for the permutated ones; therefore, we outline the proof by taking, as an
example, the transformation (4.11). Substituting the formulae for the
variables without tildes into the first equation of (4.9), we immediately
confirm the validity of the first equation of (4.7) for the tilde variables.
To confirm the second equation of (4.7), we have to use the first one twice;
more precisely, substituting variables without tildes into the second equation
of (4.9), we get

$\displaystyle\tilde{g}_{1}(\tilde{g}_{1}-1)=\tilde{g}_{3}(-\tilde{g}_{2}+s\tilde{g}_{1}-s\tilde{g}_{3})$
$\displaystyle\quad\Rightarrow\quad\tilde{g}_{1}(\tilde{g}_{2}-\tilde{g}_{3}+\tilde{s}\tilde{g}_{3})=\tilde{g}_{3}(-\tilde{g}_{2}+\tilde{s}\tilde{g}_{1}-\tilde{s}\tilde{g}_{3})\Rightarrow$
$\displaystyle\tilde{g}_{1}\tilde{g}_{2}=\tilde{g}_{3}(\tilde{g}_{1}-\tilde{g}_{2}+\tilde{s}\tilde{g}_{3})$
$\displaystyle\quad\Rightarrow\quad\tilde{g}_{1}\tilde{g}_{2}=\tilde{g}_{3}(1-(1-\tilde{s})\tilde{g}_{3}+\tilde{s}\tilde{g}_{3}).$

$\blacksquare$

######  Lemma 4.1.

Let $u(\tau)$ be the solution of equation (1.1) with $a=0$ defined by the
asymptotics (1.10). Then, the monodromy data characterizing $u(\tau)$ reads:

$\begin{gathered}\tilde{s}=0,\quad s_{0}^{0}=\mathrm{i},\quad
s_{0}^{\infty}s_{1}^{\infty}=-1,\quad
s_{0}^{\infty}g_{12}^{2}=s_{1}^{\infty}g_{21}^{2}=-\frac{\mathrm{i}(H(0)-1)^{2}}{3H(0)},\\\
\tilde{g}_{4}=g_{12}g_{21}=\frac{(H(0)-1)^{2}}{3H(0)},\quad\tilde{g}_{3}=g_{11}g_{22}=\frac{1}{3H(0)}\Big{(}H(0)-\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}\Big{)}\Big{(}H(0)-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}\Big{)},\\\
\tilde{g}_{2}=\mathrm{i}g_{21}g_{22}=-\frac{\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}{3H(0)}\Big{(}H(0)-1\Big{)}\Big{(}H(0)-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}\Big{)},\quad\tilde{g}_{1}=\mathrm{i}g_{12}g_{11}=\frac{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}}{3H(0)}\Big{(}H(0)-1\Big{)}\Big{(}H(0)-\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}\Big{)}.\\\
\end{gathered}$ (4.12)

###### Proof.

The asymptotics as $\tau\to 0$ of the solution $u(\tau)$ does not contain
logarithmic terms (cf. (1.10)); therefore, its parametrization via the
monodromy data is given in Appendix B, Theorem B.1 (where we substitute
$a=0$):

$\displaystyle u(\tau)\underset{\tau\to+0}{=}$
$\displaystyle\,\frac{\tau}{16\pi}\\!\left(\varpi_{1}(-\rho)\varpi_{2}(-\rho)\tau^{-4\rho}+\varpi_{1}(-\rho)\varpi_{2}(\rho)+\varpi_{1}(\rho)\varpi_{2}(-\rho)+\varpi_{1}(\rho)\varpi_{2}(\rho)\tau^{4\rho}\right)\left(1+o(\tau^{\delta})\right),$
(4.13)

where $\varpi_{k}(\lambda)$ for $k=1,2$ and $\lambda=\pm\rho$ are given in
equations (B.6)–(B.8), and $\delta>0$.

To match with the asymptotics (1.10), one has to assume that
$\varpi_{1}(-\rho)\varpi_{2}(-\rho)\neq 0$ and $1-4\rho=1/3$: the last
equality implies $\rho=1/6$. In the conclusions above, we used the fact that
the leading term of asymptotics (4.13) is symmetric with respect to the change
$\rho\to-\rho$, so that, in case $\rho\neq 0$, one can always assume that
$\rho>0$. Equations (B.3) now read

$\cos\Big{(}\frac{\pi}{3}\Big{)}=\frac{1}{2}=-\frac{\mathrm{i}s_{0}^{0}}{2}=1+\frac{1}{2}s_{0}^{\infty}s_{1}^{\infty},$

which confirms the fist three relations in (4.12).

Comparing the coefficients of the leading terms in equations (4.13) and
(1.10), we get

$\frac{\varpi_{1}(-1/6)\varpi_{2}(-1/6)}{16\pi}=\frac{1}{2}H(0).$ (4.14)

For $\rho=1/6$, equations (B.6)–(B.8) in Appendix B read:

$\displaystyle\varpi_{1}(\pm 1/6)=\mathbf{p}_{1}(\pm 1/6)\chi_{1}(\pm
1/6)=\mathbf{p}_{1}(\pm
1/6)\left(g_{11}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{\pm\mathrm{i}\pi/6}+g_{21}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{\mp\mathrm{i}\pi/6}\right),$
(4.15) $\displaystyle\varpi_{2}(\pm 1/6)=\mathbf{p}_{2}(\pm 1/6)\chi_{2}(\pm
1/6)=\mathbf{p}_{2}(\pm
1/6)\left(g_{12}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{\pm\mathrm{i}\pi/6}+g_{22}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{\mp\mathrm{i}\pi/6}\right),$
(4.16)

with

$\mathbf{p}_{1}(\pm 1/6)=\pm
6\mathrm{e}^{\pm\mathrm{i}\pi/12}\left(\frac{1}{2}\right)^{\pm
1/6}\frac{\Gamma(1\mp\frac{1}{3})\Gamma(1\pm\frac{1}{6})}{\Gamma(1\pm\frac{1}{3})},\qquad\mathbf{p}_{2}(\pm
1/6)=\mathbf{p}_{1}(\pm 1/6)\mathrm{e}^{\mp\mathrm{i}\pi/6},$ (4.17)

and $\Gamma(\ast)$ is the (Euler) gamma function [13]. The numbers
$\mathbf{p}(\pm 1/6)$ can be calculated explicitly,

$\mathbf{p}_{1}(1/6)=3\sqrt{2\pi}\,\mathrm{e}^{\frac{\pi\mathrm{i}}{12}},\quad\mathbf{p}_{1}(-1/6)=-2\sqrt{2\pi}\,\mathrm{e}^{-\frac{\pi\mathrm{i}}{12}},\quad\mathbf{p}_{1}(1/6)\,\mathbf{p}_{1}(-1/6)=-12\pi.$
(4.18)

Combining equations (4.14)–(4.16), one obtains

$(\mathbf{p}_{1}(-1/6))^{2}\left(g_{11}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\mathrm{i}\pi/6}+g_{21}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{\mathrm{i}\pi/6}\right)\\!\left(g_{12}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{-\mathrm{i}\pi/6}+g_{22}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{\mathrm{i}\pi/6}\right)\mathrm{e}^{\mathrm{i}\pi/6}=8\pi
H(0).$ (4.19)

Using the expression for $\mathbf{p}_{1}(-1/6)$ given in (4.18), multiplying
out equation (4.19) and exploiting (4.5) in order to remove the term
$g_{12}g_{21}$, and introducing the $\tilde{g}_{k}$ variables, one arrives at
the following equation for the monodromy data:

$\tilde{g}_{1}\mathrm{e}^{-\frac{\pi\mathrm{i}}{3}}+2(\tilde{g}_{3}-1)-\tilde{g}_{2}\mathrm{e}^{\frac{\pi\mathrm{i}}{3}}=H(0)-1.$
(4.20)

Equation (4.20) should be supplemented with two equations defining the
monodromy manifold (4.7); therefore, we obtain three equations for the three
variables $\tilde{g}_{k}$, $k=1,2,3$. In order to solve these equations,
express $\tilde{g}_{1}$ and $\tilde{g}_{2}$ from the linear equations as
linear combinations of $H(0)-1$ and $\tilde{g}_{3}-1$; then, substituting
these expressions into the quadratic equation in (4.7), one finds that the
quadratic term $(\tilde{g}_{3}-1)^{2}$ cancels, so that we get a linear
equation in $\tilde{g}_{3}-1=\tilde{g}_{4}$ which contains $H(0)$. Solving the
last equation for $\tilde{g}_{4}$, we arrive at the expression for
$\tilde{g}_{4}$ stated in (4.12). Then, the formula for $\tilde{g}_{3}$
follows from the equation $\tilde{g}_{3}-1=\tilde{g}_{4}$, and $\tilde{g}_{1}$
and $\tilde{g}_{2}$ are obtained from the linear equations mentioned in the
proof above. ∎

######  Remark 4.2.

Note that the error estimate in the asymptotics (4.13) contains an
undetermined positive parameter $\delta$. The value of this parameter in many
questions, as, in particular, demonstrated in the proof above, is not
important. For some very special cases, though, this parameter may turn out to
be an impediment to the direct application of asymptotics for the calculation
of the monodromy data; in such cases, however, it is the special properties of
the solution that may, nevertheless, help to circumvent this problem (see, for
example, [28]). The value of $\delta$ is not universal, and it depends on the
solution; of course, the local analysis of the solution allows one to find the
value of the parameter $\delta$ for particular solutions (see (4.26) below).

Below, we show how our asymptotic formula (4.13) is consistent with the
expansion for the function $H(r)$ studied in Section 2. As a matter of fact,
we present an alternative calculation for the monodromy data that were
obtained in Lemma 4.1 for the purpose of demonstrating the applicability of
those formulae in Appendix B, Theorem B.1 that were not used for the
calculation of the monodromy data. Note, however, that this latter calculation
does, in fact, use the value of $\delta$.

We commence with the proof of two identities for $\varpi_{k}(\pm 1/6)$,
$k=1,2$:

$\begin{gathered}\varpi_{1}(-1/6)\,\varpi_{2}(1/6)+\varpi_{1}(1/6)\,\varpi_{2}(-1/6)\\\
=\mathbf{p}_{1}(1/6)\,\mathbf{p}_{1}(-1/6)\left(\mathrm{e}^{\mathrm{i}\pi/6}\chi_{1}(1/6)\,\chi_{2}(-1/6)+\mathrm{e}^{-\mathrm{i}\pi/6}\chi_{1}(-1/6)\,\chi_{2}(1/6)\right)\\\
=-12\pi\sqrt{3}\,(\mathrm{i}(g_{11}g_{12}-g_{21}g_{22})+g_{12}g_{21})=0;\end{gathered}$
(4.21)

the first equality follows from equation (B.6) and the second equation in
(4.17); the second equality uses the definition (B.7) and the third equation
in (4.18); and the last relation is equivalent to equations (4.2) (with
$s_{0}^{0}=\mathrm{i}$) and (4.5). In an analogous manner, one proves the
following identity:

$\displaystyle\varpi_{1}(1/6)\,\varpi_{1}(-1/6)\,\varpi_{2}(1/6)\,\varpi_{2}(-1/6)$
$\displaystyle=$ (4.22)
$\displaystyle\underbrace{\big{(}\mathbf{p}_{1}(-1/6)\mathbf{p}_{1}(1/6)\big{)}^{2}}_{=(-12\pi)^{2}}\underbrace{\big{(}g_{11}^{2}-g_{21}^{2}-\mathrm{i}g_{11}g_{21}\big{)}}_{=\mathrm{i}s_{0}^{\infty}}\underbrace{\big{(}g_{22}^{2}-g_{12}^{2}+\mathrm{i}g_{12}g_{22}\big{)}}_{=\mathrm{i}s_{1}^{\infty}}$
$\displaystyle=144\,\pi^{2};$

the first equality is derived with the help of the definitions in Theorem B.1;
the underbraced relations follow from the third relation in equation (4.18)
and the equations (4.3) and (4.4) defining the monodromy manifold; and the
last equality is a consequence of equation (4.1) with $s_{0}^{0}=\mathrm{i}$.

Equations (4.22) and (4.14) imply that

$\varpi_{1}(1/6)\,\varpi_{2}(1/6)=\frac{144\,\pi^{2}}{\varpi_{1}(-1/6)\,\varpi_{2}(-1/6)}=\frac{18\pi}{H(0)}.$
(4.23)

Now, with the help of (4.15) and (4.16), equation (4.23) can be presented as
follows:

$\big{(}\mathbf{p}_{1}(1/6)\big{)}^{2}\left(g_{11}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{\mathrm{i}\pi/6}+g_{21}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{-\mathrm{i}\pi/6}\right)\left(g_{12}\mathrm{e}^{\mathrm{i}\pi/4}\mathrm{e}^{\mathrm{i}\pi/6}+g_{22}\mathrm{e}^{-\mathrm{i}\pi/4}\mathrm{e}^{-\mathrm{i}\pi/6}\right)\mathrm{e}^{-\mathrm{i}\pi/6}=\frac{18\pi}{H(0)}.$
(4.24)

Using the formula for $\mathbf{p}(1/6)$ given in (4.18), multiplying out the
expressions in parentheses, and introducing the contraction variables, we
rewrite equation (4.24) as

$\tilde{g}_{1}\mathrm{e}^{\frac{\pi\mathrm{i}}{3}}+2(\tilde{g}_{3}-1)-\tilde{g}_{2}\mathrm{e}^{-\frac{\pi\mathrm{i}}{3}}=\frac{1}{H(0)}-1.$
(4.25)

Equation (4.25) is consistent with equation (4.20), and, together with the
equations defining the contraction manifold, are equivalent to equations
(4.12) stated in Lemma 4.1.

Now, we show that the asymptotics (4.13) is consistent with the local
expansion of the function $u(\tau)$ that follows from equations (1.5)–(1.8)
and (2.1). For this purpose, substitute into the asymptotic expansion (4.13)
the relations for $\varpi_{k}(\pm 1/6)$, $k=1,2$, given in equations (4.14),
(4.21), and (4.23). The parameter $\delta$ is not yet specified; but, we know
the expansion for $H(r)$ given in Subsection 2.1: the latter implies, in fact,
that $\delta=3/4$. Thus, replacing $1+o\big{(}\tau^{\delta}\big{)}$ by a
corresponding series expansion, one arrives at

$u(\tau)\underset{\tau\to+0}{=}\frac{1}{2}\tau^{1/3}\left(H(0)+\frac{9}{4H(0)}\tau^{4/3}\right)\left(1+\sum_{m=1}^{\infty}\mu_{m}\tau^{4m/3}\right)=\frac{1}{2}\tau^{1/3}H\left(-\left(\frac{3}{2}\right)^{2}\tau^{4/3}\right),$
(4.26)

where $\mu_{m}$ are $\tau$-independent coefficients, and the function $H(r)$
is defined via the series (2.1). Multiplying out the expressions in
parentheses in equation (4.26) one shows that

$\displaystyle
H(0)\mu_{1}+\frac{9}{4H(0)}=-\frac{9}{4}H^{\prime}(0)=-\frac{9}{4}\left((H(0))^{2}-\frac{1}{H(0)}\right)\Rightarrow\mu_{1}=-\frac{9}{4}H(0),$
$\displaystyle
H(0)\mu_{m}+\frac{9}{4H(0)}\mu_{m-1}=(-1)^{m}\left(\frac{3}{2}\right)^{2m}a_{m},\quad
m=2,3,\dotsc,$

where the numbers $a_{m}$ are defined in Section 2. Obviously, the series
$\sum_{m=1}^{\infty}\mu_{m}\tau^{4m/3}$ is uniquely defined and convergent in
some neighbourhood of $\tau=0$. $\blacksquare$

######  Corollary 4.1.

There are three algebraic solutions of equation (1.1) (cf. (1.6)$)$ with $a=0$
that correspond to three constant solutions of equation (1.2). These solutions
and the corresponding monodromy data of equation (1.1) read:

  1. (1)

$u(\tau)=\frac{1}{2}\tau^{1/3},\qquad\qquad y(t)=t^{1/3},\qquad\qquad H(r)=1,$
$\tilde{s}=\tilde{g}_{1}=\tilde{g}_{2}=\tilde{g}_{4}=0,\qquad\tilde{g}_{3}=1,$
$s_{0}^{0}=\mathrm{i},\quad s_{0}^{\infty}=-\mathrm{i}g_{11}^{2},\quad
s_{1}^{\infty}=-\mathrm{i}g_{22}^{2},\quad g_{12}=g_{21}=0,\quad
g_{11}g_{22}=1;$

  2. (2)

$u(\tau)=\frac{1}{2}\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}\tau^{1/3},\qquad\qquad
y(t)=\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}t^{1/3},\qquad\qquad
H(r)=\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}},$
$\tilde{s}=\tilde{g}_{1}=\tilde{g}_{3}=0,\qquad\tilde{g}_{2}=-1,\quad\tilde{g}_{4}=-\mathrm{i},$
$s_{0}^{0}=\mathrm{i},\quad s_{0}^{\infty}=\mathrm{i}g_{21}^{2},\quad
s_{1}^{\infty}=\mathrm{i}g_{12}^{2},\quad g_{11}=0,\quad g_{12}g_{21}=-1,\quad
g_{21}g_{22}=\mathrm{i};$

  3. (3)

$u(\tau)=\frac{1}{2}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}\tau^{1/3},\qquad\qquad
y(t)=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}t^{1/3},\qquad\qquad
H(r)=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}},$
$\tilde{s}=\tilde{g}_{2}=\tilde{g}_{3}=0,\qquad\tilde{g}_{1}=1,\quad\tilde{g}_{4}=-\mathrm{i},$
$s_{0}^{0}=\mathrm{i},\quad s_{0}^{\infty}=\mathrm{i}g_{21}^{2},\quad
s_{1}^{\infty}=\mathrm{i}g_{12}^{2},\quad g_{22}=0,\quad g_{12}g_{21}=-1,\quad
g_{11}g_{12}=-\mathrm{i}.$

######  Remark 4.3.

The branch of $\tau^{1/3}$ in Corollary 4.1 is fixed such that it is positive
for $\tau>0$. The function $y(t)$ is calculated with the help of relation
(1.8). These three different solutions of equation (1.1) (cf. (1.6)) coincide,
of course, with the pullback of the three branches of the algebraic function
$\tfrac{1}{2}\tau^{1/3}$ (resp., $t^{1/3}$); however, from the point of view
of solutions to the ODE, they represent three different solutions, since, for
the same value of $\tau$ (resp., $t$), they have different initial values.
$\blacksquare$

######  Remark 4.4.

According to [17], the solutions enumerated in Corollary 4.1 are the only
algebraic solutions of equation (1.1) with $a=0$ (resp., equation (1.6)).
Here, we show how this fact can be deduced from our asymptotic results. The
essential singular point at infinity imposes severe restrictions on the
algebraic behaviour of solutions at this point. Our main results concerning
the asymptotic behaviour of solutions at the point at infinity are presented
in Appendix C. These asymptotic results imply, in particular, that in case of
algebraic behaviour of solutions at the point at infinity, the corresponding
monodromy data necessarily satisfy the condition $g_{12}g_{21}g_{11}g_{22}=0$
(cf. equation (C.2) with (C.3), and equation (C.24) with (C.26)). There is,
seemingly, another possibility, namely, $g_{11}g_{22}=1$ (cf. equation (C.1)
with (C.3)). The latter case, in conjunction with equation (4.5), implies that
$g_{12}g_{21}=0$, so that the condition $g_{12}g_{21}g_{11}g_{22}=0$ holds.
This last condition, with the help of the equations defining the monodromy
manifold (cf. equations (4.2)–(4.5)), can be subdivided into three sub-cases:
(1) $g_{12}g_{21}=0$; (2) $g_{11}=0$; and (3) $g_{22}=0$.

Consider sub-case (1) for which the large-$\tau$ asymptotics is stated in
Theorem 3.2 of [26].666In Theorem 3.2 of [26], set $a=0$, $\varepsilon b=+1$,
$(\varepsilon_{1},\varepsilon_{2})=(0,0)$, $s_{0}^{0}(0,0):=s_{0}^{0}$,
$s_{0}^{\infty}(0,0):=s_{0}^{\infty}$, $s_{1}^{\infty}(0,0):=s_{1}^{\infty}$,
and $g_{ij}(0,0):=g_{ij}$, $i,j\in\\{1,2\\}$. This theorem implies that the
algebraic behaviour is possible only if $s_{0}^{0}=\mathrm{i}$. Now, equations
(4.2) and (4.5) imply that $g_{12}=g_{21}=0$. This supplies the necessary
conditions for the existence of algebraic solutions in case the monodromy data
satisfy $g_{11}g_{22}=1$. The first case in Corollary 4.1 supplies the
sufficiency conditions.

With the help of equations (4.2)–(4.5), sub-case (2) gives rise to the
following values for the monodromy data: $g_{11}=0$, $g_{12}g_{21}=-1$,
$g_{21}g_{22}=\mathrm{i}$, and $s_{0}^{\infty}=\mathrm{i}g_{21}^{2}$. Even
though the values of $s_{0}^{0}$ and $s_{1}^{\infty}$ cannot be determined
directly, there is, however, a simple ruse related to symmetries. Clearly, if
$u(\tau)$ is an algebraic solution, then
$\hat{u}(\tau)=u\big{(}\tau\mathrm{e}^{2\pi\mathrm{i}}\big{)}$ is also an
algebraic solution; consequently, one can deduce the action of the
transformation $\tau\to\tau\mathrm{e}^{2\pi\mathrm{i}}$ on the monodromy
manifold. In our case, that is, $a=0$, this action, in terms of the Stokes
matrices, $S_{k}$, and the connection matrix, $G$, defined in [26], reads:
$\hat{S}_{k}^{\infty}=S_{k}^{\infty}$ and
$-\mathrm{i}S_{0}^{0}\sigma_{1}\hat{G}=G$, where
$\sigma_{1}=\left(\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right)$, and
where the ‘hat’ denotes the monodromy matrices corresponding to the solution
$\hat{u}(\tau)$, whereas the monodromy matrices without the ‘hat’ correspond
to the solution $u(\tau)$. Taking into account the definition of these
matrices [26] and assuming that the hat variables correspond to sub-case (2),
the matrix relations can be rewritten in terms of the corresponding scalar
variables as follows: $\hat{s}_{k}^{\infty}=s_{k}^{\infty}$, $k=0,1$,
$\hat{s}_{0}^{0}=s_{0}^{0}$, $g_{12}=g_{21}=0$,
$g_{11}=-\mathrm{i}\hat{g}_{21}$, and $g_{22}=-\mathrm{i}\hat{g}_{12}$. Thus,
we map sub-case (2) to sub-case (1). For sub-case (1), it is proved that
algebraic solutions exist iff $s_{0}^{0}=\mathrm{i}$, which means that the
same is true for sub-case (2). Clearly, the same matrix action links the
monodromy data corresponding to sub-cases (3) and (2), where, now, the hat
variables correspond to sub-case (3). Since sub-case (2) is studied, one can
repeat the aforementioned arguments in order to confirm the uniqueness of the
algebraic solution for sub-case (3). $\blacksquare$

##  5 The Coxeter Group

In this section, we consider some group actions on the monodromy manifold.

######  Proposition 5.1.

The projectivization of the contracted monodromy manifold for equation (1.1)
is a singular cubic surface of type $A_{3}$.

###### Proof.

The monodromy manifold for equation (1.1) in the generic case $a\in\mathbb{C}$
is defined in [26] (see [26], p. 1172, the system (33)). We introduce the same
change of variables (4.6) as for the case $a=0$ and arrive at the following
system for the contracted variables:

$\tilde{g}_{1}-\tilde{g}_{2}+\tilde{g}_{3}(1-\tilde{s})=\mathrm{e}^{-\pi
a},\qquad\tilde{g}_{3}(\tilde{g}_{3}-1)=-\tilde{g}_{1}\tilde{g}_{2}.$ (5.1)

Solving the first equation of the system (5.1) for $\tilde{g}_{2}$ and
substituting the result into the second equation, we obtain the following
cubic equation in $\mathbb{C}^{3}$ with the parameter $\mathrm{e}^{-\pi a}$,

$\tilde{g}_{1}\tilde{g}_{3}(\tilde{s}-1)=(\tilde{g}_{1})^{2}+(\tilde{g}_{3})^{2}-(\tilde{g}_{1}\mathrm{e}^{-\pi
a}+\tilde{g}_{3}).$ (5.2)

Introducing local co-ordinates in $\mathbb{CP}^{3}$,
$\\{x_{0}:x_{1}:x_{2}:x_{3}\\}$, according to the formulae,

$\tilde{g}_{3}=-\frac{x_{0}}{x_{2}},\qquad\tilde{g}_{1}=-\frac{x_{1}}{x_{2}}\mathrm{e}^{-\pi
a},\qquad\tilde{s}-1+2\cosh(\pi a)=\frac{x_{3}}{x_{2}}\,\mathrm{e}^{\pi a},$
(5.3)

one rewrites equation (5.2) as follows:

$x_{0}x_{1}x_{3}=x_{2}(x_{0}+x_{1}+x_{2})(x_{0}-ux_{1}),\quad\mathrm{where}\quad\sqrt{-u}=\mathrm{e}^{-\pi
a}.$ (5.4)

This surface has a singularity of type $A_{3}$ [6, 34]. ∎

Henceforth, till the end of this section, we proceed with the study of the
case $a=0$ $(u=-1)$; in this case, equation (5.4) reads

$x_{0}x_{1}x_{3}=x_{2}(x_{0}+x_{1}+x_{2})(x_{0}+x_{1}).$ (5.5)

Equation (5.5) contains ten $\mathbb{CP}^{3}$-lines [6, 34]. These lines can
be presented as the intersection of two hyperplanes. Three of these lines
belong to the hyperplane $x_{2}=0$, which is located at “infinity”, namely,
they can be presented as the intersection of the plane $x_{2}=0$ with the
planes $x_{0}=0$, $x_{1}=0$, and $x_{3}=0$. The monodromy co-ordinates cannot
take on infinite values; therefore, we cannot give, at least not directly, an
interpretation for these lines in terms of the monodromy data and the
corresponding solutions of equation (1.1). Consequently, for our purposes, we
resort back to $\mathbb{C}^{4}$, and denote the co-ordinates in this space as
$(x,y,z,s)$. We identify these co-ordinates with our monodromy data as
follows:

$x=\tilde{g}_{1},\quad y=-\tilde{g}_{2},\quad z=\tilde{g}_{3},\quad
s=\tilde{s}.$

In these co-ordinates, the system of equations (4.7) defining the contracted
monodromy manifold reads

$x+y+z(1-s)=1,\qquad z(z-1)=xy.$ (5.6)

The remaining seven lines of the surface (5.5) and the corresponding monodromy
co-ordinates are:

  1. 1.

$x_{0}=0$ and $x_{1}=0$, $(0,1,0,s)$;

  2. 2.

$x_{0}=0$ and $x_{1}+x_{2}=0$, $(1,0,0,s)$;

  3. 3.

$x_{1}=0$ and $x_{0}+x_{2}=0$, $(0,s,1,s)$;

  4. 4.

$x_{3}=0$ and $x_{0}+x_{1}=0$, $(x,x+1,-x,-1)$;

  5. 5.

$x_{3}=0$ and $x_{0}+x_{1}+x_{2}=0$, $(x,x-1,1-x,-1)$;

  6. 6.

$x_{1}+x_{2}=0$ and $x_{0}+x_{1}+x_{3}=0$, $(1,s(s-1),s,s)$;

  7. 7.

$x_{0}+x_{2}=0$ and $x_{0}+x_{1}+x_{3}=0$, $(s,0,1,s)$;

where $s,x\in\mathbb{C}$. Note that, in item 6 above, the dependence of $y$
with respect to $s$ is quadratic, and it remains a straight line due to the
fact that the surface (5.5) is written in terms of co-ordinates that do not
depend on $y$. These lines appear again later in this section whilst studying
the action of the Coxeter group on the contracted monodromy manifold.

Letting $z=\kappa x$ in the system (5.6), one finds a rational parametrization
of the contracted monodromy manifold in terms of the parameters
$\kappa,s\in\mathbb{C}$. (The second equation in (5.6) suggests three other
rational parametrizations; however, this fact is not important for our
considerations.) With the help of this rational parametrization, we find the
following transformations of the contracted monodromy manifold:

$\displaystyle r_{1}:\;\;(x,y,z,s)\longrightarrow(y,x,z,s),$ (5.7)
$\displaystyle r_{2}:\;\;(x,y,z,s)\longrightarrow(z,y+(x-z)s,x,s),$ (5.8)
$\displaystyle
r_{3}:\;\;(x,y,z,s)\longrightarrow\left(\frac{(2-x)z+xy}{z+y},-\frac{z-y}{z+y}y,\frac{z-y}{z+y}z,2-s\right).$
(5.9)

One can consider these transformations as acting in $\mathbb{C}^{4}$.
Straightforward calculations show that these transformations are of order $2$:

$r_{1}^{2}=r_{2}^{2}=r_{3}^{2}=1,$ (5.10)

where $1$ in (5.10) above and in (5.11) below denotes the transformation
corresponding to the identity map in $\mathbb{C}^{4}$. While the
transformations $r_{1}$ and $r_{2}$ act in $\mathbb{C}^{4}$, the
transformation $r_{3}$ is not defined on the hyperplane $z+y=0$; moreover, if
one desires to apply it twice in order to prove the last relation in (5.10),
then one has to exclude the hyperplane $z-y=0$. If one wants to consider the
action of the group generated by the three transformations (5.7)–(5.9), then
one has to remove a countable number of surfaces from $\mathbb{C}^{4}$. We
will not discuss this question further because we are primarily interested in
the action of these transformations on the surface (5.6).

Restricted to the surface (5.6), these reflections satisfy the relations

$(r_{3}r_{1})^{4}=(r_{1}r_{3})^{4}=1\qquad\text{and}\qquad(r_{3}r_{2})^{2}=(r_{2}r_{3})^{2}=1.$
(5.11)

Of course, the action of $r_{3}$ is not defined on the entirety of the surface
(5.6), so that the relations (5.11) are proved only for those points of (5.6)
where the corresponding transformations are defined. In fact, as we show at
the end of this section, one can regularize the definition of $r_{3}$ on the
surface (5.6) so that after the excision of a few lines from the surface it is
well defined.

We commence our considerations with the dihedral group $\mathcal{D}_{1,2}$
generated by $\\{r_{1},r_{2}\\}$. Let $\mathcal{N}_{1,2}$ be its normal
subgroup with generator $r_{1}r_{2}$.

######  Proposition 5.2.

There is a one-to-one correspondence between algebroid solutions of equation
(1.1) and the finite orbits of the action of $\mathcal{N}_{1,2}$ on the
monodromy manifold (5.6). The length of the finite orbits coincides with the
order of a generator of the symmetry transformations for the corresponding
algebroid solutions.

###### Proof.

Let $u(\tau,c)$ be a solution corresponding to the branching parameter
$\rho\in\mathbb{C}$ ($u(\tau,c)\underset{\tau\to 0}{\sim}c\,\tau^{1-4\rho}$,
where $c\in\mathbb{C}\setminus\\{0\\}$: see Appendix B); then the
transformation corresponding to the generator $r_{1}r_{2}$ is
$u(\tau,c)\to-\mathrm{i}u(\tau\mathrm{e}^{\pi\mathrm{i}/2},-c)=u(\tau,-c\mathrm{e}^{-2\pi\mathrm{i}\rho})$.
If $\rho\notin\mathbb{Q}$, then any finite number of such transformations give
different solutions. On the other hand, all algebroid solutions after a finite
number of such iterations are mapped to themselves. For the $m$\- and the
$n$-series, these symmetries are defined explicitly in Corollaries 3.2 and
3.3.

The only solutions that remain are those possessing logarithmic behavior as
$\tau\to 0$ [25, 26]: they have infinite orbits that are explicitly presented
in Proposition 5.4 below. ∎

The action of $\mathcal{D}_{1,2}$ on $\mathbb{C}^{4}$ does not change the
fourth coordinate. This fact allows us to treat $s$ as a parameter, and to
consider the action of $\mathcal{D}_{1,2}$ in $\mathbb{C}^{3}$ by regarding it
as the hyperplane $s=s_{0}$ in $\mathbb{C}^{4}$; in this case, we denote this
action by $\mathcal{D}_{1,2}(s_{0})$.

######  Proposition 5.3.

Define $\rho_{1}=1/2-\rho$, where $\rho$ is the branching parameter of the
algebroid solution (see Corollary 3.1 and Remark 3.3$)$. The group
$\mathcal{D}_{1,2}(s_{0})$ is finite iff $s_{0}$ is an algebraic number that
can be written in the form $s_{0}=1+2\cos(2\pi\rho_{1})$, with
$\rho_{1}\in\mathbb{Q}$ such that $0<2\rho_{1}<1$. In this case, the length of
the orbit of the normal subgroup $\mathcal{N}_{1,2}$ coincides the denominator
of $\rho_{1}$ in its representation as an irreducible fraction.

###### Proof.

We proved in Proposition 5.2 that, when acting on the contracted monodromy
manifold, $\mathcal{D}_{1,2}$ has finite orbits for the points corresponding
to the algebroid solutions. Here, we consider the action of
$\mathcal{D}_{1,2}(s_{0})$ in $\mathbb{C}^{3}$. Consider the column-vector
$\mathbf{R}=(x_{0},y_{0},z_{0})^{T}\in\mathbb{C}^{3}$; then, after $n$
iterations via $r_{1}r_{2}$, we arrive at the point
$\widehat{P}_{n}(s_{0})\mathbf{R}$, where $\widehat{P}_{n}(s_{0})$ is a
$3\times 3$ matrix with polynomial entries in $\mathbb{Z}[s_{0}]$. Assume that
$s_{0}$ corresponds to an algebroid solution whose orbit on the contracted
monodromy manifold has length $n$. In this case,
$\widehat{P}_{n}(s_{0})\mathbf{R}=\mathbf{R}$, for $\mathbf{R}$ defining a
point on the monodromy manifold. We choose the following three points of the
manifold, $\mathbf{R}_{1}=(1,0,0)^{T}$, $\mathbf{R}_{2}=(0,1,0)^{T}$, and
$\mathbf{R}_{3}=(s_{0},0,1)^{T}$; then, the matrix
$\widehat{R}:=(\mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3})$ has unit
determinant and satisfies the equation
$(\widehat{P}_{n}(s_{0})-I)\widehat{R}=0$, where $I$ is the $3\times 3$
identity matrix . Thus, for this value of $s_{0}$, $\widehat{P}_{n}(s_{0})=I$.
Since the matrix $\widehat{P}_{n}(s_{0})$ does not depend on the initial point
$\mathbf{R}$, it means that the condition
$\widehat{P}_{n}(s_{0})\mathbf{R}=\mathbf{R}$ is true for any point
$\mathbf{R}\in\mathbb{C}^{3}$.

Reverting to the proof of Proposition 5.2, since we know that the generator of
the transformation $r_{1}r_{2}$ is equivalent to the change of the parameters
defining the solution $(\rho,c)\to(\rho,-c\mathrm{e}^{-2\pi\mathrm{i}\rho})$
because
$-c\mathrm{e}^{-2\pi\mathrm{i}\rho}=c\mathrm{e}^{2\pi\mathrm{i}(1/2-\rho)}=c\mathrm{e}^{2\pi\mathrm{i}\rho_{1}}$,
we see that the finite orbits are possible only for rational $\rho_{1}$, and
the lengths of these orbits coincide with the denominators of the irreducible
representation of the numbers $\rho_{1}$ as ratios of integers.

According to Theorem B.1 (see Appendix B), the contracted Stokes multiplier
$s=1+\mathrm{i}s_{0}^{0}$ (cf. (4.6)) is related to the branching parameter
$\rho$ of the solution $u(\tau)$ as $s=1-2\cos(2\pi\rho)$, with $0<2\rho<1$;
hence, $s=1+2\cos(2\pi\rho_{1})$, where $0<2\rho_{1}<1$. Since, for the
algebroid solutions, $\rho_{1}$ is rational, the corresponding numbers $s$ are
algebraic [31], and the dihedral group $\mathcal{D}_{1,2}(s_{0})$ is finite
for this $s_{0}=s$. ∎

######  Remark 5.1.

The length of the orbit of the normal subgroup $\mathcal{N}_{1,2}$
corresponding to solution $u(\tau)$ defined via $H(r)$ (cf. Section 3, Remark
3.3 and Section 4, Lemma 4.1) equals $3$ because $s=0$, $\rho=1/6$, and
$\rho_{1}=1/3$. The length of the orbit corresponding to the solution
$u(\tau)$ holomorphic at $\tau=0$ (cf. Section 3, Remark 3.3) equals $4$
because $s=1$ and $\rho=\rho_{1}=1/4$.

Below, we define the set of minimal polynomials $q_{k}(s)$, $k\in\mathbb{N}$;
for $k=3,4,\dotsc$, these polynomials define the algebraic numbers
$s=1+2\cos(2\pi\rho_{1})$, $0<2\rho_{1}<1$, that coincide with the contracted
Stokes multipliers corresponding to the algebroid solutions. This set is
defined so that the subscript $k$ of the polynomial $q_{k}(s)$ coincides with
the denominator of $\rho_{1}$ in its representation as an irreducible
fraction, and thus with the length of the corresponding orbit of
$\mathcal{N}_{1,2}$. $\blacksquare$

Since the minimal polynomials $q_{k}(s)$ defining the algebraic numbers
$1+2\cos(2\pi\rho_{1})$ for $\rho_{1}\in\mathbb{Q}$ and $0<2\rho_{1}<1$ play
an important role in the description of the algebroid solutions, we briefly
recall the corresponding construction in the notation adopted in this paper.
The subject is well known [31], so that some details of the proofs are
omitted.

Consider the cyclotomic equation $\mathrm{e}^{2\pi\mathrm{i}\rho_{1}n}=1$. Use
the Euler formula
$\mathrm{e}^{2\pi\mathrm{i}\rho_{1}}=\cos(2\pi\rho_{1})+\mathrm{i}\sin(2\pi\rho_{1})$
to find that the cyclotomic equation is equivalent to
$T_{n}(\cos(2\pi\rho_{1}))=\cos(2\pi\rho_{1}n)=1$, where $T_{n}(x)$ is the
$n$th Chebyshev polynomial of the first kind; the explicit formulae for it can
be found in [14]:

$T_{n}(x)=\frac{n}{2}\sum_{m=0}^{\lfloor\frac{n}{2}\rfloor}(-1)^{m}\frac{(n-m-1)!}{m!(n-2m)!}(2x)^{n-2m}.$

With the aid of the Euler formula, it is easy to establish that the polynomial
$T_{n}(x)-1$ for $n\geqslant 2$ is always reducible, and, moreover, the roots
of the polynomials on the right-hand sides of the following identities are of
order two,

$\displaystyle
2q_{1}(s)\left(T_{2n+1}\left(\frac{s-1}{2}\right)-1\right)=\left(\prod_{d\smallsetminus(2n+1)}q_{d}(s)\right)^{2},$
(5.12) $\displaystyle
2q_{1}(s)q_{2}(s)\left(T_{2n+2}\left(\frac{s-1}{2}\right)-1\right)=\left(\prod_{d\smallsetminus(2n+2)}q_{d}(s)\right)^{2},\quad
n\in\mathbb{N},$ (5.13)

where $q_{1}(s)=2(\cos(2\pi\rho_{1})-1)=2(s-1)/2-2=s-3$, and
$q_{1}(s)q_{2}(s)=2(\cos(4\pi\rho_{1})-1)=4\cos^{2}(2\pi\rho_{1})-4=(s-3)(s+1)$,
so that $q_{2}(s)=s+1$. The polynomials $q_{d}(s)$ are assumed to be
irreducible over $\mathbb{Z}$. Equations (5.12) and (5.13) allow one to
recursively derive the polynomials $q_{k}(s)\in\mathbb{Z}[s]$ for all
$k\in\mathbb{N}$. If we assume that the polynomials are monic, then the
sequence $q_{k}(s)$ satisfying the system (5.12) and (5.13) is unique.

It follows (by mathematical induction) from the Gauß identity for the Euler
totient function, $\varphi(n)$,

$\sum_{d\smallsetminus n}\varphi(d)=n,$

that $\deg q_{k}(s)=\varphi(k)/2$ for $k>2$. The set of roots of the
polynomials $q_{k}(s)$, $k\in\mathbb{N}$, are, by construction, real algebraic
numbers that are dense on the segment $[-1,3]$. The Galois group of the
polynomials $q_{k}(s)$ is solvable, so that all its roots can be presented in
terms of radicals. Thus, the Stokes multipliers corresponding to the algebroid
solutions can be expressed in terms of radicals.

In [43], the authors, using identities for the Chebyshev polynomials, derive,
from the system (5.12) and (5.13), a more convenient system that allows one to
recursively obtain the polynomials $q_{k}(s)$:

$\displaystyle 2(T_{n+1}((s-1)/2)-T_{n}((s-1)/2))$
$\displaystyle=\prod_{d\smallsetminus(2n+1)}q_{d}(s),$ (5.14) $\displaystyle
2(T_{n+1}((s-1)/2)-T_{n-1}((s-1)/2))$
$\displaystyle=\prod_{d\smallsetminus(2n+2)}q_{d}(s),\qquad n\in\mathbb{N}.$
(5.15)

We list below the first $18$ polynomials $q_{k}(s):=q_{k}$ derived with the
help of equations (5.14) and (5.15):

$\displaystyle q_{1}$
$\displaystyle=s-3,\;q_{2}=s+1,\;q_{3}=s,\;q_{4}=s-1,\;q_{5}=s^{2}-s-1,\;q_{6}=s-2,\;q_{7}=s^{3}-2s^{2}-s+1\;$
$\displaystyle q_{8}$
$\displaystyle=s^{2}-2s-1,\;q_{9}=s^{3}-3s^{2}+3,\;q_{10}=s^{2}-3s+1,\;q_{11}=s^{5}-4s^{4}+2s^{3}+5s^{2}-2s-1,$
$\displaystyle q_{12}$
$\displaystyle=s^{2}-2s-2,\;q_{13}=s^{6}-5s^{5}+5s^{4}+6s^{3}-7s^{2}-2s+1,\;q_{14}=s^{3}-4s^{2}+3s+1,$
$\displaystyle q_{15}$
$\displaystyle=s^{4}-5s^{3}+5s^{2}+5s-5,\;q_{16}=s^{4}-4s^{3}+2s^{2}+4s-1,$
$\displaystyle q_{17}$
$\displaystyle=s^{8}-7s^{7}+14s^{6}+s^{5}-25s^{4}+9s^{3}+12s^{2}-3s-1,\;q_{18}=s^{3}-3s^{2}+1.$

As follows from Corollary 3.1, the boundary values of $s$, i.e., $s=-1$
($2\rho=1$) and $s=3$ ($2\rho=0$), are the roots of the polynomials $q_{1}(s)$
and $q_{2}(s)$. In fact, we know that for these values of $s$ there correspond
solutions of equation (1.1) for $a=0$ that have logarithmic behaviour as
$\tau\to 0$ [25, 26]. Thus, according to Proposition 5.3, the corresponding
dihedral group $\mathcal{D}_{1,2}(s_{0})$, $s_{0}=-1,3$, is infinite.

######  Proposition 5.4.

Let $s=-1$ or $s=3$ and the point $(x_{0},y_{0},z_{0})$ belong to
$\mathbb{C}^{3}$ or to the manifold of the monodromy data (5.6); then, the
orbits of the normal subgroup $\mathcal{N}_{1,2}(s)$ in $\mathbb{C}^{3}$ or on
the manifold of the monodromy data are infinite. The points, after
$n\in\mathbb{Z}_{\geqslant 0}$ iterations, have the following co-ordinates:

$\displaystyle s=-1;$ $\displaystyle x_{n}$
$\displaystyle=(-1)^{n}\left\lfloor 1+\frac{n}{2}\right\rfloor
x_{0}+(-1)^{n+1}\left\lfloor\frac{n+1}{2}\right\rfloor
y_{0}+n(\mathrm{mod}(2))\;z_{0},$ $\displaystyle y_{n}$
$\displaystyle=(-1)^{n}\left\lfloor\frac{n}{2}\right\rfloor
x_{0}+(-1)^{n-1}\left\lfloor\frac{n-1}{2}\right\rfloor
y_{0}+n(\mathrm{mod}(2))\;z_{0},$ $\displaystyle z_{n}$
$\displaystyle=(-1)^{n+1}\left\lfloor\frac{n+1}{2}\right\rfloor
x_{0}+(-1)^{n}\left\lfloor\frac{n}{2}\right\rfloor
y_{0}+(n+1)(\mathrm{mod}(2))\;z_{0},$ $\displaystyle s=3;$ $\displaystyle
x_{n}$
$\displaystyle=\frac{(n+1)(n+2)}{2}\,x_{0}+\frac{n(n+1)}{2}\,y_{0}+\big{(}1-(n+1)^{2}\big{)}z_{0},$
$\displaystyle y_{n}$
$\displaystyle=\frac{n(n-1)}{2}\,x_{0}+\frac{(n-2)(n-1)}{2}\,y_{0}+\big{(}1-(n+1)^{2}\big{)}z_{0},$
$\displaystyle z_{n}$
$\displaystyle=\frac{n(n+1)}{2}\,x_{0}+\frac{(n-1)n}{2}\,y_{0}+(1-n^{2})z_{0}.$

###### Proof.

By mathematical induction: the base of the induction, $n=0$, can be verified
immediately, and the inductive step is straightforward to make with the help
of the explicit formula for the transformation $r_{1}r_{2}$ (cf. (5.7) and
(5.8)). ∎

Now, consider the transformation $r_{3}$ (cf. (5.9)). This transformation is
interesting for us provided that it acts on the monodromy manifold (5.6), and
thus its action can be extended to the space of solutions; therefore, we
consider its action on the monodromy manifold rather than on $\mathbb{C}^{4}$.

If we want to apply this transformation once to a point $\mathcal{P}$ of the
monodromy manifold (5.6), then the co-ordinates of $\mathcal{P}$ should
satisfy the condition $\mathcal{P}\neq(1,0,0,s)$ or
$\mathcal{P}\neq(x,x-1,1-x,-1)$, $x,s\in\mathbb{C}$.

The first condition can, however, be regularized; in this case, both the
numerators and denominators of the proposed image of $r_{3}$ (cf. (5.9)) are
zeros. Using this fact, one can set $z+y=\varepsilon$,777Not be confused with
$\varepsilon$ in the Introduction. and rewrite the equations defining the
monodromy manifold (5.6) in the following manner:

$z=\frac{\varepsilon}{2}+\sqrt{\frac{\varepsilon}{s+1}+\frac{\varepsilon^{2}(s-3)}{4(s+1)}},\qquad
y=\frac{\varepsilon}{2}-\sqrt{\frac{\varepsilon}{s+1}+\frac{\varepsilon^{2}(s-3)}{4(s+1)}},\qquad
x=zs+1-\varepsilon.$ (5.16)

Substituting these equations into (5.9) and considering the limit
$\varepsilon\to 0$, one finds that

$r_{3}(1,0,0,s)=\left(\frac{1-s}{1+s},\frac{2}{1+s},\frac{2}{1+s},2-s\right),\quad
s\in\mathbb{C}\setminus\\{-1\\}.$

One can readily verify that this definition implies
$r_{3}^{2}(1,0,0,s)=(1,0,0,s)$, so that the third relation in (5.10) holds. We
see that the only problem occurs when $s=-1$. For $s=-1$, the monodromy
manifold consists of two lines, $(x,x-1,1-x,-1)$ and $(x,x+1,-x,-1)$, where
$x\in\mathbb{C}$. They are two different lines which are related by the
symmetry $r_{1}(x,x-1,1-x,-1)=(x_{1},x_{1}+1,-x_{1},-1)$, where $x_{1}=x-1$.
On the other hand, $r_{3}(x,x+1,-x,-1)=((2x-1)x,(2x+1)(x+1),(2x+1)x,3)$: the
last quadric curve provides a rational parametrization for the monodromy
manifold when $s=3$. Therefore, we can correctly define the action of $r_{3}$
on the first curve $(x,x-1,1-x,-1)$ only in the sense of projective geometry;
however, having in mind an application to the theory of the degenerate third
Painlevé equation, we do not consider this option. Note that, for all other
points of the monodromy manifold, any transformation $r_{3}w(r_{1},r_{2})$,
where $w$ is any word consisting of two letters $r_{1}$ and $r_{2}$, can be
regularized in a natural way; e.g., $r_{3}r_{1}(0,1,0,s)=r_{3}(1,0,0,s)$ and
$r_{3}r_{2}(0,s,1,s)=r_{3}(1,0,0,s)$, or the more complicated examples,
$r_{3}(r_{1}r_{2})^{3}(1,s(s-1),s,s)=r_{3}(1,0,0,s)$ and
$r_{3}(r_{1}r_{2})^{4}=(s,(s-1)(s^{2}-s-1),s(s-1))=r_{3}(1,0,0,s)$. In the
last two examples, one can, upon using equations (5.10), certainly find a
general formula for the transformations $r_{3}(r_{1}r_{2})^{3}$ and
$r_{3}(r_{1}r_{2})^{4}$, and then apply the limiting procedure of the type
delineated above. The latter limiting procedure, however, is significantly
more elaborate than that described by equations (5.16).

Thus, in case one would like to consider the action of the complete group
$\mathcal{G}$ (with generators $r_{1}$, $r_{2}$, and $r_{3}$) on the monodromy
manifold, one has to remove from it those points with the fourth co-ordinate
$s=-1$ and $s=3$. The previous considerations suggest the following
construction: for any $s\in\mathbb{C}\setminus\\{-1,3\\}$ and $s\neq 1$,
consider in $\mathbb{C}^{3}$, with co-ordinates $x$, $y$, and $z$, the
following two planes:

$\displaystyle\mathcal{H}_{s}:\qquad x+y+z(1-s)=1,$ (5.17)
$\displaystyle\mathcal{H}_{2-s}:\qquad x+y-z(1-s)=1.$ (5.18)

The intersection of these planes is the line $x+y=1$, which lies on the plane
$z=0$. Consider the quadric $z(z-1)=xy$: each plane $\mathcal{H}_{k}$,
$k=s,2-s$, intersects it by a conic $\mathcal{C}_{k}$. These two conics have
two common points, $(1,0,0)$ and $(0,1,0)$, in $\mathbb{C}^{3}$; however, in
$\mathbb{C}^{4}$, instead of these points, we have two pairs of points:
$(1,0,0,s)$ and $(1,0,0,2-s)$, and $(0,1,0,s)$ and $(0,1,0,2-s)$. Therefore,
the correct geometric object for the action of the group $\mathcal{G}$ in
$\mathbb{C}^{3}$ is its restriction on the disjoint sum of two conics
$\mathcal{C}_{s}\sqcup\mathcal{C}_{2-s}$. Denote this restriction as
$\mathcal{G}(s)$: this notation assumes that
$\mathcal{G}(s)\equiv\mathcal{G}(2-s)$. In the case $s=1$, the planes
$\mathcal{H}_{s}$ and $\mathcal{H}_{2-s}$ coincide and, instead of the
disjoint sum of conics, we have one conic $\mathcal{C}_{1}$.

The dihedral group $\mathcal{D}_{1,2}(s_{0})$ for $s_{0}=s$ and $s_{0}=2-s$
acts on the conics $\mathcal{C}_{s}$ and $\mathcal{C}_{2-s}$, respectively,
whilst the transformation $r_{3}$ maps the points of one conic to another,
e.g., $r_{3}(0,1,0,s)=(0,1,0,2-s)$.

######  Proposition 5.5.

The Coxeter group $\mathcal{G}(s)$ is finite iff $s$ is the Stokes multiplier
corresponding to algebroid solutions of equation (1.1) for $a=0$, or, in other
words, it is a root of some polynomial $q_{m}(s)$. In this case, $2-s$ is also
a root of some polynomial $q_{n}(s)$ and
$\text{ord}\,\mathcal{G}(s)=4\max\\{m,n\\}$.

##  6 Large-$r$ Asymptotics of $H(r)$ and Numerical Aspects

In Section 4, the $\tau\to+0$ asymptotics of the general solution $u(\tau)$ of
equation (1.1) is used in order to determine the monodromy data corresponding
to the function $H(r)$. This data constitutes the set of parameters that
enables one to determine, with the help of the results derived in [26, 27],
the asymptotics as $r\to-\infty$ of the function $H(r)$ and the corresponding
integral $I(r):=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$. For
the convenience of the reader, all the necessary asymptotic results from [26,
27] are collected in Appendices B and C below. In this section, we present and
compare the asymptotic and numerical results for several solutions
corresponding to different choices of the initial value $H(0)$.

Before we present the corresponding asymptotic formulae, let us comment on the
numerical calculations. The function $H(r)$ and the corresponding monodromy
data are defined via the initial value $H(0)$; on the other hand, equation
(1.2) defining $H(r)$ is singular at $r=0$. Strictly speaking, one has to take
a step from $r=0$ to $r=r_{1}$: this step should be smaller than the radius of
convergence of the series (cf. (2.15) and (2.16)) representing $H(r)$, and
then calculate, with the help of this series, the initial data for $H(r)$ at
$r=r_{1}$. Theoretically, this calculation can be executed with arbitrary
precision. Our calculations are performed via Maple 16 and 17. We found that,
in case we want to calculate only the function $H(r)$ for initial data at
$r=0$ given by Gaussian rationals, then, in many (but not all!) cases, the
standard Maple procedure for the numerical solution of ODEs was able to
correctly calculate the corresponding solution; at least visually the plots
obtained by the ‘simplified’ procedure and the ‘correct’ method coincide. This
might be occurring because (cf. equation (1.4))
$H^{\prime}(0)=H(0)^{2}-1/H(0)$ appears to be a much more complicated Gaussian
rational than $H(0)$, thus Maple treats it in floating-point arithmetic, hence
making this relation only approximately valid, and applies general numerical
procedures. Clearly, for generic Cauchy data specified at singular points of
an ODE, the regular solution does not exist.

Theorem C.1 (see Appendix C) implies the following asymptotics for $H(r)$:

$H(r)\underset{r\to-\infty}{=}1-\sqrt{6\nu_{1}}\cdot\frac{\cos\big{(}\psi(r)+o\big{(}r^{-\delta}\big{)}\big{)}}{\sqrt[4]{-3r}},\qquad\delta>0,$
(6.1)

where

$\displaystyle\psi(r)=2\sqrt{-3r}+\frac{\nu_{1}}{2}\ln{(-3r)}+\nu_{1}\ln(24)+\frac{3\pi}{4}-\frac{3\pi\mathrm{i}}{2}\nu_{1}-\frac{\mathrm{i}}{2}\ln(2\pi)+\mathrm{i}\ln\left(\tilde{g}_{1}\sqrt{\nu_{1}}\,\Gamma(\mathrm{i}\nu_{1})\right),$
(6.2)
$\displaystyle\nu_{1}:=\frac{\ln{\tilde{g}_{3}}}{2\pi},\qquad\left|\text{Im}\,\nu_{1}\right|<\frac{1}{6}=0.1666\ldots.$
(6.3)

Here and below, the following natural conventions for the branches of multi-
valued functions are assumed: (i) the branches of all roots of positive
numbers are positive; (ii) the branch of a root of a parameter, which may take
positive values, is fixed to be positive, and then further defined via
analytic continuation; and (iii) the branches of logarithms of positive
numbers are real.

The parameter $\nu_{1}$ is uniquely defined via equation (6.3). It is related
to the parameter $\tilde{\nu}$ in Appendix C via the relation
$\mathrm{i}\nu_{1}=\tilde{\nu}+1$. The branch of $\sqrt{\nu_{1}}$ can be
chosen as per the above conventions; however, the particular choice for the
branch of $\sqrt{\nu_{1}}$ is not important, provided that the following
natural branch matching is assumed:
$\sqrt{6\nu_{1}}=\sqrt{6}\cdot\sqrt{\phantom{!}\\!\nu_{1}}$ (cf. (6.1) and
(6.2)).

The asymptotics (6.1) is not valid for $H(0)=1$,
$\mathrm{e}^{\pm\frac{2\pi\mathrm{i}}{3}}$ because the monodromy parameters
$\tilde{g}_{1}$ and $\tilde{g}_{3}$ are not defined for these values of
$H(0)$. We can, however, formally consider that, for $H(0)=1$, the asymptotics
remains valid, since it is known (cf. Corollary 4.1, item (1)) that
$\tilde{g}_{3}=1$, which implies that $\nu_{1}=0$ if we assume that the cosine
function remains finite, in which case $H(r)\to 1$ as $r\to-\infty$, which is
consistent with the fact that $H(r)=1$ for all $r$.

An additional restriction on the initial value $H(0)$ is provided by equation
(6.3): this condition not only fixes the branch of the logarithm, but also
imposes a condition on $H(0)$.

For the same conditions as for asymptotics (6.1), the following asymptotic
formula is valid:

$\int_{r}^{0}\frac{\mathrm{d}r}{\sqrt{-r}H(r)}\underset{r\to-\infty}{=}2\sqrt{-r}+2\nu_{1}\ln(2+\sqrt{3})+\mathrm{i}\ln\left(\frac{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}H(0)-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}-H(0)\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}\right)+\mathrm{E}(r),$
(6.4)

where

$\mathrm{E}(r)=\frac{\sqrt{6\nu_{1}}}{2}\cdot\frac{\sin\big{(}\psi(r)+o\big{(}r^{-\delta}\big{)}\big{)}}{\sqrt[4]{-3r}}.$
(6.5)

While the branch of the right-most logarithmic term in equation (6.2) is not
important, the branch of the right-most logarithmic term in equation (6.4) is
fixed by the condition $|\text{Im}\,\ln(\cdot)|<\pi$; the last condition does
not impose any additional restrictions on the initial value $H(0)$ since
$|\text{Im}\,\ln(\cdot)|=\pi$ only for $H(0)=\infty$. In terms of the initial
value $H(0)$, the important condition (6.3) for the validity of the asymptotic
formulae (6.1) and (6.4) reads:

$\left|\text{arg}\left(H(0)+\frac{1}{H(0)}+1\right)\right|<\frac{\pi}{3}.$

In the following, we present two examples of the application of the asymptotic
formulae (6.1) and (6.4), and compare them with the corresponding numerical
plots for $H(r)$ and $I(r)$. These functions are finite at $r=0$, whilst the
denominators of their asymptotics contain the factor $\sqrt[4]{-3r}$;
therefore, the plots of the functions $H(r)$ and $I(r)$ and their asymptotics
are compared outside of some small neighbourhoods of the origin, which are
specified in the figure captions.

###  6.1 Example 1: $H(0)=-\frac{1}{30}-\mathrm{i}$

For this value of $H(0)$, $\nu_{1}=-0.185823\ldots-\mathrm{i}0.0001892\ldots$,
so that the condition (6.3) is satisfied. For the Maple calculations we choose
the parameter Digits=50, and the procedure that provides maximum precision for
this value of Digits. Each plot of the numeric solutions and the corresponding
integrals presented below is based on the calculation of $500$ points. We also
choose the initial point $r_{1}=-10^{-6}$. In principle, for the calculation
of $H(r)$, one can choose a smaller value for the number of digits and larger
values for $r_{1}$, Digits=20 and $r_{1}=-0.001$, say; however, these
parameters do not produce ‘correct’ numerical precision for the integral
$\smallint_{r}^{0}\tfrac{1}{\sqrt{-r}H(r)}\mathrm{d}r$. Our choice for these
parameters is close to the optimal values, i.e., an increase in the accuracy
of calculations is not noticeable on the plots of the corresponding functions;
for example, using, say, the parameter values Digits=100 and $r_{1}=-10^{-9}$,
the Maple calculations produce plots that are visually indistinguishable from
those presented in Figs. 3–7.888 Maple $16$, on a notebook with 4Gb RAM and
processor Intel(R) Core(TM) i7-3517U (3rd Generation), makes $50$-digits
calculation in $84\mathrm{s}$, whilst $100$-digits calculation takes
$103\mathrm{s}$. In preparing this work, we redid the calculations with the
help Maple $2017$ on a more modern notebook with 16Gb RAM and processor
Intel(R) Core(TM) i7-12700H (12th Generation): these calculations were
executed in $60\mathrm{s}$ and $65\mathrm{s}$, respectively.

Figure 3: The red and black plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic values of the function $H(r)$ for
$r\leqslant-0.6$ corresponding to the initial value $H(0)=-1/30-\mathrm{i}$.
Figure 4: The red and black plots are, respectively, the imaginary parts of
the numeric and large-$r$ asymptotic values of the function $H(r)$ for
$r\leqslant-0.5$ corresponding to the initial value $H(0)=-1/30-\mathrm{i}$.
Figure 5: The red and black plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic values of
$I=\smallint_{r}^{0}\tfrac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-7}$ corresponding to the initial value
$H(0)=-1/30-\mathrm{i}$. On this scale, both plots almost coincide. In Figure
6, the reader will see a more detailed comparison of these plots. Figure 6:
The red and black plots are, respectively, the real parts of the numeric and
large-$r$ asymptotic values of
$\smallint_{r}^{0}\tfrac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r-2\sqrt{-r}$ for
$r\leqslant-10^{-7}$ corresponding to the initial value
$H(0)=-1/30-\mathrm{i}$. The blue line is the real part of the asymptotics
(6.4) modulo the parabola $2\sqrt{-r}$ and $\mathrm{E}(r)$. Figure 7: The red
and black plots are, respectively, the imaginary parts of the numeric and
large-$r$ asymptotic values of
$\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for $r\leqslant-0.1$
corresponding to the initial value $H(0)=-1/30-\mathrm{i}$. The blue line is
the imaginary part of the asymptotics (6.4) modulo $\mathrm{E}(r)$.

###  6.2 Example 2: $H(0)=60-\mathrm{i}100$

For this value of $H(0)$, $\nu_{1}=0.583250\ldots-\mathrm{i}0.162814\ldots$,
so that the condition (6.3) is satisfied. Here, however, we are closer to the
boundary of the interval of validity (6.3) for the asymptotics (6.1). For the
Maple calculations, we choose the same settings as in Example 1. The results
of these calculations are presented in Figs. 8–12: these $50$-digits
calculations take approximately $133\text{s}$ on an older notebook (see
footnote 8).

Figure 8: The red and black plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic values of the function $H(r)$ for
$r\leqslant-0.6$ corresponding to the initial value $H(0)=60-\mathrm{i}100$
Figure 9: The red and black plots are, respectively, the imaginary parts of
the numeric and large-$r$ asymptotic values of the function $H(r)$ for
$r\leqslant-0.1$ corresponding to the initial value $H(0)=60-\mathrm{i}100$.
Figure 10: The red and black plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic values of
$\smallint_{r}^{0}\tfrac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-6}$ corresponding to the initial value
$H(0)=60-\mathrm{i}100$. Both plots virtually coalesce into one curve;
however, one notices that this curve has some thick segments coloured in red
from one side and in black from the other. A more detailed comparison of these
plots is presented in Figure 11. Figure 11: The red and black plots are,
respectively, the real parts of the numeric and large-$r$ asymptotic values of
$\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r-2\sqrt{-r}$ for
$r\leqslant-10^{-2}$ corresponding to the initial value
$H(0)=60-\mathrm{i}100$. The blue line is the real part of the asymptotics
(6.4) modulo the parabola $2\sqrt{-r}$ and $\mathrm{E}(r)$. Figure 12: The red
and black plots are, respectively, the imaginary parts of the numeric and
large-$r$ asymptotic values of
$\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for $r\leqslant-0.1$
corresponding to the initial value $H(0)=60-\mathrm{i}100$. The blue line is
the imaginary part of the asymptotics (6.4) modulo $\mathrm{E}(r)$.

######  Remark 6.1.

In principle, the asymptotics (6.1) and (6.4) are valid even beyond the
condition (6.3), that is, $|\text{Im}\,\nu_{1}|<1/2$; however, going farther
and farther beyond the condition (6.3), the visual correspondence between the
solution and its asymptotics is attained at larger and larger values of $|r|$.
The correction terms to the asymptotics (6.1) and (6.4) (see Appendix C)
improve the correspondence, but do not alter the general tendency: the
correspondence between the numeric and asymptotic values for $H(r)$ and $I(r)$
deteriorates for finite, though quite large, values of $r$. As one approaches
$|\text{Im}\,\nu_{1}|\approx 2/5$, the values of $r$ for which the visual
correspondence is expected appear to be so large that it becomes unfeasible to
calculate the solution for such values of $r$. This tendency is partially
illustrated upon comparing Figs. 3 and 4 with Figs. 8 and 9, respectively, and
the corresponding figures with the plots for the integrals: the visual
correspondence worsens when the boundary value of the condition (6.3) is
approached. Example 6 below illustrates at which values of $r$ one may expect
to observe that the asymptotics (6.1) and (6.4) faithfully describe the
large-$r$ behaviour of $H(r)$ and $I(r)$.

In order to cope with the problem indicated above, we derive in [27] yet
another asymptotic formula which will be discussed below. $\blacksquare$

We now turn our attention to the numerical illustration of the asymptotic
results for the function $H(r)$ that follow from Theorem C.2 (see Appendix C):

$H(r)\underset{r\to-\infty}{=}1-\frac{3}{2\sin^{2}\left(\frac{1}{2}\hat{\psi}(r)+o\big{(}r^{-\delta}\big{)}\right)},\qquad\delta>0,$
(6.6)

where

$\displaystyle\hat{\psi}(r)=2\sqrt{-3r}+\left(\nu_{1}+\frac{\mathrm{i}}{2}\right)\ln\big{(}24\sqrt{-3r}\big{)}-\frac{\pi}{4}-\frac{3\pi\mathrm{i}}{2}\nu_{1}-\frac{\mathrm{i}}{2}\ln(2\pi)+\mathrm{i}\ln\big{(}\tilde{g}_{1}\Gamma(\mathrm{i}\nu_{1})\big{)},$
(6.7)
$\displaystyle\nu_{1}:=\frac{\ln{\tilde{g}_{3}}}{2\pi},\qquad\text{Im}\,\nu_{1}\in(-1,0)\setminus\\{-1/2\\}.$
(6.8)

We draw the attention of the reader to the fact that, even though the
parameter $\nu_{1}$ in (6.8) is defined by the same formula as $\nu_{1}$ in
(6.3), its imaginary part is fixed in another interval. Clearly, both
asymptotics (6.1) and (6.6) have intersecting domains of validity. The
asymptotics for the integral $I(r)$ reads:

$\int_{r}^{0}\frac{\mathrm{d}r}{\sqrt{-r}H(r)}\underset{r\to-\infty}{=}2\sqrt{-r}+(2\nu_{1}+\mathrm{i})\ln(2+\sqrt{3})+\pi(2k-1)+\mathrm{i}\ln\left(\frac{{H(0)\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}}-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}-H(0)\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}\right)+\mathcal{E}(r),$
(6.9)

where $k\in\mathbb{Z}$,

$\mathcal{E}(r)=-\mathrm{i}\ln\left(\frac{\sin\big{(}\frac{1}{2}\hat{\psi}(r)+\theta_{0}+o\big{(}r^{-\delta}\big{)}\big{)}}{\sin\big{(}\frac{1}{2}\hat{\psi}(r)-\theta_{0}+o\big{(}r^{-\delta}\big{)}\big{)}}\right),\qquad\theta_{0}=-\frac{\pi}{2}+\frac{\mathrm{i}}{2}\ln\big{(}2+\sqrt{3}\big{)}.$
(6.10)

######  Remark 6.2.

The number $k=k(H(0))\in\mathbb{Z}$ cannot be determined via the isomonodromy
technique developed in [27] because, in fact, asymptotics of the function
$\exp\left(\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r\right)$ is
studied there. It looks like an interesting problem to determine the integer
$k$ in terms of the initial value $H(0)$. In the numerical examples considered
below, we find $k$ by comparing the numerical solution for $H(r)$ with its
asymptotics (6.9). Since $k$ is an integer, its experimental determination in
this situation does not represent a problem. The values of $k$ are given in
the corresponding figure captions. $\blacksquare$

######  Remark 6.3.

If $\text{Im}\,\nu_{1}\in(-1,-5/6)$, which corresponds to the range of
validity of the asymptotics (6.6) and (6.9) (cf. condition (6.8)), then one
can fix the branch of $\nu_{1}$ such that $\text{Im}\,\nu_{1}\in(0,1/6)$, so
that the asymptotics (6.1) and (6.4) can be used; moreover, the closer
$\text{Im}\,\nu_{1}$ is to $-1$ or to $0$, the better (in the sense of
approximating $H(r)$ at finite values) work the asymptotics (6.1) and (6.4).
$\blacksquare$

######  Remark 6.4.

This remark concerns the term $\mathcal{E}(r)$ in the asymptotic formula (6.9)
(cf. (6.10)). It is assumed that one has to take a continuous branch of the
logarithmic function as $r$ varies from $0$ to $-\infty$. This problem is
important in numerical calculations because Maple calculates only the
principal branch of the logarithmic function; therefore, plotting the
asymptotic formula (6.10) on a large-$r$ interval almost always produces a
saw-like plot. To cope with this problem, one can present the logarithm in
(6.10) as an integral of the difference of cotangent functions. This idea
works on small-length intervals; however, the calculation of this integral on
the segment $[-10,0]$, say, already consumes a considerable amount of time,
several times longer than the numerical calculation of the solution of the
Painlevé equation itself! Consequently, in lieu of the integral, we define the
continuous branch of the logarithm as the solution of the differential
equation for the above-mentioned integral. It appears that Maple numerically
evaluates this solution much faster than its explicit form represented by the
integral in terms of cotangents. The execution time of this fast calculation,
however, where only $20$-digits of accuracy were maintained, appears to be $5$
to $8$ times longer than the numerical calculation of the integral on the
left-hand side of equation (6.9) based on its direct calculation (with
$120$-digits of accuracy) via the Painlevé transcendent! There is, thus, a
significant difference between the numerical calculations using the asymptotic
formulae (6.4) and (6.9): plots of the first asymptotic formulae (6.4) are
calculated almost immediately, whilst the second asymptotics (6.9) is
interesting for theoretical studies until a fast algorithm for the calculation
of continuous branches of the logarithmic functions will be implemented into
Maple (and analogous codes). $\blacksquare$

Below, we consider several examples that illustrate some features of the
asymptotic formulae (6.6) and (6.9).

###  6.3 Example 3: $H(0)=-0.148+\mathrm{i}0.191$

For this value of $H(0)$, $\nu_{1}=0.0249933\ldots-\mathrm{i}0.329580\ldots$,
so that the condition (6.8) is satisfied, and we can use the asymptotic
formulae (6.6) and (6.9). For the numerical calculation of the solution $H(r)$
and its related integral $I(r)$ via Maple, we consider the initial values for
$H(r)$ at $r_{1}=10^{-8}$ and use 4 terms of the corresponding Taylor series
expansions for the calculation of the initial values, $H(r_{1})$ and
$H^{\prime}(r_{1})$, and the parameter Digits $=120$. For the calculation of
the corresponding asymptotics, Digits is set equal to $20$. In principle, for
the numerical calculation of $H(r)$ and $I(r)$, one can take a smaller number
of digits, $60$, say, and the resulting calculation with $60$ digits produces
visually the same pictures.

Figs. 13, 15, 17, and 19 demonstrate that the large-$r$ asymptotics also give
quite good approximations for the corresponding functions even for small
values of $r$. We took $800$ points for the calculation of these plots. The
calculation from scratch of the numerical solution presented in Figure 17 with
an older notebook (see footnote 8) takes $20\text{s}$, whilst the calculation
of its asymptotics represented by the blue line of the same figure takes
$170\text{s}$ (cf. Remark 6.4). In order to produce the numerical solution and
the corresponding integral shown in Figs. 14, 16, 18, and 20 by the red
curves, which are virtually concealed under the blue curves, a newer notebook
(cf. footnote 8) requires $188\text{s}$ of execution time. The blue curves
representing the asymptotic values were generated in approximately
$1015\text{s}$ (cf. Remark 6.4). For the generation of these plots, we
increased the number of points from $800$ to $1837$ because the plots have
sharp peaks and the heights of the peaks depend substantially on the number
and position of the points inside the peaks.

Figure 18 resembles a mole’s dwelling that can be reached with the aid of two
ladders.999The mole seems to be the main personnage in the Carlo Goldoni
comedy _Il servitore di due padroni_ , or, perhaps, works as a double agent.
The slopes of all the steps and risers of the left staircase are negative. The
right staircase possesses steps with negative slopes and risers with positive
slopes. Obviously, the sign of the slopes coincides with the sign of
$-\mathrm{Re}\,(H(r))$. If we look at Figure 14, we observe, in fact, that the
function $\mathrm{Re}\,(H(r))$ is negative on the segments near its local
minima located near the origin; however, these minima are growing
monotonically and slowly, as long as one moves in the negative direction,
farther and farther away from the origin. What is not clearly seen from the
figure, however, is the fact that the calculations show that the last segment
where $H(r)<0$ occurs for $r\in(-414.68789\ldots,-398.07403\ldots)$, which
corresponds to the right wall (first riser) of the mole’s dwelling. The left
wall of the mole’s dwelling is the image of a segment around a local minimum
at $r_{\mathrm{min}}=-482.3271\ldots$ where
$H(r_{\mathrm{min}})=0.0022342881\ldots$. Due to the very small values of
$|H(r)|$ on these segments, the walls look almost vertical, although, in fact,
they have negative and positive slopes for the left and right walls,
respectively.

Figure 13: The red and blue plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic (cf. (6.6)) values of the function $H(r)$ for
$r\leqslant-0.1$ corresponding to the initial value
$H(0)=-0.148+\mathrm{i}0.191$. Figure 14: Extended version of Figure 13 (both
plots almost coincide). Figure 15: The red and blue plots are, respectively,
the imaginary parts of the numeric and large-$r$ asymptotic (cf. (6.6)) values
of the function $H(r)$ for $r\leqslant-0.01$ corresponding to the initial
value $H(0)=-0.148+\mathrm{i}0.191$. Figure 16: Extended version of Figure 15
(both plots almost coincide). Figure 17: The red and blue plots are,
respectively, the real parts of the numeric and large-$r$ asymptotic (cf.
(6.9) with $k=0$) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-8}$, where $H(r)$ is the solution with initial value
$H(0)=-0.148+\mathrm{i}0.191$. In black-and-white the curves are
indistinguishable; in colour, though, one notices minor discrepancies between
the curves. Figure 18: Extended version of Figure 17 (both plots almost
coincide). Figure 19: The red and blue plots are, respectively, the imaginary
parts of the numeric and large-$r$ asymptotic (cf. (6.9) with $k=0$) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-3}$, where $H(r)$ is the solution with initial value
$H(0)=-0.148+\mathrm{i}0.191$. In black-and-white some minor discrepancies
between the curves are noticeable. Figure 20: The red and blue plots are,
respectively, the imaginary parts of the numeric and large-$r$ asymptotic (cf.
(6.9)) values of $I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-3}$ corresponding to the solution $H(r)$ with initial value
$H(0)=-0.148+\mathrm{i}0.191$. On a coloured figure one can see that the three
red peaks on the segment $[-600,-400]$ are slightly (less than $5$%) longer
than the blue ones, whilst the differences between the lengths of the other
corresponding red and blue peaks are virtually indistinguishable.

In Example 3, we demonstrated that, for some initial vales of $H(r)$, the
leading terms of the large-$r$ asymptotics approximate very closely the
corresponding Painlevé function and its related integral starting from very
small values of $r$ ($r<1$). In Example 4 below, we show that there are
markedly different initial values of $H(r)$ for which $H(r)$ and its related
integral have behaviour similar to that seen in Example 3; however, an
approximation for small values of $r$ becomes considerably worse, even though
it remains satisfactory in the qualitative sense; moreover, in order to
achieve the correct plots, one has to be much more cautious with the numerical
settings.

###  6.4 Example 4: $H(0)=-100-\mathrm{i}300$

For this value of $H(0)$, $\nu_{1}=0.741160\ldots-\mathrm{i}0.300731\ldots$,
so that the condition (6.8) is satisfied, and we can use the asymptotic
formulae (6.6) and (6.9). For the numerical calculation of the solution and
its related integral via Maple, we consider the initial values for $H(r)$ at
$r_{1}=10^{-9}$. For the calculation of the initial values of $H(r)$ and its
corresponding integral, we use 6 terms of the Taylor expansion for $H(r)$ (cf.
Section 2, equations (2.1), (2.2), and (2.17)). For visualisation purposes, 4
terms for $H(r)$ and 2 terms for $I(r)$ of the Taylor series expansion are, in
principle, sufficient. We set the parameter Digits $=180$, and the same value
$180$ was set for the calculation of asymptotics. This large value $180$ for
the parameter Digits is important for the calculation of the imaginary part of
the asymptotic formulae, because, as explained above, the logarithmic term in
the asymptotics (6.10) for the integral is calculated via a solution of an
appropriate differential equation. Somehow, in Example 3, for the calculation
of the asymptotics, we kept Digits equal to $20$ and did not notice visual
differences while increasing its value. In this example, however, we found a
difference when the value of Digits was increased from 20 to $100$. To verify
that the plots remain stable, we increased Digits up to $180$: the plots
presented below correspond to this value of the parameter Digits. In
principle, $80$ to $90$ digits for this calculation suffices.

For the generation of the plots with ‘peaks’, we found that the most
appropriate value for the number of points was $5237$, while in Example 3,
this value was kept at $1837$. In this example, we were obliged to keep it
that high because the height of the three highest peaks were not stable until
the number of points reached the value $5237$, after which, these heights
underwent only minor variations. The calculation with that many digits and
number of points increases the required time by about $20\%$, which, in our
case, is not crucial.

In Example 3, we explained that the ‘staircase-structure’ of the plot for
$\mathrm{Re}\,I(r)$ is a consequence of the fact that the tail of the numeric
plot for $\mathrm{Re}\,H(r)$ is floating slowly upwards from the third to the
second quadrant as $r\to-\infty$; the same behaviour, of course, is replicated
in its asymptotic plot. If the tails of both plots float above the negative
real axis almost simultaneously, then the numeric and asymptotic mole
dwellings almost coincide (cf. Figure 18). The rate of floating of the tails,
however, depends on the initial value $H(0)$. It may happen (and in many cases
it does) that, although both plots are very close to one another, as in the
present case, one tail (the red one) floats before the other (the blue one);
in Figure 22, the red tail floats at $r\approx-55$, prior to the fifth peak,
while the blue tail floats at $r\approx-120$, prior to the seventh peak. Due
to scaling, this fact is not clearly seen in Figure 22, but we verified it
numerically. Thus, the upward-trending float of the asymptotic tail lags the
numeric one by two peaks, so that the ‘asymptotic’ mole’s dwelling appears to
be two steps lower than the ‘numeric’ dwelling (cf. Figure 27). This example
demonstrates that the integer $k$ in the asymptotic formula (6.9) for $I(r)$
appears not only because this formula originates from the asymptotics of the
function $\varphi(r)$, which is defined modulo $2\pi k$ (cf. Appendix B), but
also because it is related to the quality of the approximation of the numeric
plot by its asymptotics at finite values of $r$. Note that the parameter $k$
undergoes a shift of $2$ when one approximates the small-$r$ part of the
numeric solution and its large-$r$ part via the asymptotics (6.9) (cf. Figs.
27 and 25, respectively).

Figure 21: The red and blue plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic (cf. (6.6)) values of the function $H(r)$ for
$r\leqslant-0.1$ corresponding to the initial value $H(0)=-100-\mathrm{i}300$.
Figure 22: Extended version of Figure 21, where only the first two peaks are
shown. On this plot, the first peak virtually coincides with the vertical
axis. Figure 23: The red and blue plots are, respectively, the imaginary parts
of the numeric and large-$r$ asymptotic (cf. (6.6)) values of the function
$H(r)$ for $r\leqslant-0.1$ corresponding to the initial value
$H(0)=-100-\mathrm{i}300$. Figure 24: Extended version of Figure 23. On this
plot, the first down-up peak of Figure 23 almost coincides with the vertical
axis. Figure 25: The red and blue plots are, respectively, the real parts of
the numeric and large-$r$ asymptotic (cf. (6.9) with $k=3$) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-8}$ corresponding to the solution $H(r)$ with initial value
$H(0)=-100-\mathrm{i}300$. Figure 26: A close-up of the segment of the plot on
Figure 25 corresponding to $-280\leqslant r\leqslant-80$ where the numeric and
asymptotic plots merge at $r\approx-120$. One can actually distinguish two
different intertwining (for $r<-120$) curves which are very close to each
other. Figure 27: A close-up of the part of the plot of the numerical solution
on Figure 25 for $-160<r<-10^{-9}$ and the corresponding asymptotic plot (cf.
(6.9) with—attention!—$k=1$). One can actually distinguish two separate
intertwining curves which are very close to one another. The ‘numeric’ mole’s
dwelling corresponds to $r\in(-80,-55)$, while the ‘asymptotic’ dwelling is
located two steps below with $r$-coordinate in $(-160,-120)$. Figure 28: The
red and blue plots are, respectively, the imaginary parts of the numeric and
large-$r$ asymptotic (cf. (6.9): $k$ is not important here) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for $-20\leqslant
r\leqslant-10^{-9}$ corresponding to the solution $H(r)$ with initial value
$H(0)=-100-\mathrm{i}300$. Figure 29: Extended version of Figure 28 for
$-560\leqslant r\leqslant-10^{-9}$. Here, the first peak on Figure 28
virtually coincides with the vertical axis, and is therefore not clearly
visible. The first 6 icicles of the numerical solution are longer than the
corresponding asymptotic icicles; however, from the 7th icicle onward, the
asymptotic icicles are longer, but the difference between the corresponding
icicles decreases rapidly.

###  6.5 Example 5: $H(0)=-\mathrm{i}300$

For this value of $H(0)$, $\nu_{1}=0.732934\ldots-\mathrm{i}0.249469\ldots$,
so that the condition (6.8) is satisfied, and we can use the asymptotic
formulae (6.6) and (6.9). For the numerical calculation of the solution and
its related integral via Maple, we used the same settings as in Example 4. In
this case, the plots look similar to those in Example 4. The difference in
this example is that, for $r<0$, $\mathrm{Re}\,H(r)>0$; however, for small
values of $r$, $\mathrm{Re}\,H(r)$ is very close to zero; for example, at the
first minimum $r_{\mathrm{min}}=-0.485998\ldots$,
$H(r_{\mathrm{min}})=0.024587\ldots$ (cf. Figure 30). As in Example 4, the
approximation for $H(r)$ via its large-$r$ asymptotics is not as good as that
for the initial data for which $10^{-3}<|H(0)|<10^{3/2}$. For small values of
$r$, though, there are two segments wherein the asymptotics for
$\mathrm{Re}\,H(r)$ is negative (cf. Figure 30). These facts imply that the
numerical plot for $I(r)$ looks similar to the plot of Figure 10 (no mole’s
dwelling (!)); however, the plot for the asymptotics does, indeed, contain two
descending steps to the numerical plot of $I(r)$ until they merge. So, in lieu
of the mole’s dwelling, we have an ‘asymptotic pigeon hole’. On the segment
$[-5.5,0]$, the asymptotic formula does not even approximate qualitatively the
solution.

When we increase the value of $|\mathrm{Im}\,H(0)|$, the asymptotic pigeon
hole shifts to the left; for $H(0)=-\mathrm{i}10^{5}$, say, the asymptotic
pigeon hole is located on the segment $[-79,-59]$, and, as a result, the
asymptotics does not approximate $I(r)$ on $[-59,0]$. Our numerical
experiments show that, in this case, a good approximation for $I(r)$ on the
segment $[-79,-3]$ is attained by a reflection of the asymptotic plot with
respect to a straight line passing through two points at the bottom of the
asymptotic pigeon hole. These two points are not uniquely defined; therefore,
one may vary the location of these points in order to achieve a better
approximation.

Figure 30: The red and blue plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic (cf. (6.6)) values of the function $H(r)$ for
$r\leqslant-0.1$ corresponding to the initial value $H(0)=-\mathrm{i}300$. The
numeric solution (in red) is positive, while the asymptotic solution (in blue)
has two segments where it is negative. Figure 31: The numerical solution (in
red) merges with the asymptotic (cf. (6.9) with—attention!—$k=3$) solution (in
blue) on the segment $(-6,-5.5)$. The right-most point of both plots is
$r=-10^{-9}$.

###  6.6 Example 6: $H(0)=-0.2+\mathrm{i}0.045$

For this value of $H(0)$, $\nu_{1}=0.049319\ldots-\mathrm{i}0.459650\ldots$,
so that the condition (6.8) is satisfied, and we can use the asymptotic
formulae (6.6) and (6.9). For the numerical calculation of the solution and
its related integral via Maple, we used the same settings as in Example 4. The
plots presented in Figs. 32–39 were generated in $957\text{s}$ with the help
of the newer notebook described in footnote 8.

It is instructive to compare this example with Example 3: the initial values
in both examples are close, but the behaviour of the solutions is very
different. This occurs because, in Example 6, $\textrm{Im}\,\nu_{1}$ is much
closer to $-0.5$. This last fact also causes the appearance of very sharp
peaks in the plots for the real and imaginary parts of $H(r)$. These peaks
create an additional problem for the visualization of asymptotics because we
were not able to obtain the right heights for the first peaks in our plots:
not even 15000 points for constructing the plots was sufficient! Our strategy
is two-fold: (i) we consider two close-up plots for $\textrm{Re}\,H(r)$ and
$\textrm{Im}\,H(r)$ (cf. Figs. 32 and 34) in order to determine the correct
heights of the first two peaks; and (ii) to construct the extended plots (cf.
Figs. 33 and 35) we randomly choose the number of points in order to get the
heights of the first two peaks to be as close as possible, like in the close-
up figures. For item (i), we used 5237 and 1800 points for the construction of
the plots in Figs. 32 and 34, respectively. In this case, the heights of the
first two peaks of the real part of the numerical solution were found to be
approximately 182 and 121, respectively. On the extended version of Figure 32
(cf. Figure 33), the best possible heights that we were able to achieve were
163 and 119, respectively, with the number of points equal to 6237. To get
good correspondence between the peaks on the close-up and extended figures for
the imaginary part of the numerical solution, we constructed the extended plot
with the help of 10800 points. For the corresponding asymptotic plots, 2200
points was enough for both cases. The real part of $I(r)$ does not have any
sharp peaks, so we used 1800 points. The imaginary part of $I(r)$ has only one
sharp peak for small negative $r$, and we achieved its right height with 5237
points for both the numerical and asymptotic plots.

An intriguing feature of this example is that, practically, it is not possible
to reach the mole’s dwelling by numerically calculating $I(r)$. Our
calculation reveals that the last zero of $\mathrm{Re}\,H(r)$ is located at
$\approx-2.625\times 10^{24}$: the right wall of the mole’s dwelling is
located at this point, and what is termed ‘asymptotics’ only begins here!
Therefore, practically speaking, it is not possible to numerically generate a
plot for the solution and verify that, finally, the leading term of
asymptotics is given by a shifted parabola $2\sqrt{-r}$, i.e., the non-
oscillatory part of the asymptotics (6.4).101010The mole has burrowed far too
deep! We see, however, that the asymptotic formula (6.9) very well describes
the right staircase leading to the mole’s dwelling starting from the first
step; therefore, the asymptotics (6.9) is correct, from which it follows that,
for very large negative values of $r$, the solution should behave like the
shifted parabola $2\sqrt{-r}$. This illustrates the theoretical application of
the asymptotic formula (6.9) mentioned in Remark 6.4; without knowledge of the
asymptotics, one may assume that the right staircase constitutes the
asymptotic behaviour of $I(r)$ for all large negative $r$.

Figure 32: The red and blue plots are, respectively, the real parts of the
numeric and large-$r$ asymptotic (cf. (6.6)) values of the function $H(r)$ for
$r\leqslant-0.1$ corresponding to the initial value
$H(0)=-0.2+\mathrm{i}0.045$. Figure 33: Extended version of Figure 32 where
the first two peaks of the plot are shown. On this plot, the first peak almost
coincides with the vertical axis. Figure 34: The red and blue plots are,
respectively, the imaginary parts of the numeric and large-$r$ asymptotic (cf.
(6.6)) values of the function $H(r)$ for $r\leqslant-0.1$ corresponding to the
initial value $H(0)=-0.2+\mathrm{i}0.045$. Figure 35: Extended version of
Figure 34. On this plot, the first down-up peak of Figure 34 virtually
coincides with the vertical axis. Figure 36: The red and blue plots are,
respectively, the real parts of the numeric and large-$r$ asymptotic (cf.
(6.9) with $k=0$) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for
$r\leqslant-10^{-8}$ corresponding to the solution $H(r)$ with initial value
$H(0)=-0.2+\mathrm{i}0.045$. The red plot is overlapped by the blue plot, and
is therefore not visible. On the close-up Figure 37 that follows, one can
distinguish the red colour. Figure 37: A close-up of a part of the plot of the
numerical solution in Figure 36 for $-20\leqslant r\leqslant-10^{-8}$, and the
corresponding asymptotic plot (cf. (6.9) with $k=0$). On the coloured picture,
one can see that the steeper slopes from below are in red while those from
above are in blue. Figure 38: The red and blue plots are, respectively, the
imaginary parts of the numeric and large-$r$ asymptotic (cf. (6.9)) values of
$I=\smallint_{r}^{0}\frac{1}{\sqrt{-r}H(r)}\,\mathrm{d}r$ for $-20\leqslant
r\leqslant-10^{-9}$ corresponding to the solution $H(r)$ with initial value
$H(0)=-0.2+\mathrm{i}0.045$. Figure 39: Extended version of Figure 38 for
$r\leqslant-10^{-5}$. In this figure, the first peak of Figure 38 almost
coincides with the vertical axis, and is therefore not clearly visible. At the
end-points of the first few peaks, one can distinguish small red dots.

######  Remark 6.5.

Question: Where is the mole’s dwelling in Figure 36? Answer: No one can see
it, but it exists! As explained in Example 3, the location of the dwelling
(its right wall) corresponds to the last zero of the function
$\mathrm{Re}\,H(r)$. Observing Figure 33, one may have a concern as to whether
or not this last zero does, in fact, exist, because we see that the asymptotic
formula (6.6) works very well within the given plot, and we expect that it
should replicate, with minor changes, as $r\to-\infty$. The major tendency of
the large-$r$ behaviour of the solution that follows from Figure 33 is that
the heights of the peaks becoming lower and the distances between them
increase. Another tendency in the transformation of the plot which is more
subtle, however, is that its tail is floating up slowly above the negative
real semi-axis as one proceeds to the the left ($r\to-\infty$): finally,
$\mathrm{Re}\,H(r)\to 1$; therefore, the last zero of $\mathrm{Re}\,H(r)$
should exist.

In order to comprehend these tendencies, one has to consider the asymptotics
of the asymptotic formula (6.6): equation (6.6), in fact, can be simplified
and transformed into an expression resembling the “regular asymptotics” (6.1);
so, in this sense, the latter asymptotics serves as “the asymptotics of
asymptotics”. This is reasonable terminology, because the beginning of the
plot where the first asymptotics (6.6) is already working, and the tail of the
plot, where it is, of course, still working, look radically different! At the
same time, though, we explain below that, practically, one cannot visualize
this difference!

The plots in Figs. 32 and 33 show that the asymptotics accurately approximates
$\mathrm{Re}\,H(r)$; as a result, for the calculation of the last zero of
$\mathrm{Re}\,H(r)$, we can use the asymptotics of this function instead of
its numerical evaluation. This is important because, as is evident from the
result (6.11) stated below, such a calculation is not possible due to the
“astronomical distances” required for this purpose. According to our
calculations, the last zero of $\text{Re}\,H(r)$ is located at111111Recall
that $10^{24}$ is an estimate for the number of stars in the observable
universe.

$r_{0}=-2.6279340765216450944920718115\ldots\times 10^{24}.$ (6.11)

On the plot of $\mathrm{Re}\,I(r)$, after the point, $r_{0}$, one steps onto
the floor of the mole’s dwelling, from which the left stairway to heaven
begins. In order to evaluate the depth where the dwelling is located, one has
to count the number of zeros of $\text{Re}\,H(r)$. In Figure 33, we see that
the plot has a quasi-periodic structure: by quasi-period we mean a part of the
plot located between two neighbouring peaks. Each peak “stands” on two legs,
so that one leg of a peak belongs to the left quasi-period, whereas the other
leg belongs to the right quasi-period; therefore, each leg intersects the
negative real axis, and gives rise to one zero of $\text{Re}\,H(r)$. Thus,
each quasi-period has at least two zeros. There is one more point that
requires verification, namely, the maximum point on the “bridge” connecting
the legs. This point is always negative until one arrives at the quasi-period
with the last zero. It is evidently not a trivial matter to establish this for
the Painlevé function; however, it can be proved for its asymptotics (6.6).
Numerically, for the first two quasi-periods shown in Figure 32, we found that
$r_{\max,1}=-3.756\ldots$, with $H(r_{\max,1})=-0.46471\ldots$, and
$r_{\max,2}=-14.118\ldots$, with $H(r_{\max,2})=-0.47042\ldots$. So, the
floating process of the tail evolves as follows: the legs are lifting up and
the distance between them is growing at the same time that the “bridge”
between the legs straightens out so that its maximum point is moving down.
Thus, the shape of the bridge changes from convex to concave, the legs
disappear, the maximum turns smoothly to the minimum, and the floating process
continues: the tail (on some appropriate scale) becomes similar to the plot
shown in Figure 14. This transformation of the plot is progressing very slowly
on astronomically large distances for $|r|$; however, it can still be observed
with the help of the asymptotic formula (6.6). At this stage of the evolution,
the transformation of the plot continues: the spikes become more and more
“plateaued”, so that, finally, the tail, after an appropriate re-scaling,
resembles the black plot in Figure 1. Practically, the graph of this part of
the tail cannot be plotted even with the help of asymptotics, because the
distance between the spikes, as well as the spikes themselves, are becoming
progressively more and more stretched along the negative semi-axis, unless
some special scaling for $r$ is used which changes concomitant with the growth
of $|r|$.

Reverting back to the determination of the mole’s dwelling, we note that the
pair of zeros between two neighbouring peaks of $\mathrm{Re}\,H(r)$ correspond
to one step down in the right staircase. If $2N$ denotes the number of zeros,
then there are $N$ steps down in the right staircase. The depth of each step-
down, starting from the second one, should be $\pi$, so that the depth of the
mole’s dwelling, for large $N$, is $\pi N+\mathcal{O}(1)$. On the other hand,
this depth can be calculated (cf. equations (6.9) and (6.10)) as follows,
$\mathrm{Re}\big{(}(\hat{\psi}(r_{0})-\theta_{0})/2+(\hat{\psi}(r_{0})+\theta_{0})/2\big{)}-2\sqrt{-r_{0}}+o(1)$,
so that, for large $|r_{0}|$, mole’s dwelling is located at a depth of about
$2\sqrt{-3r_{0}}-2\sqrt{-r_{0}}+\mathcal{O}\big{(}\ln\sqrt{-3r_{0}}\big{)}$. A
more careful estimation requires the $\mathcal{O}(1)$ contribution which, on
the scale of our distances (6.11), is comparable to the logarithmic error
estimate; but, for the purposes of the rough evaluation we are looking for,
such contributions do not matter. Numerically, for the $y$-co-ordinate of the
mole’s dwelling, we get the following estimate:121212Taking into account the
co-ordinates of the mole’s dwelling (6.11), (6.12), its size, and the fact
that the left staircase goes up, far beyond the level of the negative semi-
axis, we may suspect that the strange mole from Example 6 fell from the
heavens some time ago.

$-2(\sqrt{3}-1)\sqrt{-r_{0}}=-2.373441069108\ldots\times 10^{12}.$ (6.12)

We expect that $10$ to $11$ digits after the decimal point in (6.12) are
correct. A more careful estimate for the depth of the mole’s dwelling can be
obtained with the help of the so-called stair-stringer (6.14) discussed in the
following subsection. $\blacksquare$

###  6.7 Stair-Stringers

In conclusion, we would like to formulate some observations concerning the
asymptotic behaviour of $I(r)$. These observations require further
investigations and more careful formulations. In Examples 1 and 2, the real
part of $I(r)$ looks like a staircase. In Examples 3–6, we observe two
staircases connected with the mole’s dwelling. The appearance of the mole’s
dwelling on the plot of $I(r)$ was explained in terms of the floating of the
tail of the plot of the function $H(r)$ above the negative-$r$ semi-axis. We
conjecture that the straight line
$H_{-}(r)=-1/2=\operatorname{Re}\big{(}\mathrm{e}^{\pm
2\pi\mathrm{i}/3}\big{)}$ plays the role of an unstable attracting line, and
the appearance of the mole’s dwelling and its location can be explained via
the interaction of the stable attracting line, $H_{+}(r)=+1$, and the unstable
attracting line, $H_{-}(r)$.131313The numbers $1$ and $\mathrm{e}^{\pm
2\pi\mathrm{i}/3}$ are the roots of the non-differentiated part of equation
(1.2) (cf. Corollary 4.1).

By the term _stair-stringer_ we mean the non-oscillatory part of the leading
term of asymptotics describing the staircases. The stair-stringer for the left
staircase, denoted by $Str_{l}$, can be deduced immediately from the
asymptotics (6.4), namely,

$Str_{l}=2\sqrt{-r}+2\mathrm{Re}(\nu_{1})\ln(2+\sqrt{3})+2\pi
k-\mathrm{Im}\left(\ln\left(\frac{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}H(0)-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}-H(0)\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}\right)\right).$
(6.13)

To get a formula for the stair-stringer for the right staircase, denoted by
$Str_{r}$, is more complicated; however, the assiduous reader of this section
should be able to derive the following formula:

$\displaystyle Str_{r}=$
$\displaystyle-2(\sqrt{3}-1)\sqrt{-r}-\mathrm{Re}(\nu_{1})\ln\sqrt{-r}+\mathrm{Re}(\nu_{1})\ln\left(\frac{(2+\sqrt{3})^{2}}{(2\sqrt{3})^{3}}\right)-\frac{3\pi}{2}\mathrm{Im}(\nu_{1})$
(6.14)
$\displaystyle+\frac{\pi}{4}+2\pi(k-1)-\mathrm{Im}\left(\ln\left(\frac{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}H(0)-\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}{\mathrm{e}^{\frac{2\pi\mathrm{i}}{3}}-H(0)\mathrm{e}^{-\frac{2\pi\mathrm{i}}{3}}}\right)\right)+\mathrm{Im}\left(\ln\left(\tilde{g}_{1}\Gamma(\mathrm{i}\nu_{1})\right)\right).$

In equations (6.13) and (6.14), $k\in\mathbb{Z}$ is an explicit manifestation
of the $2\pi$-indeterminacy of the imaginary part of the asymptotics (6.4) and
(6.9).

The stair-stringer $Str_{l}$ does not require additional visualization since
it is a major part of the leading term of asymptotics which has already been
verified in Examples 1 and 2. The “intermediate” stair-stringer $Str_{r}$ is
illustrated with the help of Figs. 40 and 41: the integer $k$ in both of these
figures is equal to $1$, while $k=0$ for $Str_{r}$ corresponding to Example 6,
which is not presented here.

Figure 40: The red curve represents the numerical values of
$\mathrm{Re}\,I(r)$ corresponding to the initial value
$H(0)=-0.4+\mathrm{i}0.78$. The Maple settings for the generation of this
curve are similar to those of Example 6. The parameter $\nu_{1}$ for this
solution equals $-0.396664\dotsc-\mathrm{i}0.198139\dotsc$. The black line
represents $Str_{r}$ with $k=1$, and the lowest (green) line is the unstable
attracting parabola. Figure 41: The red curve represents the numerical values
of $\mathrm{Re}\,I(r)$ corresponding to the initial value
$H(0)=-100-\mathrm{i}200$. The Maple settings for the generation of this curve
are similar to those of Example 4. The parameter $\nu_{1}$ for this solution
equals $0.686841\dotsc-\mathrm{i}0.323156\dotsc$. The black line represents
$Str_{r}$ with $k=1$, and the upper (green) line is the unstable attracting
parabola.

The asymptotic behaviour as $r\to-\infty$ of almost all solutions of equation
(1.2) (in particular, all solutions considered in this work) can be described
as follows: there exists some $r$ which could be small or large, depending on
a solution, such that solutions are attracted to the parabola $2\sqrt{-r}$ and
remain oscillatory at a distance (cf. (6.13)) $Str_{l}-2\sqrt{-r}$ which can
be small or large, depending on a particular solution. In this sense, we can
call $r\to 2\sqrt{-r}$ the _stable attracting parabola_. Many solutions, prior
to their behaviours being equilibrated by the stable attracting parabola, are
being “captured” by the _unstable attracting parabola_ , that is,
$r\to-2(\sqrt{3}-1)\sqrt{-r}$. For any $r=r_{0}$, there is a solution which is
attracted by the unstable parabola on the distance $|r_{0}|$. These solutions
are oscillating near $Str_{r}$. Contrary to the case for $Str_{l}$, the
distance between the unstable parabola and $Str_{r}$ has logarithmic growth;
however, this growth is not the reason why the unstable parabola cannot hold
the solution on the infinite-$r$ interval. Our formal explanation for this
behaviour of the solutions, which relies on the notion of ‘asymptotics of
asymptotics’, is given in Remark 6.5; it implies that the tail of the plot for
$\textrm{Re}\,H(r)$ is floating above the negative real semi-axis, and
therefore the number of down-steps should be finite.

The function $\mathrm{Re}\,I(r)$ can serve as a mathematical model describing
the mole’s behaviour. This model gives a simpler and even more convincing
explanation for the instability of the right parabola. The mole is moving
leftward-downward leaving behind a trajectory resembling the right staircase;
the deeper and deeper the mole burrows, the less and less food it finds, so
that the right staircase becomes less and less concave. Finally, the mole
realizes that there is no food. At this stage, the mole constructs its
dwelling, and, after a brief respite gathering its thoughts, the mole,
regardless of its nature, realizes that it is necessary to change its
direction of movement, and, finally, does so! Consequently, the mole starts to
build the left staircase. Initially, the mole moves up in a near-vertical
manner; however, since it is difficult to move vertically, its trajectory
becomes “wavy”. After a few steps, the mole sees more and more food in front
of itself, thus its horizontal motion becomes longer and longer, whilst its
vertical movement continues to shrink; hence, the left staircase becomes less
and less convex. Depending on the particular situation, regulated by the value
of $H(0)$, the location of the mole’s dwelling will be different. We can also
interpret the steps of the staircase as tunnels made by the mole, the lengths
of which can be regulated with the help of a scaling parameter, $c>0:r\to cr$.
What about the imaginary part of $I(r)$? We see (cf. Figure 20) that each
icicle corresponds to a step/tunnel in its own right. We know that the mole is
making repositories for its food; so, in each tunnel there is a single food
repository, and the lengths of the icicles can be interpreted as numbers that
are proportional to the food supply accrued in the corresponding repositories.
In accordance with this model, the largest repository of food is located
precisely in the mole’s dwelling.

A more interesting model describing the “underground mole’s geometry” could be
related to a generalization of the function $I(r)$ that depends on two
variables. It seems plausible, therefore, to “dig” for such a function amongst
the integrals associated with the higher Painlevé equations depending on two
variables, namely, the second member of a hierarchy related to the degenerate
third Painlevé equation (1.1).

##  Appendix A Appendix: The Function $g_{2}(r)$

In Section 2, the definition of the generating functions $g_{k}(r)$,
$k=1,2,\dotsc$, are given, and the first function of this sequence,
$g_{1}(r)$, is constructed. The second function, $g_{2}(r)$, is constructed
here, and we also explain how one calculates $P_{n}^{{}^{\prime}}(-1)$ with
the help of this result.

The differential equation for $g_{2}(r)$ is

$\big{(}rg_{2}^{{}^{\prime}}(r)\big{)}^{\prime}=3g_{2}(r)-\frac{1}{3}+\frac{1}{3}\big{(}I_{1}(2\sqrt{3r})\big{)}^{2},$
(A.1)

where $I_{1}$ is the modified Bessel function of order $1$. This ODE is an
inhomogeneous modified Bessel equation of order $0$. The small-$r$ expansion
of $g_{2}(r)$ does not contain logarithmic terms, and its leading term is
$-r/3$. This fact allows us to uniquely specify the proper solution of this
equation:

$g_{2}(r)=\frac{1}{9}-\frac{1}{9}I_{0}(2\sqrt{3r})+\frac{2}{3}\int_{0}^{r}\left(I_{0}(2\sqrt{3r})K_{0}(2\sqrt{3x})-K_{0}(2\sqrt{3r})I_{0}(2\sqrt{3x})\right)\left(I_{1}(2\sqrt{3x})\right)^{2}\mathrm{d}x.$
(A.2)

The following expansion is not widely known; however, it is very helpful in
our study:

$\left(I_{1}(2\sqrt{3x})\right)^{2}=3x\sum_{n=0}^{\infty}\frac{\binom{2n+2}{n}(3x)^{n}}{((n+1)!)^{2}},$
(A.3)

where $\tbinom{m}{k}=\tfrac{m!}{k!(m-k)!}$ is the binomial coefficient.

Since, in fact, we need the Taylor series expansion of $g_{2}(r)$, we solve
equation (A.1) with the help of power series. Substituting the expansion

$g_{2}(r)=\sum_{k=1}^{\infty}c_{k}\,r^{k},$ (A.4)

where the coefficients $c_{k}$, $k=1,2,\ldots$, are independent of $r$, into
equation (A.1), one deduces the following recurrence relation,

$(k+1)^{2}c_{k+1}=3c_{k}+\frac{3^{k-1}}{(k!)^{2}}\binom{2k}{k-1},\quad
c_{1}=-\frac{1}{3}.$

Solving the last relation, we find that

$c_{n}=\frac{3^{n-2}}{(n!)^{2}}\left(-1+\sum_{k=1}^{n-1}\binom{2k}{k-1}\right),\quad
n=1,2,\ldots.$ (A.5)

######  Remark A.1.

The integer sequence $\sum_{k=1}^{n-1}\binom{2k}{k-1}$, $n=2,3,\ldots$,
coincides with sequence A057552 in [24].141414See, also, the sequence A279561
in OEIS enumerating the number of length-$n$ inversion sequences avoiding the
patterns $101$, $102$, $201$, and $210$. Shifting $n$ to $n-2$ in a formula
given in [24], we find that

$\sum_{k=1}^{n-1}\binom{2k}{k-1}=\frac{\binom{2n-2}{n-1}}{2n}\left(4n-2-n\,{}_{2}F_{1}(1,-n+1;-n+3/2;1/4)\right)-\frac{1}{2},\quad
n\in\mathbb{N},$

where ${}_{2}F_{1}(1,-n+1;-n+3/2;1/4)$ is the value of the Gauss
hypergeometric function at $x=1/4$, and $\tfrac{\binom{2n-2}{n-1}}{n}$ is the
$(n-1)$th Catalan number. It is interesting to note that Maple gives a more
complicated presentation for this sum:

$\sum_{k=1}^{n-1}\binom{2k}{k-1}=-\frac{1}{2}+\frac{\mathrm{i}\sqrt{3}}{6}-\binom{2n}{n-1}\,{}_{3}F_{2}(1,n+1,n+1/2;n,n+2;4),$

where ${}_{3}F_{2}(1,n+1,n+1/2;n,n+2;4)$ is the value of the generalized
hypergeometric function at $x=4$. $\blacksquare$

######  Corollary A.1.

The function (A.2) is the generating function for the sequence $c_{n}$ _(_ cf.
(A.5)_)_.

###### Proof.

The function (A.2) is obtained from the general solution by specifying the
initial conditions $g_{2}(0)=0$ and $g_{2}^{\prime}(0)=-1/3$. Since $r=0$ is
the singular point of the equation, we, strictly speaking, have to prove that
this solution can be developed into a power series in $r$ of the form (A.4).
The proof is straightforward: one employs the expansions for
$I_{0}(2\sqrt{3r})$, $I_{1}(2\sqrt{3r})$, and $K_{0}(2\sqrt{3r})$ at $r=0$.
The expansions for the functions $I_{0}(2\sqrt{3r})$ and $I_{1}(2\sqrt{3r})$
are convergent power series. The expansion for $K_{0}(2\sqrt{3r})$ reads

$K_{0}(2\sqrt{3r})=-I_{0}(2\sqrt{3r})\ln\sqrt{3r}+\sum_{m=0}^{\infty}\frac{\psi(1+m)(3r)^{m}}{(m!)^{2}},\quad\psi(1+m)=1+\frac{1}{2}+\ldots+\frac{1}{m}-\gamma,$

where $\gamma=0.57721\ldots$ is the Euler-Mascheroni constant. Then, the
integral in (A.2) can be decomposed into two parts: the first part is the
integral with logarithms, that is,

$\frac{1}{2}I_{0}(2\sqrt{3r})\int_{0}^{r}(\ln(r)-\ln(x))I_{0}(2\sqrt{3x})\left(I_{1}(2\sqrt{3x})\right)^{2}\mathrm{d}x,$
(A.6)

whilst the second one is the integral which contains functions that can be
expanded in power series in the variable of integration and $r$. Substituting
into the second integral the corresponding power series, one can integrate it
successively and obtain a power series with rational (!) coefficients,
because, in this expansion, there appear the differences $\psi(m+1)-\psi(l+1)$
for integers $m$ and $l$ [13]. The integral (A.6) should be integrated by
parts; then, the explicitly integrated term vanishes, while the remaining
integral contains only those functions that can be developed into Taylor
series due to the expansion (A.3). ∎

######  Remark A.2.

We have executed the scheme discussed in the proof of Corollary A.1
explicitly; however, the corresponding formula for the numbers $c_{n}$ which
we obtained via substitution of the corresponding series into the integral
(A.2) is quite cumbersome. We were not able to convert the latter formula to
the simple expression (A.5), but we confirmed numerically that both formulae
coincide. $\blacksquare$

######  Proposition A.1.

$(-1)^{\lfloor\frac{n+1}{2}\rfloor}3^{\nu_{3}(n+1)}P_{n}^{{}^{\prime}}(-1)=3^{b_{n}-2}\left(\frac{n}{2}-\frac{5}{4}-(-1)^{n}\frac{3}{4}+\sum_{k=1}^{n-1}\binom{2k}{k-1}\right),\quad
n\in\mathbb{N},$ (A.7)

where the sequences $\nu_{3}(n+1)$ and $b_{n}$ are defined in Conjecture 2.1
and equation (2.19), respectively.

###### Proof.

We now calculate the coefficients $c_{n}$ with the help of the ansatz (2.25).
Substitute into ansatz (2.25) $a_{0}=-(1-\varepsilon)^{1/3}$, and consider the
expansion of $a_{n}$ as $\varepsilon\to 0$. The numbers $c_{n}$ coincide with
the coefficients of this expansion at $\varepsilon^{2}$; the result of this
calculation reads:

$c_{n}=(-1)^{\left\lfloor\frac{n-1}{2}\right\rfloor+1}\frac{3^{\nu_{3}(n+1)}}{(n!|n!|_{3})^{2}}\left(P_{n}^{{}^{\prime}}(-1)+\frac{1}{3}\left(\frac{n}{2}-\frac{1}{4}-(-1)^{n}\frac{3}{4}\right)P_{n}(-1)\right).$

Taking into account the formula for $P_{n}(-1)$ given in (2.28), and equation
(A.5), we, after a straightforward calculation, arrive at equation (A.7). ∎

##  Appendix B Appendix: Asymptotics as $\tau\\!\to\\!0$ of $u(\tau)$ and
$\varphi(\tau)$ for $a\\!\in\\!\mathbb{C}$

The result presented in Theorem B.1 below is based on Theorem 3.4 of [26].
Here, we formulate only the key asymptotic result of Theorem 3.4 of [26] that
concerns asymptotics as $\tau\to+0$ ($\arg(\tau)=0$), which means that the
parameters $\varepsilon_{1}$ and $\varepsilon_{2}$ appearing in Theorem 3.4
must be set equal to $0$. This, in turn, allows one to simplify the notation
in Theorem 3.4 of [26], namely, we use
$(a,s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},g_{11},g_{12},g_{21},g_{22})$
instead of
$(a,s_{0}^{0}(0,0),s_{0}^{\infty}(0,0),s_{1}^{\infty}(0,0),g_{11}(0,0),g_{12}(0,0),g_{21}(0,0),g_{22}(0,0))$
for the monodromy co-ordinates. Furthermore, we simplify the notation for the
parameters in the asymptotic formulae: (i)
$\varpi_{k}^{\natural}(0,0;\lambda)$ is changed to $\varpi_{k}(\lambda)$,
$k=1,2$; (ii) $\chi_{k}(\mathbf{g};\lambda)$ is denoted here as
$\chi_{k}(\lambda)$; and (iii)
$\mathbf{p}(a,\lambda):=\mathbf{p}_{1}(\lambda)$ and
$\mathbf{p}(-a,\lambda)\mathrm{e}^{-\mathrm{i}\pi\lambda}:=\mathbf{p}_{2}(\lambda)$.
Moreover, we used standard identities for the (Euler) gamma function in order
to simplify the expression for $\mathbf{p}(z_{1},z_{2})$ that appears in
Theorem 3.4 of [26]. Since the Hamiltonian function corresponding to equation
(1.1) is not studied in this work, its small-$\tau$ asymptotics is omitted in
Theorem B.1 below; at the same time, though, Theorem B.1 contains the
small-$\tau$ asymptotis of the function $\varphi(\tau)$. The latter
asymptotics was not included in the formulation of Theorem 3.4 of [26], even
though, in fact, it was obtained in the course of the proof of Theorem 3.4.

Recall that, for any solution $u(\tau)$ of equation (1.1), the function
$\varphi(\tau)$ is defined as the indefinite integral

$\varphi{{}^{\prime}}(\tau)=\frac{2a}{\tau}+\frac{b}{u(\tau)}.$ (B.1)

This function appears in the parametrization of the coefficients of the
corresponding isomonodromy system (see [26], Proposition 1.2) in the form
$\mathrm{e}^{\mathrm{i}\varphi(\tau)}$; therefore, as long as the monodromy
data for the isomonodromy system are given, the function $\varphi(\tau)$ is
fixed modulo $2\pi$, or, in other words, the constant of integration in (B.1)
is defined via the monodromy data modulo $2\pi$, i.e., from the isomonodromy
point of view the function $\varphi(\tau)$ is an “almost definite integral”.
This fact allows one to calculate some particular integrals that are related
to the function $u(\tau)$ [29].

There is a one-to-one correspondence between the pair of functions
$(u(\tau),\mathrm{e}^{\mathrm{i}\varphi(\tau)})$, where the pair
$(u(\tau),\varphi(\tau))$ is a solution of the system (1.1), (B.1), and the
monodromy data of the first equation of the system (12) in [26]. It is the
monodromy data of this equation that is denoted as
$(a,s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},g_{11},g_{12},g_{21},\linebreak[1]g_{22})\in\mathbb{C}^{8}$.
These complex co-ordinates satisfy a set of algebraic equations where, instead
of $a$, the variable $\mathrm{e}^{\pi a}$ appears; this system of equations in
$\mathbb{C}^{8}$ defines a monodromy manifold. By treating $\mathrm{e}^{\pi
a}$ as a parameter, this manifold can also be considered as an algebraic
variety in $\mathbb{C}^{7}$. We call a point of this manifold the monodromy
data corresponding to $(u(\tau),\mathrm{e}^{\mathrm{i}\varphi(\tau)})$, or,
for brevity, the monodromy data corresponding to the pair
$(u(\tau),\varphi(\tau))$, with the _proviso_ that under the function
$\varphi(\tau)$ is understood the class of functions defined by the
equivalence relation $\varphi\equiv\varphi+2\pi k,k\in\mathbb{Z}$.

######  Theorem B.1.

Let $(u(\tau),\varphi(\tau))$ be a solution of the system (1.1), (B.1) for
$\varepsilon b>0$ corresponding to the monodromy data $(a,s_{0}^{0}$,
$s_{0}^{\infty}$, $s_{1}^{\infty}$, $g_{11},g_{12},g_{21},g_{22})$. Suppose
that

$\lvert\operatorname{Im}(a)\rvert\\!<\\!1,\qquad\rho\\!\neq\\!0,\qquad\lvert\operatorname{Re}(\rho)\rvert\\!<\\!1/2,\qquad
g_{11}g_{22}\\!\neq\\!0,$ (B.2)

where

$\cos(2\pi\rho)\\!:=\\!-\frac{\mathrm{i}s_{0}^{0}}{2}\\!=\\!\cosh(\pi
a)\\!+\\!\frac{1}{2}s_{0}^{\infty}s_{1}^{\infty}\mathrm{e}^{\pi a}.$ (B.3)

Then, $\exists$ $\delta\\!>\\!0$ such that

$u(\tau)\underset{\tau\to+0}{=}\,\frac{\tau b\mathrm{e}^{\pi
a/2}}{16\pi}\\!\left(\varpi_{1}(\rho)\tau^{2\rho}+\varpi_{1}(-\rho)\tau^{-2\rho}\right)\\!\left(\varpi_{2}(\rho)\tau^{2\rho}+\varpi_{2}(-\rho)\tau^{-2\rho}\right)\\!\left(1+o\big{(}\tau^{\delta})\right),$
(B.4)

and

$\mathrm{e}^{\mathrm{i}\varphi(\tau)}\underset{\tau\to+0}{=}\mathrm{e}^{\mathrm{i}\pi}2^{\mathrm{i}a}\tau^{\mathrm{i}2a}\\!\left(\frac{\varpi_{2}(\rho)\tau^{2\rho}\\!+\\!\varpi_{2}(-\rho)\tau^{-2\rho}}{\varpi_{1}(\rho)\tau^{2\rho}\\!+\\!\varpi_{1}(-\rho)\tau^{-2\rho}}\right)\\!\left(1+o\big{(}\tau^{\delta}\big{)}\right),$
(B.5)

where

$\displaystyle\varpi_{k}(\lambda)=\mathbf{p}_{k}(\lambda)\chi_{k}(\lambda),\qquad
k=1,2,$ (B.6)
$\displaystyle\chi_{k}(\lambda)=g_{1k}\mathrm{e}^{\mathrm{i}\pi(\lambda+1/4)}+g_{2k}\mathrm{e}^{-\mathrm{i}\pi(\lambda+1/4)},$
(B.7)
$\displaystyle\mathbf{p}_{k}(\lambda)=\mathrm{e}^{-(-1)^{k}\frac{\mathrm{i}\pi}{2}\lambda}\left(\frac{\varepsilon
b}{2}\right)^{\lambda}\frac{\Gamma(1-2\lambda)}{\Gamma(1+2\lambda)}\frac{\Gamma\big{(}1+\lambda-(-1)^{k}\mathrm{i}a/2\big{)}}{\lambda},$
(B.8)

and $\Gamma(\ast)$ is the gamma function [13].

##  Appendix C Appendix: Asymptotics as $\tau\\!\to\\!+\infty$ of $u(\tau)$
and $\varphi(\tau)$ for $a\\!\in\\!\mathbb{C}$

In one of our previous works (see [27], Appendix B), we noted that the phase
shift in the large-$\tau$ asymptotics of the function $u(\tau)$ reported in
[26] contained a mistake which was corrected in Appendix B of [27].
Subsequently, the author of [28] (see Section 7 of [28]) uncovered an
additional inconsistency in [26] which, “hereditarily”, transferred to the
paper [27]: this error does not affect the asymptotcs of the function
$u(\tau)$; it does, however, impact the asymptotics of the function
$\varphi(\tau)$, since the term $a\ln\tau$ was not accounted for in its small-
and large-$\tau$ asymptotics. The latter asymptotics for $\varphi(\tau)$ were
not included in list of the results obtained in the aforementioned papers.
Nevertheless, we wrote the paper [29] which dealt with the asymptotics of an
integral for the meromorphic solution of equation (1.1), where it was
explained how to rectify the asymptotics for $\varphi(\tau)$ in case it is
extracted from [26], and the particular example considered in [29] was
verified numerically. The paper [29] contains the asymptotics of the function
$\varphi(\tau)$ for the meromorphic solution together with the correction
term, which was stated, but not derived; however, it was checked numerically
for the particular example considered therein. In this appendix, we present
the asymptotics with the correction term for the general solution and also
give the derivation for the correction term.

Furthermore, during the course of the numerical calculations for the present
work, we found one more “mysterious” misprint in the large-$\tau$ asymptotics
of the function $u(\tau)$ (see [26], Section 3, p. 1174, Theorem 3.1): the
term $-\tfrac{\pi\mathrm{i}}{2}$ in the definition of
$z(\varepsilon_{1},\varepsilon_{2})$ must be changed to
$+\tfrac{\pi\mathrm{i}}{2}$. Since our original calculations presented in [26]
do, in fact, contain the term $+\tfrac{\pi\mathrm{i}}{2}$ , it is clear that
this is a typographical error. This error in the sign appears, sadly, to have
leapfrogged to the asymptotics obtained in the subsequent work [27], because
the method used therein (see [27], the paragraph at the top of p. 43) allows
us to determine the corresponding phase shift modulo $\pi\mathrm{i}$. In order
to determine, ultimately, whether or not $\pi\mathrm{i}$ should be added, we
matched the asymptotics obtained in [26] with those of [27]. Moreover, in
[28], where this discrepancy with $\pi\mathrm{i}$ was noticed, and where, in
Subsections 8.1 and 8.2, even though the correct formulae are written and used
for the numerical calculations, the general result for the asymptotics of
$u(\tau)$ stated on p. 47 still contains the wrong sign for
$\tfrac{\pi\mathrm{i}}{2}$ in the formula for $z$ in equation (8.7)!

In this appendix, we restate the main results obtained in the papers [26] and
[27] but with all the corrections outlined above included. As in Appendix B,
we present here only asymptotics on the positive real semi-axis and exclude
all transformation issues, i.e., we set $\varepsilon_{1}=\varepsilon_{2}=0$
and simplify the notation for the monodromy data and the parameters used in
Theorem 3.1 of [26] and in Theorems 2.1–2.3 and B.1 of [27], namely,
$\linebreak[1](s_{0}^{0}(0,0),s_{0}^{\infty}(0,0),s_{1}^{\infty}(0,0),g_{11}(0,0),g_{12}(0,0),g_{21}(0,0),g_{22}(0,0),\tilde{\nu}(0,0),z(0,0)):=(s_{0}^{0},s_{0}^{\infty},s_{1}^{\infty},g_{11},g_{12},g_{21},g_{22},\tilde{\nu},z).$

######  Remark C.1.

Here and throughout the paper we use non-single-valued functions such as
roots, logarithms, fractional powers, and power functions with complex
exponent to construct single-valued asymptotics. We use the natural agreement
that, for real positive argument, all these functions are positive. The branch
of such a function for complex argument, in case it appears several times in a
formula or within a group of related formulae, is assumed to be the same
within the formula or the group. In case a choice of the branch is not
discussed, it means that its particular choice does not affect the value of
the resulting formula(e); for example, the branch of $\ln(z)$ in
$\sinh(\ln(z))$ or the branch of $\sqrt{\tilde{\nu}+1}$ in equation (C.3)
below. Here, as in the paper, $\Gamma(\ast)$ is the gamma function [13].
$\blacksquare$

Since the Hamiltonian function corresponding to equation (1.1) is not studied
in this work, its large-$\tau$ asymptotics stated in Theorem 3.1 of [26] is
not included in Theorem C.1 below; instead, it is supplanted by the
asymptotics of the function $\varphi(\tau)$: with this, the asymptotic
description of the isomonodromy deformation for the first equation of the
system (12) in [26] is complete.

######  Theorem C.1.

Let $(u(\tau),\varphi(\tau))$ be a solution of the system (1.1), (B.1) for
$\varepsilon b>0$ corresponding to the monodromy data $(a,s_{0}^{0}$,
$s_{0}^{\infty}$, $s_{1}^{\infty}$, $g_{11},g_{12},g_{21},g_{22})$. Define

$\tilde{\nu}\\!+\\!1\\!:=\\!\frac{\mathrm{i}}{2\pi}\ln(g_{11}g_{22}),$ (C.1)

and suppose that

$g_{11}g_{12}g_{21}g_{22}\\!\neq\\!0,\,\qquad\,\lvert\operatorname{Re}(\tilde{\nu}\\!+\\!1)\rvert\\!<\\!1/6.$
(C.2)

Then, $\exists$ $\delta\\!>\\!0$ such that

$u(\tau)\underset{\tau\to+\infty}{=}\,\frac{\varepsilon(\varepsilon
b)^{2/3}}{2}\tau^{1/3}\\!\left(1+\frac{2\sqrt{\tilde{\nu}+1}\mathrm{e}^{3\pi\mathrm{i}/4}}{3^{1/4}(\varepsilon
b)^{1/6}\tau^{1/3}}\cosh\Big{(}\mathrm{i}\theta(\tau)+(\tilde{\nu}+1)\ln\theta(\tau)+z+o\big{(}\tau^{-\delta}\big{)}\Big{)}\right),$
(C.3)

where

$\displaystyle\theta(\tau):=3^{3/2}(\varepsilon b)^{1/3}\tau^{2/3},$ (C.4)
$\displaystyle z:=\,\frac{\ln
2\pi}{2}+\frac{\pi\mathrm{i}}{2}-\frac{3\pi\mathrm{i}}{2}(\tilde{\nu}+1)+\mathrm{i}a\ln(2+\sqrt{3})+(\tilde{\nu}+1)\ln
12-\ln\\!\left(g_{11}g_{12}\sqrt{\tilde{\nu}+1}\,\Gamma(\tilde{\nu}+1)\right),$
(C.5)

and

$\begin{gathered}\varphi(\tau)\underset{\tau\to+\infty}{=}\,3(\varepsilon
b)^{1/3}\tau^{2/3}+2a\ln\big{(}\tau^{2/3}\big{)}-a\ln\left((\varepsilon
b)^{1/3}/4\right)+\pi-2\pi(\tilde{\nu}+1)\\\
+\mathrm{i}\ln\big{(}g_{11}^{2}\big{)}-2\mathrm{i}(\tilde{\nu}+1)\ln(2+\sqrt{3})+\rm{E}_{\varphi}(\tau),\qquad\rm{E}_{\varphi}(\tau)=o\big{(}\tau^{-\delta}\big{)}.\end{gathered}$
(C.6)

######  Remark C.2.

The explicit expression for the correction term $\mathrm{E}_{\varphi}(\tau)$
(cf. equation (C.6)) is obtained in Proposition C.1 below. $\blacksquare$

######  Remark C.3.

Here, we discuss the second restriction in (C.2). The methodology used in [26]
is such that the function $\cosh(\cdot)$ in the asymptotics (C.3) is obtained
during the course of the appropriate estimates as the half-sum of two
exponentials: if $\mathrm{Re}(\tilde{\nu}+1)\neq 0$, then, one of the
exponentials has a power-like growth, and the other has a power-like decay.
Each exponential should represent a leading term of asymptotics at the
corresponding stage of the derivation which leads to the second restriction in
(C.2). In principle, had we required that only the growing exponential
represent the leading term of asymptotics, then we would have obtained a less
restrictive condition, namely, $|\mathrm{Re}(\tilde{\nu}+1)|<1/2$. Had we
followed the latter scheme of the derivation, we would have had to consider
three different conditions on $\mathrm{Re}(\tilde{\nu}+1)$, i.e.,
$\mathrm{Re}(\tilde{\nu}+1)=0$ and $\pm\mathrm{Re}(\tilde{\nu}+1)\in(0,1/2)$,
and would have derived three separate asymptotic formulae which, in our case,
are amalgamated in the unique formula (C.3).

This fact can be revealed from a completely different perspective: if we
address the complete asymptotic expansion given in equation (C.7) below, then
we can verify that, in fact, the smallest exponential in the leading term of
this expansion is larger than the largest exponential in the second correction
term when the second condition in (C.2) is valid.

The value $1/6$ is quite clearly observed in our numerical examples presented
in Section 6 where, in the first two examples
$|\mathrm{Re}(\tilde{\nu}+1)|<1/6$, whilst in the other examples, this
condition is violated. According to our numerical studies, the solutions with
the monodromy data satisfying this condition swiftly reach their asymptotic
behaviour provided that none of the factors in the product of the first
condition in (C.2) is close to zero. By “not close to zero” we mean that
$|g_{i,j}|\geqslant 0.1$; however, the specification of this numerical aspect
requires further investigation.

If one understands the $\cosh(\cdot)$ term as a unique function, then it plays
the role of the leading term of asymptotics (as its growing exponential) for a
wider range of monodromy data, namely, $|\text{Re}(\tilde{\nu}+1)|<1/2$. When
$|\text{Re}(\tilde{\nu}+1)|$ increases farther and farther away from the value
$1/6$, the corresponding solutions reach their asymptotic behaviour at larger
and larger values of $\tau$. Application of the higher-order terms of the
expansion (C.7) is helpful, but not radically, though. A much better
correspondence between the asymptotic and numeric results for finite values of
$\tau$ for the case $|\text{Re}(\tilde{\nu}+1)|\geqslant 1/6$ is achieved with
the help of the asymptotics given in Theorem C.2 below. $\blacksquare$

######  Remark C.4.

The branch of $\ln\big{(}g_{11}^{2}\big{)}$ in the asymptotics (C.6) can not
be fixed because the function $\varphi(\tau)$ enters into the corresponding
isomonodromy system in the form $\mathrm{e}^{\mathrm{i}\varphi(\tau)}$ (see
[26], Propositions 1.1 and 1.2); therefore, the real part of the asymptotics
for $\varphi(\tau)$ is defined modulo $2\pi$.

On the other hand, the function $u(\tau)$ has movable zeros with leading term
$\pm\mathrm{i}b(\tau-\tau_{0})$, and equation (B.1) implies that
$\varphi(\tau)$ has movable logarithmic singularities with leading term
$\mp\mathrm{i}\ln(\tau-\tau_{0})$. Even with a branch cut on the complex
$\tau$-plane from the origin to the point at infinity, the analytic
continuation of $\varphi(\tau)$ depends on the path of continuation: we thus
get infinitely many branches of $\varphi(\tau)$ that differ by $2\pi k$,
$k\in\mathbb{Z}$. As a matter of fact, the asymptotics (C.3) is valid not only
on the positive semi-axis, but also in the larger domain
$\mathcal{D}=\\{\tau\in\mathbb{C};\mathstrut\,|\text{Im}\,\tau^{2/3}|<h,h\in\mathbb{R}_{+}\\}$:
for large $|\tau|$, the width of $\mathcal{D}$ is growing as
$\mathcal{O}\big{(}|\tau|^{1/3}\big{)}$. There may be some logarithmic
singularities of $\varphi(\tau)$ in the interior of this domain along with a
few non-homotopic paths going around them. The locations of such singularities
for finite values of $\tau$ are not known, and, consequently, the
corresponding asymptotics “are not cognizant” of the paths along which
$\varphi(\tau)$ is continued; therefore, the formula for the asymptotics
should contain an ambiguity in order to account for all possible analytic
continuations of $\varphi(\tau)$ along non-homotopic paths, if any, which is
retained in the ambiguity in the choice of the imaginary part of
$\ln\big{(}g_{11}^{2}\big{)}$.

During the course of our numerical studies, we noticed one additional
mechanism for the $2\pi$-ambiguity in the asymptotic formula $\varphi(\tau)$.
This mechanism is related to the fact that our plots are made starting from
finite—and quite small—values of $\tau$. For such values of $\tau$, the
asymptotics does (sometimes) not approximate the solution $u(\tau)$ as well as
it does for very large values of $\tau$; therefore, if, in some segment,
$u(\tau)$ is close, but not equal, to zero, its asymptotics may vanish in this
segment, which provides a $2\pi$-discrepancy in the approximation of the
function $\varphi(\tau)$ via its asymptotics (cf. Section 6, Example 4).

This remark brings to the forefront the fact that the formulation of a general
rule which would fix the $2\pi$-ambiguity in the $\varphi(\tau)$ asymptotics,
applicable for all solutions, is a nontrivial problem; however, for some
classes of the solutions, such a rule could, feasibly, be formulated. Assume
one considers the positive semi-axis as the path of analytic continuation for
$\varphi(\tau)$: on this semi-axis, there exists a class of solutions,
$u(\tau)$, with strictly positive real part. Suppose one finds the correct
branch of $\ln\big{(}g_{11}^{2}\big{)}$ for one solution from this class;
then, since the asymptotics of $u(\tau)$ is given explicitly, one can
determine the proper branch for all solutions from this class. $\blacksquare$

In order to obtain the explicit formula for $\mathrm{E}_{\varphi}(\tau)$, we
are going to use equation (B.1): for this purpose, the first few terms of the
complete asymptotic expansion for the function $1/u(\tau)$ are required. The
asymptotic expansion we need was obtained by Shimomura [35] with the help of a
unique method, together with asymptotic expansions for the other Painlevé
transcendents. Here, we present a “visualization” for the first few terms of
this expansion for the functions $u(\tau)$ and $1/u(\tau)$, and suggest some
conjectures for their coefficients:

$u(\tau)\underset{\tau\to+\infty}{=}\frac{\varepsilon(\varepsilon
b)^{2/3}}{2}\tau^{1/3}\left(1+\sum_{k=1}^{\infty}\frac{1}{\tau^{k/3}}\sum_{j=-k}^{j=k}a_{k,j}w^{j}\right),\qquad
w:={\tau}^{\frac{2}{3}(\tilde{\nu}+1)}\mathrm{e}^{\mathrm{i}\theta(\tau)}.$
(C.7)

Comparing the expansion (C.7) with the asymptotics (C.3), we find that

$a_{1,0}=0,\qquad a_{1,\pm
1}=\frac{\sqrt{\tilde{\nu}+1}\,\mathrm{e}^{3\pi\mathrm{i}/4}}{3^{1/4}(\varepsilon
b)^{1/6}}\mathrm{e}^{\pm\left((\tilde{\nu}+1)\ln\big{(}3^{3/2}(\varepsilon
b)^{1/3}\big{)}+z\right)},\qquad
a_{1,1}\,a_{1,-1}=-\frac{\mathrm{i}(\tilde{\nu}+1)}{\sqrt{3}(\varepsilon
b)^{1/3}}.$ (C.8)

######  Remark C.5.

The finite inner sum in the expansion (C.7) is considered as a unique function
(the $k$th correction term): in this case, we get an asymptotic expansion of
$u(\tau)$ for $|\text{Re}(\tilde{\nu}+1)|<1/2$. One can consider different
types of summations for the double series in (C.7) and obtain different
asymptotic formulae for $u(\tau)$. $\blacksquare$

To derive the explicit formula for the leading-order correction
$\rm{E}_{\varphi}(\tau)$, the coefficients $a_{1,\pm 1}$, $a_{2,\pm 1}$, and
$a_{3,0}$ must be known explicitly: the coefficients $a_{1,\pm 1}$ are given
in equations (C.8); therefore, it remains to determine $a_{2,\pm 1}$ and
$a_{3,0}$. It is known that, in order to obtain the coefficients $a_{k,j}$ for
$k\geqslant 2$ and $j\neq\pm 1$ and $a_{k-2,\pm 1}$ in terms of the lower-
order coefficients $a_{l,j}$ with $l<k$, one can substitute the expansion
(C.7), with the terms up to order $\tau^{-k/3}$ retained, into equation (1.1)
and equate to zero (in the obtained expression) the first $k-1$ higher-order
coefficients as $\tau\to+\infty$. To get the last two coefficients ‘at level
$k$’, i.e., $a_{k,\pm 1}$, one has to repeat the analogous procedure, but keep
in the expansion (C.7) the terms of two more orders, namely, $\tau^{-(k+1)/3}$
and $\tau^{-(k+2)/3}$. To calculate, for example, the coefficient $a_{3,0}$,
one has to keep in the expansion (C.7) the
$\mathcal{O}\big{(}\tau^{-4/3}\big{)}$ terms.

It is convenient to introduce new variables:

$\varkappa=\tilde{\nu}+1,\qquad\alpha=2\mathrm{i}\sqrt{3}\,a.$

In terms of these variables, the coefficients of orders
$\mathcal{O}\left(\tau^{-2/3}\right)$ and $\mathcal{O}\left(\tau^{-1}\right)$
read:151515In all formulae with $\pm$’s, one has to choose, on both the left-
and right-hand sides, either the upper or the lower sign.

$\displaystyle a_{2,\pm 2}=\frac{a_{1,\pm 1}^{2}}{3},\quad a_{2,\pm 1}=0,\quad
a_{2,0}=\frac{\alpha+12\varkappa}{6\mathrm{i}\sqrt{3}(\varepsilon b)^{1/3}},$
(C.9) $\displaystyle a_{3,\pm 3}=\frac{a_{1,\pm 1}^{3}}{12},\quad a_{3,\pm
2}=a_{3,0}=0,\quad a_{3,\pm 1}=\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}}{6^{3}(\varepsilon
b)^{1/3}}\big{(}3(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})-3\mp
12\alpha\mp 80\varkappa\big{)}.$ (C.10)

We have explicitly calculated the coefficients of the expansion (C.7) up to
the level $k=10$; based on these calculations, we formulate below some
conjectures regarding the coefficients $a_{k,j}$. Since the formulae become
progressively more cumbersome for higher values of the level $k$, we present
below the coefficients for levels $k=4,5,6$ in order to demonstrate our
hypotheses: these examples might be useful for the proofs of our conjectures.

$\begin{gathered}a_{4,\pm 4}=\frac{a_{1,\pm 1}^{4}}{54},\qquad a_{4,\pm
3}=a_{4,\pm 1}=a_{4,0}=0,\\\ a_{4,\pm 2}=\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}^{2}}{2^{2}3^{4}(\varepsilon
b)^{1/3}}\big{(}3\alpha^{2}+24\varkappa\alpha+30\varkappa^{2}+1\mp 12\alpha\mp
54\varkappa\big{)}.\end{gathered}$ (C.11) $\begin{gathered}a_{5,\pm
5}=\frac{5a_{1,\pm 1}^{5}}{1296},\qquad a_{5,\pm 4}=a_{5,\pm 2}=a_{5,0}=0,\\\
a_{5,\pm 3}=\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}^{3}}{2^{5}3^{4}(\varepsilon
b)^{1/3}}\big{(}9(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})+1\mp
36\alpha\mp 138\varkappa\big{)},\\\ a_{5,\pm 1}=-\frac{a_{1,\pm
1}}{2^{7}3^{5}(\varepsilon
b)^{2/3}}\left(9(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})^{2}\mp(24\alpha^{3}+48\alpha^{2}\varkappa-912\alpha\varkappa^{2}-3440\varkappa^{3})\right.\\\
\left.-90\alpha^{2}-1296\varkappa\alpha-4168\varkappa^{2}\pm(216\alpha+240\varkappa)+81\right).\end{gathered}$
(C.12) $\begin{gathered}a_{6,\pm 6}=\frac{a_{1,\pm 1}^{6}}{1296},\qquad
a_{6,\pm 5}=a_{6,\pm 3}=a_{6,\pm 1}=0,\\\
a_{6,0}=-\frac{\mathrm{i}\sqrt{3}}{2^{3}3^{6}\varepsilon
b}\left(\alpha^{3}+18\alpha^{2}\varkappa+72\alpha\varkappa^{2}+60\varkappa^{3}-12\alpha-90\varkappa\right),\\\
a_{6,\pm 4}=\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}^{4}}{2^{3}3^{6}(\varepsilon
b)^{1/3}}\big{(}6(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})-1\mp
24\alpha\mp 84\varkappa\big{)},\\\ a_{6,\pm 2}=-\frac{a_{1,\pm
1}^{2}}{2^{5}3^{6}(\varepsilon
b)^{2/3}}\left(9(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})^{2}-171\mp(48\alpha^{3}+396\alpha^{2}\varkappa+576\alpha\varkappa^{2}-880\varkappa^{3})\right.\\\
\left.-30\alpha^{2}-816\varkappa\alpha-2770\varkappa^{2}\pm(384\alpha+1408\varkappa)\right).\end{gathered}$
(C.13)

The coefficients calculated above suggest the following

######  Conjecture C.1.

If $k\in\mathbb{N}$ and $j\in\mathbb{Z}$, $|j|\leqslant k$, have different
parity, then $a_{k,j}=0$, and

$a_{k,\pm k}=\frac{k\,a_{1,\pm 1}^{k}}{2^{k-1}3^{k-1}}.$

In addition, we calculated the coefficients for the levels $k=7$, $8$, $9$,
and $10$ in order to arrive at the following

######  Conjecture C.2.

For $k\geqslant 4$,

$\displaystyle a_{k,\pm(k-2)}=$
$\displaystyle\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}^{k-2}}{2^{k}3^{k}(\varepsilon
b)^{1/3}}\left(3(k-2)^{2}(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2})-5(k-2)^{2}+24(k-2)-24\right.$
(C.14) $\displaystyle\left.\mp 12(k-2)^{2}\alpha\mp
6\big{(}5(k-2)^{2}+8(k-2)\big{)}\varkappa\right).$

######  Remark C.6.

The coefficients $a_{3,\pm 1}$ look similar in form to (C.14), but are
different. An analogous situation occurs with the Taylor coefficients for the
function $H(r)$ studied in Section 2, where the first few members of some
sequences of the coefficients do not conform to the general formula for the
sequence. The quickest way of proving these conjectures, and other analogous
formulae, is to use the technique of generating functions, presented, in
particular, in Section 2. $\blacksquare$

Using the expansion (C.7) for $u(\tau)$, we find a similar expansion for
$1/u(\tau)$:

$\displaystyle\frac{b}{u(\tau)}\underset{\tau\to+\infty}{=}$
$\displaystyle\frac{2(\varepsilon b)^{1/3}}{\tau^{1/3}}-\frac{2(\varepsilon
b)^{1/3}}{\tau^{2/3}}\left(a_{1,-1}w^{-1}+a_{1,1}w\right)+\frac{1}{\tau}\left(\frac{4}{3}(\varepsilon
b)^{1/3}\big{(}a_{1,-1}^{2}w^{-2}+a_{1,1}^{2}w^{2}\big{)}-\frac{2}{3}a\right)$
(C.15) $\displaystyle+$
$\displaystyle\sum_{k=4}^{\infty}\frac{1}{\tau^{k/3}}\sum_{j=-k+1}^{j=k-1}b_{k,j}w^{j},\qquad\quad
w={\tau}^{\frac{2}{3}(\tilde{\nu}+1)}\mathrm{e}^{\mathrm{i}\theta(\tau)},$

where the coefficients $b_{k,j}$ have similar expressions in terms of
$a_{1,\pm 1}$ as do the $a_{k,j}$’s. For our goal of determining the leading
term of asymptotics for $\rm{E}_{\varphi}(\tau)$, the coefficient $b_{4,0}$ is
important: it can be determined with the help of equations (C.9) and (C.10);
in fact, we determined all the coefficients at level $k=4$:

$\displaystyle b_{4,0}$ $\displaystyle=b_{4,\pm 2}=0,\qquad b_{4,\pm
3}=-\frac{5}{6}(\varepsilon b)^{1/3}a_{1,\pm 1}^{3},$ $\displaystyle b_{4,\pm
1}$ $\displaystyle=\mp\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm
1}}{108}\big{(}3(\alpha^{2}+8\varkappa\alpha+10\varkappa^{2}-1)\pm 12\alpha\pm
40\varkappa\big{)}.$

######  Proposition C.1.

$\mathrm{E}_{\varphi}(\tau)\underset{\tau\to+\infty}{=}\frac{2\sqrt{\tilde{\nu}+1}\,\mathrm{e}^{-3\pi\mathrm{i}/4}}{3^{3/4}(\varepsilon
b)^{1/6}\tau^{1/3}}\sinh\big{(}\mathrm{i}\theta(\tau)+(\tilde{\nu}+1)\ln\theta(\tau)+z\big{)}+\mathcal{O}\left(\frac{\mathrm{e}^{\pm
2\mathrm{i}\theta(\tau)}}{\tau^{\frac{2}{3}(1-2|\mathrm{Re}(\tilde{\nu}+1)|)}}\right)+\mathcal{O}\left(\frac{1}{\tau^{2/3}}\right),$
(C.16)

where $\theta(\tau)$ and $z$ are defined in Theorem C.1.

###### Proof.

For $\tau>0$, define the function $f(\tau)$ via

$\frac{b}{u(\tau)}=\frac{2(\varepsilon
b)^{1/3}}{\tau^{1/3}}-\frac{2(\varepsilon
b)^{1/3}}{\tau^{2/3}}\left(a_{1,-1}w^{-1}+a_{1,1}w\right)-\frac{2a}{3\tau}+f(\tau).$
(C.17)

This definition suggests the following estimates as $\tau\to+\infty$,

$f(\tau)\underset{\tau\to+\infty}{=}\mathcal{O}\left(\frac{w^{2}}{\tau}\right)+\mathcal{O}\left(\frac{w^{-2}}{\tau}\right)+\mathcal{O}\left(\frac{1}{\tau^{5/3}}\right),$
(C.18)
$\int_{+\infty}^{\tau}f(\tau)\,\mathrm{d}\tau\underset{\tau\to+\infty}{=}\mathcal{O}\left(\frac{\mathrm{e}^{\pm
2\mathrm{i}\theta(\tau)}}{\tau^{\frac{2}{3}(1-2|\text{Re}(\tilde{\nu}+1)|)}}\right)+\mathcal{O}\left(\frac{1}{\tau^{2/3}}\right),$
(C.19)

where the contour of integration is assumed to be taken along the positive
semi-axis, or, more generally, in the domain $\mathcal{D}$ described in Remark
C.4. The right-most estimates in the asymptotics (C.18) and (C.19) are
proportional to
$b_{5,0}=\big{(}\alpha^{2}-3(4\varkappa+\alpha)^{2}\big{)}/\big{(}2^{2}3^{3}(\varepsilon
b)^{1/3}\big{)}$.

Now, integrating equation (B.1) from $\tau_{0}$ to $\tau$ along the contour in
$\mathcal{D}$, and taking into account the definition of the function
$f(\tau)$ (cf. equation (C.17)), we obtain the following equation:

$\varphi(\tau)-\varphi(\tau_{0})=3(\varepsilon
b)^{1/3}\big{(}\tau^{2/3}-\tau_{0}^{2/3}\big{)}+\frac{4a}{3}\ln\frac{\tau}{\tau_{0}}-\int_{\tau_{0}}^{\tau}\frac{2(\varepsilon
b)^{1/3}}{\tau^{2/3}}\left(a_{1,-1}w^{-1}+a_{1,1}w\right)\,\mathrm{d}\tau+\int_{\tau_{0}}^{\tau}f(\tau)\,\mathrm{d}\tau.$
(C.20)

Substituting into (C.20) the asymptotics for $\varphi(\tau)$ given in (C.6)