
the primitive
$\eta\mapsto\int^{\eta}h_{\eta}(\eta^{\prime},y,c)\,\mathrm{d}\eta^{\prime}$
onto a half-plane with a vertical boundary. Since
$\operatorname{Re}(h(\eta,y,c))=0$ on the boundary of each end domain because
it is a subset of $K^{\prime}\subset K$, the vertical boundary of the image is
exactly the imaginary axis, and hence there can be no other points with
$\operatorname{Re}(h(\eta,y,c))=0$ in the interior of each end domain. This
proves that $K=K^{\prime}$, which establishes the key properties of the level
curve $K$ asserted in the statement of the lemma.

Introducing a contour arc $\Sigma_{0,2}$ joining $\eta=0$ with
$\eta=\mathrm{i}s_{2}$ and an unbounded arc $\Sigma_{1,\infty}$ with finite
endpoint $\eta=\mathrm{i}s_{1}$ such that
$\Sigma_{h}:=\Sigma_{2,3}\cup\Sigma_{0,2}\cup\Sigma_{0,1}\cup\Sigma_{1,\infty}$
is a simple contour, a function $\eta\mapsto h(\eta,y,c_{1}(y))$ is well-
defined up to an integration constant possibly depending on $y$ by contour
integration of $h_{\eta}(\eta,y,c_{1}(y))$ in the simply connected domain
${\mathbb{C}\setminus\Sigma_{h}}$. Since $h_{\eta}(\eta,y,c)$ is integrable at
all three of the roots $\eta=\mathrm{i}s_{j}$, we can and will choose the
integration constant so that
$\operatorname{Re}(h(\mathrm{i}s_{3},y,c_{1}(y)))=0$. It follows from (2.2)
and the reality of the residue of $h_{\eta}(\eta,y,c_{1}(y))$ at $\eta=\infty$
that $\operatorname{Re}(h(\eta,y,c_{1}(y)))$ is a function harmonic on the
larger domain $\mathbb{C}\setminus(\Sigma_{2,3}\cup\Sigma_{0,1})$. Recalling
that $\Sigma_{2,3}$ and $\Sigma_{0,1}$ have been chosen to agree with arcs of
$K^{\prime}=K$, the function
$\eta\mapsto\operatorname{Re}(h(\eta,y,c_{1}(y)))$ is continuous except at
${\eta=0}$. ∎

### Acknowledgements

R.J. Buckingham was supported by the National Science Foundation under Grant
DMS-2108019. P.D. Miller was supported by the National Science Foundation
under Grants DMS-1812625 and DMS-2204896.

## References

  * [1] Bothner T., Miller P.D., Rational solutions of the Painlevé-III equation: large parameter asymptotics, Constr. Approx. 51 (2020), 123–224, arXiv:1808.01421.
  * [2] Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-II functions: noncritical behaviour, Nonlinearity 27 (2014), 2489–2578, arXiv:1310.2276.
  * [3] Buckingham R.J., Miller P.D., Large-degree asymptotics of rational Painlevé-IV solutions by the isomonodromy method, Constr. Approx. 56 (2022), 233–443, arXiv:2008.00600.
  * [4] Buckingham R.J., Miller P.D., On the algebraic solutions of the Painlevé-III $(\rm D_{7})$ equation, Phys. D 441 (2022), 133493, 22 pages, arXiv:2202.04217.
  * [5] Clarkson P.A., The third Painlevé equation and associated special polynomials, J. Phys. A 36 (2003), 9507–9532.
  * [6] Deift P., Zhou X., A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993), 295–368.
  * [7] Dubrovin B.A., Theta-functions and nonlinear equations, Russian Math. Surveys 36 (1981), 11–92.
  * [8] Fokas A.S., Its A.R., Kapaev A.A., Novokshenov V.Yu., Painlevé transcendents: The Riemann–Hilbert approach, Math. Surveys Monogr., Vol. 128, American Mathematical Society, Providence, RI, 2006.
  * [9] Its A.R., Kapaev A.A., The nonlinear steepest descent approach to the asymptotics of the second Painlevé transcendent in the complex domain, in MathPhys Odyssey, 2001, Prog. Math. Phys., Vol. 23, Birkhäuser, Boston, MA, 2002, 273–311, arXiv:nlin/0108054.
  * [10] Jenkins J.A., Univalent functions and conformal mapping, Ergeb. Math. Grenzgeb. (3), Vol. 18, Springer, Berlin, 1958.
  * [11] Kitaev A.V., Vartanian A.H., Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I, Inverse Problems 20 (2004), 1165–1206, arXiv:math.CA/0312075.
  * [12] Ohyama Y., Kawamuko H., Sakai H., Okamoto K., Studies on the Painlevé equations. V. Third Painlevé equations of special type ${\rm P}_{\rm III}({\rm D}_{7})$ and ${\rm P}_{\rm III}({\rm D}_{8})$, J. Math. Sci. Univ. Tokyo 13 (2006), 145–204.
  * [13] Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A., NIST digital library of mathematical functions, Release 1.1.10 of 2023-06-15, aviable at https://dlmf.nist.gov/.
  * [14] Shimomura S., Boutroux ansatz for the degenerate third Painlevé transcendents, arXiv:2207.11495.
  * [15] Strebel K., Quadratic differentials, Ergeb. Math. Grenzgeb. (3), Vol. 5, Springer, Berlin, 1984.