1.2.1. Symmetries

In the setting that $\Psi(X,T;\mathbf{G},{B})$ arises from the joint near-field/high-order limit of rogue-wave solutions of (1.1), the parameter B > 0 has the interpretation of the amplitude of the background wave supporting the rogue waves. However, it is not hard to see that the dependence on B > 0 can be scaled out of Ψ by the scaling invariance $\Psi(X,T;\mathbf{G},{B}) \mapsto {B}^{-1} \Psi({B}^{-1}X, {B}^{-2} T;\mathbf{G},{B})$ of the focusing NLS equation (1.3).

Proposition 1.2 (Scaling symmetry)

Given G with $\det(\mathbf{G})=1$ and $\mathbf{G}^*=\sigma_2\mathbf{G}\sigma_2$, for each B > 0 and $(X,T)\in\mathbb{R}^2$, we have

(1.8)\begin{align} \Psi(X,T; \mathbf{G}, {B}) = {B} \Psi({B} X, {B}^2 T; \mathbf{G}, 1). \end{align}

We give a proof in Appendix A. In this paper we make use of Proposition 1.2 and take B = 1. Accordingly, we write

(1.9)\begin{align} \Psi(X,T;\mathbf{G}):= \Psi(X,T;\mathbf{G},{B}=1) \end{align}

to denote the special solution under study, and similarly we generally omit B = 1 from the argument list of P going forward. On the other hand, the dependence of $\Psi(X,T;\mathbf{G})$ on the $2\times 2$ matrix G is nontrivial.

We proceed with two observations that concern the symmetries with respect to reflections in the X variable and in the T variable.

Similarly, we have

The proofs of Proposition 1.3 and Proposition 1.4 are in Appendix A. The next observation we make concerns a useful normalization of the parameters $a,b$. Indeed, we have the following result, which is also proved in Appendix A.

Proposition 1.5 (Normalized parameters)

For all $(X,T)\in\mathbb{R}^2$ and $a,b\in\mathbb{C}$ with $ab\neq 0$,

(1.10)\begin{align} \Psi(X,T;\mathbf{G}(a,b))={\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\Psi\left(X,T;\mathbf{G}\left(\mathfrak{a},\mathfrak{b}\right)\right), \end{align}

where

(1.11)\begin{align} \mathfrak{a}:=\frac{|a|}{\sqrt{|a|^2+|b|^2}}\quad\text{and}\quad \mathfrak{b}:=\frac{|b|}{\sqrt{|a|^2+|b|^2}} \end{align}

satisfy $\mathfrak{a},\mathfrak{b} \gt 0$ with $\mathfrak{a}^2+\mathfrak{b}^2=1$.

Therefore, up to a phase factor, there is just one real parameter in the family of solutions $\Psi(X,T;\mathbf{G})$ with ab ≠ 0, which one could take as $\mathfrak{a}\in (0,1)$, or equivalently as an angle $\eta\in (0,\frac{1}{2}\pi)$ for which $\mathfrak{a}=\cos(\eta)$ and $\mathfrak{b}=\sin(\eta)$. The coordinate η was used, for example, in the analysis of [Reference Li and Miller24, Section 2.3]. Combining Proposition 1.4 and Proposition 1.5 for T = 0 shows that

(1.12)\begin{align} \begin{split} \Psi(X,0;\mathbf{G}(a,b))&={\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\Psi(X,0;\mathbf{G}(\mathfrak{a},\mathfrak{b}))\\ &={\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\Psi(X,0;\mathbf{G}(\mathfrak{a},\mathfrak{b}))^*\\ &={\mathrm{e}}^{-2{\mathrm{i}}\arg(ab)}\Psi(X,0;\mathbf{G}(a,b))^* \end{split} \end{align}

because $\mathbf{G}(\mathfrak{a},\mathfrak{b})$ is a real matrix. It follows that $X\mapsto {\mathrm{e}}^{{\mathrm{i}}\arg(ab)}\Psi(X,0;\mathbf{G}(a,b))$ is a real-valued function of $X\in\mathbb{R}$.

The normalized parameters $\mathfrak{a} \gt 0$ and $\mathfrak{b} \gt 0$ with $\mathfrak{a}^2+\mathfrak{b}^2=1$ will be used in the proofs of our asymptotic results to be described in Section 1.3 below. So that they are available later we record here the following four standard matrix factorizations of the central factor $\mathbf{G}(\mathfrak{a},\mathfrak{b})$ in the jump matrix (1.4), which have been further manipulated to have a diagonal matrix as the leftmost factor:

(1.13)\begin{align} \mathbf{G}(\mathfrak{a},\mathfrak{b})=\begin{bmatrix}\mathfrak{a} & \mathfrak{b}\\-\mathfrak{b} & \mathfrak{a}\end{bmatrix}=\mathfrak{a}^{\sigma_3}\begin{bmatrix}1 & 0\\-\mathfrak{ab} & 1\end{bmatrix}\begin{bmatrix}1 & \displaystyle\frac{\mathfrak{b}}{\mathfrak{a}}\\0 & 1\end{bmatrix},\quad \text{(\unicode{x201C}LDU\unicode{x201C})}, \end{align}

(1.14)\begin{align} \mathbf{G}(\mathfrak{a},\mathfrak{b})=\begin{bmatrix}\mathfrak{a} & \mathfrak{b}\\-\mathfrak{b} & \mathfrak{a}\end{bmatrix}=\mathfrak{a}^{-\sigma_3}\begin{bmatrix}1 & \mathfrak{ab}\\0 & 1\end{bmatrix}\begin{bmatrix}1 & 0\\\displaystyle-\frac{\mathfrak{b}}{\mathfrak{a}} & 1\end{bmatrix},\quad\text{(\unicode{x201C}UDL\unicode{x201D})}, \end{align}

(1.15)\begin{align} \mathbf{G}(\mathfrak{a},\mathfrak{b})=\begin{bmatrix}\mathfrak{a} & \mathfrak{b}\\-\mathfrak{b} & \mathfrak{a}\end{bmatrix}=\mathfrak{a}^{\sigma_3}\begin{bmatrix}1&0\\ \displaystyle\frac{\mathfrak{a}^3}{\mathfrak{b}} & 1\end{bmatrix} \begin{bmatrix}0&\displaystyle \frac{\mathfrak{b}}{\mathfrak{a}}\\\displaystyle -\frac{\mathfrak{a}}{\mathfrak{b}} & 0\end{bmatrix} \begin{bmatrix}1 & 0\\\displaystyle\frac{\mathfrak{a}}{\mathfrak{b}} & 1\end{bmatrix},\quad \text{(\unicode{x201C}LTL\unicode{x201D})}, \end{align}

(1.16)\begin{align} \mathbf{G}(\mathfrak{a},\mathfrak{b})=\begin{bmatrix}\mathfrak{a} & \mathfrak{b}\\-\mathfrak{b} & \mathfrak{a}\end{bmatrix}=\mathfrak{a}^{-\sigma_3}\begin{bmatrix}1 & \displaystyle -\frac{\mathfrak{a}^3}{\mathfrak{b}}\\0 & 1\end{bmatrix} \begin{bmatrix}0 & \displaystyle\frac{\mathfrak{a}}{\mathfrak{b}}\\\displaystyle-\frac{\mathfrak{b}}{\mathfrak{a}} & 0\end{bmatrix}\begin{bmatrix}1 & \displaystyle-\frac{\mathfrak{a}}{\mathfrak{b}}\\0 & 1\end{bmatrix},\quad \text{(\unicode{x201C}UTU\unicode{x201D})}. \end{align}

1.2.2. $\Psi(X,0;\mathbf{G})$ near $X$ = 0

It is straightforward to solve Riemann–Hilbert Problem 1 explicitly when $(X,T)=(0,0)$. Indeed, one can verify directly that the solution is:

(1.17)\begin{align} \mathbf{P}(\Lambda;0,0,\mathbf{G}) =\begin{cases} \mathbf{G}^{-1},& |\Lambda| \lt 1,\\ \mathbf{G}^{-1}{\mathrm{e}}^{-2{\mathrm{i}}\Lambda^{-1}\sigma_3}\mathbf{G}{\mathrm{e}}^{2{\mathrm{i}}\Lambda^{-1}\sigma_3},&|\Lambda| \gt 1. \end{cases} \end{align}

Then, it follows from this formula assuming $|\Lambda| \gt 1$ that $\mathbf{P}(\Lambda;0,0,\mathbf{G})=\mathbb{I} + (2{\mathrm{i}}\sigma_3 - 2{\mathrm{i}}\mathbf{G}^{-1}\sigma_3\mathbf{G})\Lambda^{-1} + O(\Lambda^{-2})$ as $\Lambda\to\infty$. Therefore (1.6) yields the following.

Theorem 1.6 (Value at the origin)

(1.18)\begin{align} \Psi(0,0;\mathbf{G})=4\left(\mathbf{G}^{-1}\sigma_3\mathbf{G}\right)_{12} = 8\mathfrak{a}\mathfrak{b}{\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}=\frac{8a^*b^*}{|a|^2+|b|^2}. \end{align}

Following [Reference Li and Miller24, Section 2.3.3], it is then systematic to calculate derivatives of $\mathbf{P}(\Lambda;X,0,\mathbf{G})$ with respect to X at X = 0. For instance, setting $\mathbf{F}(\Lambda;X):=\mathbf{P}_X(\Lambda;X,0,\mathbf{G})\mathbf{P}(\Lambda;X,0,\mathbf{G})^{-1}$, one sees that $\Lambda\mapsto\mathbf{F}(\Lambda;X)$ is analytic for $|\Lambda|\neq 1$, that $\mathbf{F}(\Lambda;X)\to \mathbf{0}$ as $\Lambda\to\infty$, and that $\mathbf{P}_+(\Lambda;X,0,\mathbf{G})=\mathbf{P}_-(\Lambda;X,0,\mathbf{G})\mathbf{V}(\Lambda;X)$ implies that also

(1.19)\begin{align} \mathbf{F}_+(\Lambda;X)-\mathbf{F}_-(\Lambda;X)=\mathbf{P}_-(\Lambda;X,0,\mathbf{G})\mathbf{V}_X(\Lambda;X)\mathbf{V}(\Lambda;X)^{-1}\mathbf{P}_-(\Lambda;X,0,\mathbf{G})^{-1},\quad |\Lambda|=1. \end{align}

It follows that (using the Plemelj formula and taking into account the clockwise orientation of the jump contour)

(1.20)\begin{align} \begin{split} \mathbf{F}(\Lambda;X)&=-\frac{1}{2\pi{\mathrm{i}}}\oint_{|\mu|=1}\frac{\mathbf{P}_-(\mu;X,0,\mathbf{G})\mathbf{V}_X(\mu;X)\mathbf{V}(\mu;X)^{-1}\mathbf{P}_-(\mu;X,0,\mathbf{G})^{-1}}{\mu-\Lambda}\, \mathrm{d}\mu\\ &=\frac{1}{2\pi{\mathrm{i}}\Lambda}\oint_{|\mu|=1}\mathbf{P}_-(\mu;X,0,\mathbf{G})\mathbf{V}_X(\mu;X)\mathbf{V}(\mu;X)^{-1}\mathbf{P}_-(\mu;X,0,\mathbf{G})^{-1}\, \mathrm{d}\mu + O(\Lambda^{-2}) \end{split} \end{align}

as $\Lambda\to\infty$, where on both lines the integration contour has counterclockwise orientation, and where $\mathbf{P}_-(\mu;X,0,\mathbf{G})$ refers to the boundary value taken from the interior of the unit circle. Since according to (1.4) the jump matrix is given by $\mathbf{V}(\Lambda;X):={\mathrm{e}}^{-{\mathrm{i}}(\Lambda X + 2\Lambda^{-1})\sigma_3}\mathbf{G}{\mathrm{e}}^{{\mathrm{i}} (\Lambda X+2\Lambda^{-1})\sigma_3}$, we get that

(1.21)\begin{align} \mathbf{V}_X(\Lambda;0)\mathbf{V}(\Lambda;0)^{-1}=-{\mathrm{i}}\Lambda\sigma_3 +{\mathrm{i}}\Lambda {\mathrm{e}}^{-2{\mathrm{i}}\Lambda^{-1}\sigma_3}\mathbf{G}\sigma_3\mathbf{G}^{-1}{\mathrm{e}}^{2{\mathrm{i}}\Lambda^{-1}\sigma_3}, \end{align}

and according to (1.17) we have $\mathbf{P}_-(\Lambda;0,0,\mathbf{G})=\mathbf{G}^{-1}$. Therefore, as $\Lambda\to\infty$,

(1.22)\begin{align} \mathbf{F}(\Lambda;0)=\frac{1}{2\pi{\mathrm{i}}\Lambda}\oint_{|\mu|=1}\left[-{\mathrm{i}}\mu\mathbf{G}^{-1}\sigma_3\mathbf{G} + {\mathrm{i}}\mu \mathbf{G}^{-1}{\mathrm{e}}^{-2{\mathrm{i}}\mu^{-1}\sigma_3}\mathbf{G}\sigma_3\mathbf{G}^{-1}{\mathrm{e}}^{2{\mathrm{i}}\mu^{-1}\sigma_3}\mathbf{G}\right]\, \mathrm{d}\mu + O(\Lambda^{-2}). \end{align}

The first term vanishes by Cauchy’s theorem, and the second term can be evaluated by residues at $\mu=\infty$ using the expansion ${\mathrm{e}}^{\pm 2{\mathrm{i}}\mu^{-1}\sigma_3}=\mathbb{I} \pm 2{\mathrm{i}}\sigma_3\mu^{-1}-2\mathbb{I}\mu^{-2} + O(\mu^{-3})$ as $\mu\to\infty$. The result is that

(1.23)\begin{align} \mathbf{F}(\Lambda;0)=\left[4{\mathrm{i}}\mathbf{G}^{-1}\sigma_3\mathbf{G}\sigma_3\mathbf{G}^{-1}\sigma_3\mathbf{G}-4{\mathrm{i}}\sigma_3\right]\Lambda^{-1}+O(\Lambda^{-2}),\quad\Lambda\to\infty. \end{align}

Differentiation of (1.6) then yields

(1.24)\begin{align} \Psi_X(0,0;\mathbf{G}) = 2{\mathrm{i}}\lim_{\Lambda\to\infty}\Lambda\frac{\partial P_{12}}{\partial X}(\Lambda;0,0,\mathbf{G}) = 2{\mathrm{i}}\lim_{\Lambda\to\infty}\Lambda F_{12}(\Lambda;0) = -8\left[\mathbf{G}^{-1}\sigma_3\mathbf{G}\sigma_3\mathbf{G}^{-1}\sigma_3\mathbf{G}\right]_{12}. \end{align}

Explicit evaluation using (1.5) then yields the following result.

Theorem 1.7 (Derivative at the origin)

(1.25)\begin{align} \Psi_X(0,0;\mathbf{G})=32{\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\mathfrak{a}\mathfrak{b} (\mathfrak{b}^2-\mathfrak{a}^2)=32a^*b^*\frac{|b|^2-|a|^2}{(|a|^2+|b|^2)^2}. \end{align}

1.2.3. Exceptional parameter values and $L^2$-norm

Another interesting result for the family $\Psi(X,T;\mathbf{G})$ of exact solutions of (1.3) has to do with special values of the parameters.

We give a proof of Proposition 1.8 in Appendix A. The proof relies on the fact that Riemann–Hilbert Problem 1 can be solved explicitly in either of the cases a = 0 or b = 0. It follows from Proposition 1.1 that the function $\Psi(X,T;\mathbf{G})$ depends continuously on the parameters (a, b) for fixed $(X,T)\in\mathbb{R}^2$, so Proposition 1.8 also implies the pointwise limit $\Psi(X,T;\mathbf{G})\to 0$ as a → 0 or b → 0 for each $(X,T)\in\mathbb{R}^2$, and this convergence can be generalized to be uniform over (X, T) ranging over any given compact set in $\mathbb{R}^2$. To avoid trivial cases, from this point on in the paper we therefore assume that $a,b$ are complex numbers with ab ≠ 0.

Another result is that for general $\mathbf{G}(a,b)$ with ab ≠ 0, $\Psi(X,T;\mathbf{G})$ lies in $L^2(\mathbb{R})$ as a function of X with an $L^2(\mathbb{R})$-norm that is independent of the parameter matrix $\mathbf{G}=\mathbf{G}(a,b)$. Namely, we prove the following theorem.

We prove Theorem 1.9 in Section 2.4 essentially as a corollary of Theorem 1.18 below. When combined with the degeneration property given in Proposition 1.8, the independence of the L ²-norm of $\Psi(X,T,\mathbf{G})$ from the matrix $\mathbf{G}=\mathbf{G}(a,b)$ asserted in Theorem 1.9 leaves us with an interesting conundrum! While $\|\Psi(\diamond,T;\mathbf{G}(a,b))\|_{L^2(\mathbb{R})}=\sqrt{8}$ for any nonzero $a,b\in\mathbb{C}$, we have

(1.26)\begin{align} \lim_{a\to 0}\Psi(X,T;\mathbf{G}(a,b)) = 0\quad\text{and}\quad \lim_{b\to 0}\Psi(X,T;\mathbf{G}(a,b)) = 0 \end{align}

pointwise for any $(X,T)\in\mathbb{R}^2$. A mechanism for this limiting behaviour could be that the $L^2$-mass of the wave packet $\Psi(X,T;\mathbf{G}(a,b))$ at any given time T escapes to $X=\pm\infty$ in the limit a → 0 or b → 0, or alternately, the wave packet spreads in the same limit so as to preserve the L ²-norm while still decaying pointwise or perhaps even uniformly to zero. In fact, it turns out that a combination of both of these mechanisms is at play. The explanation of this phenomenon lies in a double-scaling limit in which $X\to+\infty$ while also a → 0 or b → 0 at suitably related rates. The details can be found in our next paper on the subject, [Reference Bilman and Miller14].

1.2.4. Differential equations

It is straightforward to derive three different first-order systems of differential equations satisfied by the matrix function

(1.27)\begin{align} \mathbf{W}(\Lambda;X,T,\mathbf{G}):=\mathbf{P}(\Lambda;X,T,\mathbf{G}){\mathrm{e}}^{-{\mathrm{i}} (\Lambda X+\Lambda^2T+2\Lambda^{-1})\sigma_3} \end{align}

by following the procedure in [Reference Bilman, Ling and Miller8, Section 3.2.1], which though written with the special case (1.7) in mind, in fact does not depend at all on the details of the matrix $\mathbf{G}(a,b)$ and hence applies to general rogue waves of infinite order. It follows that the matrix $\mathbf{W}(\Lambda;X,T,\mathbf{G})$ satisfies the three Lax equations

(1.28)\begin{align} \frac{\partial \mathbf{W}}{\partial X} = \mathbf{X}\mathbf{W},\quad\frac{\partial \mathbf{W}}{\partial T}=\mathbf{T}\mathbf{W},\quad\frac{\partial \mathbf{W}}{\partial\Lambda}=\boldsymbol{\Lambda}\mathbf{W}, \end{align}

wherein the coefficient matrices X, T, and $\boldsymbol{\Lambda}$ are explicitly represented in terms of the coefficients $\mathbf{P}^{[j]}=\mathbf{P}^{[j]}(X,T;\mathbf{G})$ in the convergent Laurent expansion

(1.29)\begin{align} \mathbf{P}(\Lambda;X,T,\mathbf{G})=\mathbb{I}+\sum_{j=1}^\infty\mathbf{P}^{[j]}(X,T;\mathbf{G})\Lambda^{-j},\quad |\Lambda| \gt 1 \end{align}

or alternatively in terms of the Taylor coefficients $\mathbf{P}_0:=\mathbf{P}(0;X,T,\mathbf{G})$ etc., of $\mathbf{P}(\Lambda;X,T,\mathbf{G})$ at $\Lambda=0$. Thus:

(1.30)\begin{align} \mathbf{X} = -{\mathrm{i}}\Lambda\sigma_3 + {\mathrm{i}} [\sigma_3,\mathbf{P}^{[1]}] = \begin{bmatrix}-{\mathrm{i}}\Lambda & \Psi\\-\Psi^* & {\mathrm{i}}\Lambda\end{bmatrix}, \end{align}

(1.31)\begin{align} \mathbf{T}=-{\mathrm{i}}\Lambda^2\sigma_3 + {\mathrm{i}} [\sigma_3,\mathbf{P}^{[1]}]\Lambda + {\mathrm{i}}[\mathbf{P}^{[1]},\sigma_3\mathbf{P}^{[1]}] + {\mathrm{i}}[\sigma_3,\mathbf{P}^{[2]}] = \begin{bmatrix} -{\mathrm{i}}\Lambda^2 +\frac{1}{2}{\mathrm{i}} |\Psi|^2 & \Lambda\Psi + \frac{1}{2}{\mathrm{i}}\Psi_X\\ -\Lambda\Psi^* +\frac{1}{2}{\mathrm{i}}\Psi^*_X & {\mathrm{i}}\Lambda^2 -\frac{1}{2}{\mathrm{i}} |\Psi|^2\end{bmatrix}, \end{align}

and

(1.32)\begin{align} \boldsymbol{\Lambda}=\begin{bmatrix}-2{\mathrm{i}} T\Lambda -{\mathrm{i}} X +{\mathrm{i}} T|\Psi|^2\Lambda^{-1} & 2T\Psi +(X\Psi+{\mathrm{i}} T\Psi_X)\Lambda^{-1}\\-2T\Psi^*+(-X\Psi^*+{\mathrm{i}} T\Psi_X^*)\Lambda^{-1} & 2{\mathrm{i}} T\Lambda +{\mathrm{i}} X-{\mathrm{i}} T|\Psi|^2\Lambda^{-1}\end{bmatrix} + 2{\mathrm{i}} \mathbf{P}_0\sigma_3\mathbf{P}_0^{-1}\Lambda^{-2}. \end{align}

The global existence of $\mathbf{P}(\Lambda;X,T,\mathbf{G})$ from Riemann–Hilbert Problem 1 recorded in Proposition 1.1 then guarantees that the three Lax equations (1.28) are mutually compatible. In particular, the compatibility condition $\mathbf{X}_T-\mathbf{T}_X + [\mathbf{X},\mathbf{T}]=\mathbf{0}$ implies the following basic result which has already been mentioned.

On the other hand, the compatibility condition $\mathbf{X}_\Lambda-\boldsymbol{\Lambda}_X + [\mathbf{X},\boldsymbol{\Lambda}]=\mathbf{0}$ is a system of ordinary differential equations with respect to X (in which $T\in\mathbb{R}$ plays the role of a parameter) that was shown in [Reference Bilman, Ling and Miller8, Section 3.2.1] to be related to the second equation in the Painlevé-III hierarchy of Sakka [Reference Sakka32] when T ≠ 0 and to the Painlevé-III (D₆) equation itself when T = 0. More explicitly, we have the following.

Theorem 1.11 ( $\Psi(X,0)$ and the Painlevé-III (D₆) equation [Reference Bilman, Ling and Miller8, Corollary 4])

The function $u(x)$ defined for $x\in\mathbb{R}\cup({\mathrm{i}}\mathbb{R})$ by

(1.33)\begin{align} u(x):=2\left(\frac{ \mathrm{d}}{ \mathrm{d} x}\log\left(x^2\Psi(-\tfrac{1}{8}x^2,0;\mathbf{G})\right)\right)^{-1} \end{align}

is a solution of the Painlevé-III (D₆) equation

(1.34)\begin{align} \frac{ \mathrm{d}^2u}{ \mathrm{d} x^2}=\frac{1}{u}\left(\frac{ \mathrm{d} u}{ \mathrm{d} x}\right)^2-\frac{1}{x}\frac{ \mathrm{d} u}{ \mathrm{d} x} + \frac{4\Theta_0 u^2 + 4(1-\Theta_\infty)}{x}+4u^3-\frac{4}{u} \end{align}

in the case that both formal monodromy parameters vanish: $\Theta_0=\Theta_\infty=0$.

Corollary 1.12 (Behaviour of $u(x)$ near $x=0$)

The function $u(x)$ defined by (1.33) is an odd function of $x$ that is analytic at the origin with Taylor expansion

(1.35)\begin{align} u(x)=x + \frac{u'''(0)}{3!}x^3 + O(x^5),\quad x\to 0,\quad u'''(0)=3(\mathfrak{b}^2-\mathfrak{a}^2)=3\frac{|b|^2-|a|^2}{|a|^2+|b|^2}. \end{align}

Since the value of $\Psi(0,0;\mathbf{G})$ is known from Theorem 1.6, it is straightforward to invert (1.33) to explicitly express $\Psi(X,0;\mathbf{G})$ in terms of u:

(1.36)\begin{align} \Psi(X,0;\mathbf{G})=\Psi(0,0;\mathbf{G})\exp\left(2\int_0^x\left[\frac{1}{u(y)}-\frac{1}{y}\right]\, \mathrm{d} y\right),\quad X=-\frac{1}{8}x^2. \end{align}

We note that $-3 \lt u'''(0) \lt 3$. In fact, when $\Theta_0=\Theta_\infty=0$ there is for each $\omega\in (-3,3)$ a unique solution of (1.34) analytic at the origin with $u(0)=0$, $u'(0)=1$, $u''(0)=0$, and $u'''(0)=\omega$. This family of solutions of the Painlevé-III (D₆) equation has not only been associated with limits of sequences of solutions of the focusing NLS equation [Reference Bilman and Buckingham7, Reference Bilman, Ling and Miller8], but has also appeared in the description of self-similar boundary layers in the sharp-line Maxwell-Bloch equations [Reference Li and Miller24].

Remark 1.13. The parametrization of solutions of (1.34) is discussed for example in [Reference Barhoumi, Lisovyy, Miller and Prokhorov3, Section 4.6] (see also [Reference van der Put and Saito44]). Solutions are parametrized by points on a certain monodromy manifold characterized by a cubic equation in three variables. In particular, the monodromy manifold for the Painlevé-III (D₆) equation (1.34) with parameters $\Theta_0=\Theta_\infty=0$ consists of those $(x_1,x_2,x_3)\in \mathbb{C}^3$ for which

(1.37)\begin{align} x_1x_2x_3+x_1^2+x_2^2+2x_1+2x_2+1=0. \end{align}

This is a smooth complex surface except at two points obtained by also enforcing that the gradient of the left-hand side vanishes: $(x_1,x_2,x_3)=(0,-1,2)$ and $(x_1,x_2,x_3)=(-1,0,2)$. Because the Stokes phenomenon is trivial both at $\Lambda=0$ and $\Lambda=\infty$ (nontrivial Stokes phenomenon would be evident in additional jump conditions for Riemann–Hilbert Problem 1 on two contours approaching the origin and two contours approaching $\Lambda=\infty$), the coordinates of the Painlevé-III (D₆) solution arising for T = 0 from Riemann–Hilbert Problem 1 are determined from the matrix G alone (which plays the role of a connection matrix in the isomonodromy theory) by $(x_1,x_2,x_3)=(G_{11}G_{22}-1,-G_{11}G_{22},2)=(\mathfrak{a}^2-1,-\mathfrak{a}^2,2)$. Since $0 \lt \mathfrak{a}^2 \lt 1$, this is a segment of a line in $\mathbb{C}^3$ parametrized by $G_{11}G_{22}=\mathfrak{a}^2$ that is fully contained within the monodromy manifold (1.37). The line passes through both singular points at parameter values $G_{11}G_{22}=\mathfrak{a}^2=0$ and $G_{11}G_{22}=\mathfrak{a}^2=1$, which are the endpoints of the relevant segment. Thus, the endpoints of the segment correspond to normalized parameters $(\mathfrak{a},\mathfrak{b})=(0,1)$ or $(\mathfrak{a},\mathfrak{b})=(1,0)$ respectively. The singular points on the monodromy manifold then both correspond to the trivial solution $\Psi(X,T;\mathbf{G})\equiv 0$ according to Proposition 1.8. However, each interior point of the segment yields a distinct nontrivial solution $u(x)=u(x;\mathfrak{a})$ of the Painlevé-III (D₆) equation (1.34) with $\Theta_0=\Theta_\infty=0$ related to $\Psi(X,0;\mathbf{G})$ via (1.33) and its inverse (1.36). Note that the presence of the logarithmic derivative in (1.33) cancels the constant phase factor ${\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}$ in $\Psi(X,0;\mathbf{G}(a,b))$ (see (1.12)) so that while Ψ depends on the parameters (a, b), u indeed depends only on the normalized parameters $(\mathfrak{a},\mathfrak{b})$.

The compatibility condition $\mathbf{T}_\Lambda-\boldsymbol{\Lambda}_T + [\mathbf{T},\boldsymbol{\Lambda}]=\mathbf{0}$ is a system of ordinary differential equations with respect to T in which $X\in\mathbb{R}$ is a parameter. This system was written out in general in [Reference Bilman, Ling and Miller8, Eqn. (119)], and here we expand on a remark made in that paper concerning a symmetric special case (see also [Reference Suleimanov33]). Suppose that b = a. Then according to Proposition 1.3, $X\mapsto\Psi(X,T;\mathbf{G}(a,a))$ is an even function for all $T\in\mathbb{R}$, which in light of real analyticity at X = 0 implies also that

(1.38)\begin{align} \Psi_X(0,T;\mathbf{G}(a,a))=0 \end{align}

as is consistent with the conclusion of Theorem 1.7 when also T = 0. Thus X = 0 is an axis of symmetry of $\Psi(X,T;\mathbf{G}(a,a))$, and one may expect some simplification of the compatibility condition $\mathbf{T}_\Lambda-\boldsymbol{\Lambda}_T+[\mathbf{T},\boldsymbol{\Lambda}]=\mathbf{0}$ yielding ordinary differential equations satisfied by $T\mapsto \Psi(X,T;\mathbf{G})$. To see this, we remark that Proposition A.1 in Appendix A implies that $\mathbf{P}(0;0,T,\mathbf{G}(a,a))=-\sigma_3\mathbf{P}(0;0,T,\mathbf{G}(a,a))\sigma_1$, and therefore for some function $s(T)\neq 0$ the unit-determinant matrix $\mathbf{P}_0=\mathbf{P}(0;0,T,\mathbf{G}(a,a))$ has the form

(1.39)\begin{align} \mathbf{P}_0=\begin{bmatrix} s(T) & -s(T)\\ (2s(T))^{-1} & (2s(T))^{-1}\end{bmatrix}\implies 2{\mathrm{i}}\mathbf{P}_0\sigma_3\mathbf{P}_0^{-1}=2{\mathrm{i}}\begin{bmatrix}0 & n(T)\\n(T)^{-1} & 0\end{bmatrix},\quad n(T):=2s(T)^2. \end{align}

Using this and (1.38), and setting X = 0, the matrix coefficients T and $\boldsymbol{\Lambda}$ defined in (1.31) and (1.32) respectively take the simplified form

(1.40)\begin{align} \mathbf{T}=-{\mathrm{i}}\Lambda^2\sigma_3 +\begin{bmatrix}0 & \Psi\\-\Psi^* & 0\end{bmatrix}\Lambda +\frac{1}{2}{\mathrm{i}} |\Psi|^2\sigma_3 \end{align}

and

(1.41)\begin{align} \boldsymbol{\Lambda}=-2{\mathrm{i}} T\Lambda\sigma_3 +\begin{bmatrix}0 & 2T\Psi\\-2T\Psi^* & 0\end{bmatrix}+{\mathrm{i}} T|\Psi|^2\Lambda^{-1}\sigma_3 +\frac{2{\mathrm{i}}}{\Lambda^2}\begin{bmatrix} 0 & n(T)\\n(T)^{-1} & 0\end{bmatrix}, \end{align}

where $\Psi=\Psi(T):=\Psi(0,T;\mathbf{G}(a,a))$. In [Reference Kitaev and Vartanian22] a Lax pair is presented for the partially-degenerate Painlevé-III equation (D₇ type, with parameters $\mathscr{A},\mathscr{B}\in\mathbb{C}$ and $\varepsilon=\pm 1$)

(1.42)\begin{align} \frac{ \mathrm{d}^2\mathfrak{u}}{ \mathrm{d} t^2}=\frac{1}{\mathfrak{u}}\left(\frac{ \mathrm{d}\mathfrak{u}}{ \mathrm{d} t}\right)^2 -\frac{1}{t}\frac{ \mathrm{d}\mathfrak{u}}{ \mathrm{d} t} + \frac{-8\varepsilon\mathfrak{u}^2+2\mathscr{A}\mathscr{B}}{t}+\frac{\mathscr{B}^2}{\mathfrak{u}},\quad\mathfrak{u}=\mathfrak{u}(t), \end{align}

that, after a Fabry transformation, resembles the compatible linear system $\mathbf{W}_T=\mathbf{T W}$ and $\mathbf{W}_\Lambda=\boldsymbol{\Lambda} \mathbf{W}$ with the coefficient matrices given in (1.40) and (1.41). Hence we may expect that $\Psi(T)$ is related to a solution of the Painlevé-III (D₇) equation. To complete the correspondence, we change variables and make a gauge transformation by

(1.43)\begin{align} \lambda:= T^\frac{1}{4}\Lambda,\quad t:=T^{\frac{1}{2}},\quad \mathbf{W}=t^{\frac{1}{2}\sigma_3}\mathbf{V}, \end{align}

and then the resulting system takes the form

(1.44)\begin{align} \frac{\partial\mathbf{V}}{\partial\lambda}=\left(-2{\mathrm{i}} t\lambda\sigma_3 + 2t\begin{bmatrix} 0 & t^{-\frac{1}{2}}\Psi\\-t^\frac{3}{2}\Psi^* & 0\end{bmatrix}-\frac{1}{\lambda}\left(-{\mathrm{i}} t^2|\Psi|^2\right)\sigma_3 +\frac{1}{\lambda^2}\begin{bmatrix}0 & 2{\mathrm{i}} t^{-\frac{1}{2}}n\\ 2{\mathrm{i}} t^\frac{3}{2}n^{-1} & 0\end{bmatrix}\right)\mathbf{V} \end{align}

and

(1.45)\begin{align} \frac{\partial\mathbf{V}}{\partial t}=\left(-{\mathrm{i}}\lambda^2\sigma_3 + \lambda\begin{bmatrix}0 & t^{-\frac{1}{2}}\Psi\\ -t^\frac{3}{2}\Psi^* & 0\end{bmatrix} + \left(\frac{1}{2}{\mathrm{i}} t|\Psi|^2-\frac{1}{2t}\right)\sigma_3 -\frac{1}{2t\lambda}\begin{bmatrix}0 & 2{\mathrm{i}} t^{-\frac{1}{2}}n\\2{\mathrm{i}} t^\frac{3}{2}n^{-1} & 0\end{bmatrix}\right)\mathbf{V}, \end{align}

in which we now view Ψ, $\Psi^*$, and n as functions of t. This system has exactly the form of [Reference Kitaev and Vartanian22, Eqn. (12)] provided we make the correspondences

(1.46)\begin{align} D=t^{\frac{3}{2}}\Psi^*,\quad \frac{2{\mathrm{i}} A}{\sqrt{-AB}}=t^{-\frac{1}{2}}\Psi, \end{align}

(1.47)\begin{align} {\mathrm{i}} \mathscr{A}+\frac{1}{2}+\frac{2t AD}{\sqrt{-AB}}=-{\mathrm{i}} t^2|\Psi|^2,\quad \frac{{\mathrm{i}} \mathscr{A}}{2t}-\frac{AD}{\sqrt{-AB}}=\frac{1}{2}{\mathrm{i}} t|\Psi|^2-\frac{1}{2t}, \end{align}

(1.48)\begin{align} \widetilde{\alpha}=2{\mathrm{i}} t^{-\frac{1}{2}}n,\quad {\mathrm{i}} t B=2{\mathrm{i}} t^\frac{3}{2}n^{-1}. \end{align}

where the notation of [Reference Kitaev and Vartanian22] is on the left-hand sideFootnote ² and $A,B,C,D,\widetilde{\alpha}$ are functions of t while $\mathscr{A}$ is a constant. Using the product of the identities in (1.46) to eliminate $AD/\sqrt{-AB}$ shows that the two equations in (1.47) actually coincide, and yield $\mathscr{A}=\frac{1}{2}{\mathrm{i}}$. Then, according to [Reference Kitaev and Vartanian22, Lemma 2.1], the second parameter $\mathscr{B}$ is determined from (1.48) up to a sign by $\mathscr{B}=4\varepsilon$. The corresponding solution of (1.42) is then given by $\mathfrak{u}(t)=\varepsilon t\sqrt{-A(t)B(t)}$. This proves the following, which is also easy to verify directly from the compatibility condition for the system (1.44)–(1.45).

Theorem 1.14 ( $\Psi(0,T)$ for $b=a$ and the Painlevé-III (D₇) equation)

Fix $a\in\mathbb{C}$ nonzero, and let $\varepsilon=\pm 1$. The function

(1.49)\begin{align} \mathfrak{u}(t):=\frac{1}{4}{\mathrm{i}}\varepsilon t\Psi\cdot\left(\Psi^*+t\frac{ \mathrm{d}\Psi^*}{ \mathrm{d} t}\right),\quad t\in\mathbb{R},\quad\Psi=\Psi(0,t^2;\mathbf{G}(a,a)) \end{align}

satisfies the Painlevé-III (D₇) equation in the form (1.42) with parameters $\mathscr{A}=\frac{1}{2}{\mathrm{i}}$ and $\mathscr{B}=4\varepsilon$.

This result was known to Suleimanov [Reference Suleimanov33], although to extract it from his paper one must take the independent variable to be $t=T^\frac{1}{2}$ instead of T and correct some constants. Another simple identity satisfied by Ψ that follows from the compatibility condition for (1.44)–(1.45) is

(1.50)\begin{align} \left|\Psi+t\frac{ \mathrm{d}\Psi}{ \mathrm{d} t}\right|^2=16,\quad t\in\mathbb{R},\quad\Psi=\Psi(0,t^2;\mathbf{G}(a,a)). \end{align}

Corollary 1.15 (Behaviour of $\mathfrak{u}(t)$ near $t=0)$

The function $\mathfrak{u}(t)$ defined by (1.49) is an odd function of $t$ that is analytic at the origin with Taylor expansion

(1.51)\begin{align} \mathfrak{u}(t)=\mathfrak{u}'(0)t + O(t^3),\quad t\to 0,\quad \mathfrak{u}'(0)=4{\mathrm{i}}\varepsilon. \end{align}

According to [Reference Kitaev and Vartanian23], the conditions on $\mathfrak{u}(t)$ asserted in Corollary 1.15 actually uniquely determine the solution of (1.42) because the formal monodromy parameter has the special value $\mathscr{A}=\frac{1}{2}{\mathrm{i}}$. This value of $\mathscr{A}$ is special (so is $\mathscr{A}=\pm\frac{1}{2}{\mathrm{i}} + k$ for any $k\in\mathbb{Z}$) in that there is a one-parameter family of solutions that are analytic and vanishing at t = 0, but the solution becomes unique once oddness is asserted. On the other hand, for general $\mathscr{A}$ there is only one solution analytic and vanishing at t = 0.

1.2.5. Fredholm determinant representation of $\Psi(X,0;\mathbf{G})$

In [Reference Bilman, Ling and Miller8, Section 3.2.1] various transformations of (1.34) to other forms of the Painlevé-III equation were noted, including a transformation to the parameter-free and fully-degenerate Painlevé-III (D₈) equation, which takes the form

(1.52)\begin{align} \frac{ \mathrm{d}^2U}{ \mathrm{d} z^2} = \frac{1}{U}\left(\frac{ \mathrm{d} U}{ \mathrm{d} z}\right)^2-\frac{1}{z}\frac{ \mathrm{d} U}{ \mathrm{d} z}+\frac{4U^2+4}{z}. \end{align}

Here we report a new piece of information, which is that the relevant solution of (1.52) is expressible in terms of a Fredholm determinant of an integrable operator [Reference Deift19, Reference Its, Izergin, Korepin and Slavnov20]. This in turn leads to an explicit representation of $\Psi(X,0;\mathbf{G})$ in terms of the same determinant.

Provided that a > 0 and $b\in{\mathrm{i}}\mathbb{R}$, the matrix $\hat{\mathbf{S}}(\xi;z)$ defined in terms of $\mathbf{P}(\Lambda;X,0,\mathbf{G})$ by

(1.53)\begin{align} \hat{\mathbf{S}}(\xi;z):=\mathbf{P}(\Lambda;X,0,\mathbf{G}){\mathrm{e}}^{(2z)^{1/2}({\mathrm{i}}\xi +\xi^{-1})\sigma_3},\quad X=-{\mathrm{i}} z,\quad \Lambda=-2{\mathrm{i}} (2z)^{-\frac{1}{2}}\xi, \end{align}

satisfies a related Riemann–Hilbert problem associated with the Painlevé-III (D ₈) equation with specialized monodromy parameters $y_1=b/\sqrt{|a|^2+|b|^2}$, $y_2={\mathrm{i}} a/\sqrt{|a|^2+|b|^2}$, and $y_3=0$ (see [Reference Barhoumi, Lisovyy, Miller and Prokhorov3, Riemann–Hilbert Problem 9.2] and the discussion in Section 9.3 of that paper). Comparing (1.6) with [Reference Barhoumi, Lisovyy, Miller and Prokhorov3, Eqns. (9.11) and (9.14)] shows that

(1.54)\begin{align} \Psi(X,0;\mathbf{G})=\frac{{\mathrm{i}}}{2}\frac{U'(z)}{U(z)},\quad z={\mathrm{i}} X \end{align}

where U(z) is a solution of the Painlevé-III (D ₈) equation (1.52). The parameters y ₁ and y ₂ are purely imaginary numbers constrained by $y_1^2+y_2^2+1=0$ (a reduction for $y_3=0$ of the monodromy cubic $y_1y_2y_3+y_1^2+y_2^2+1=0$ parametrizing solutions of (1.52)), and they may be further parametrized by a single quantity $m\in{\mathrm{i}}\mathbb{R}+\mathbb{Z}$ according to

(1.55)\begin{align} y_1=\frac{{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\pi m}}{\sqrt{1+{\mathrm{e}}^{2\pi{\mathrm{i}} m}}},\quad y_2=\frac{{\mathrm{i}}}{\sqrt{1+{\mathrm{e}}^{2\pi{\mathrm{i}} m}}}. \end{align}

The generalization of this family of solutions U(z) to allow for arbitrary $m\in\mathbb{C}\setminus (\mathbb{Z}+\frac{1}{2})$ was proven in [Reference Barhoumi, Lisovyy, Miller and Prokhorov3] to correspond to a certain double-scaling limit of high-even-order rational solutions of the Painlevé-III (D ₆) equation when examined near the origin in the complex plane (the only fixed singularity of the equation). The Painlevé-III (D ₆) equation has two essential parameters, one of which is quantized for rational solutions ( $n=\frac{1}{2}(\Theta_0-\Theta_\infty+1)\in\mathbb{Z}$) and the other of which is arbitrary and corresponds to the value of $m=\frac{1}{2}(\Theta_0+\Theta_\infty-1)$.

Going further, the particular solution U(z) with parameter $m\in\mathbb{C}\setminus(\mathbb{Z}+\frac{1}{2})$ is shown in [Reference Barhoumi, Lisovyy, Miller and Prokhorov3, Corollary 3.4] to be explicitly related to the Fredholm determinant of the scalar Bessel kernel:

(1.56)\begin{align} U(z)-\frac{1}{U(z)}=R(z):=-2{\mathrm{i}}-\frac{1}{2}\frac{ \mathrm{d}}{ \mathrm{d} z}z\frac{ \mathrm{d}}{ \mathrm{d} z}\log \left(D_{\varkappa(m)}(32{\mathrm{i}} z)\right), \end{align}

wherein $\varkappa(m):=(1+{\mathrm{e}}^{2\pi{\mathrm{i}} m})^{-1}=-y_2^2=a^2/(|a|^2+|b|^2)$ and

(1.57)\begin{align} D_\varkappa(r):=\det(\mathrm{Id}-\varkappa \mathcal{K}_r) \end{align}

is the Fredholm determinant of the integral operator $\mathcal{K}_r:L^2[0,r]\to L^2[0,r]$ with Bessel kernel

(1.58)\begin{align} K(x,y):=\frac{\sqrt{x}J_1(\sqrt{x})J_0(\sqrt{y})-J_0(\sqrt{x})\sqrt{y}J_1(\sqrt{y})}{2(x-y)}. \end{align}

It can be shown that $D_\varkappa(r)$ is entire in $\varkappa$ and analytic for $r\in\mathbb{C}$ of sufficiently small modulus. To get a representation of $\Psi(X,0;\mathbf{G})$, we first solve (1.56) for U(z):

(1.59)\begin{align} U(z)=\frac{1}{2}\left(R(z)\pm\sqrt{R(z)^2+4}\right), \end{align}

and hence from (1.54)

(1.60)\begin{align} \Psi(X,0;\mathbf{G})=\frac{{\mathrm{i}}}{2}\frac{U'(z)}{U(z)}=\pm\frac{{\mathrm{i}} R'(z)}{2\sqrt{R(z)^2+4}}. \end{align}

We now have to resolve the sign of the square root; first, according to [Reference Barhoumi, Lisovyy, Miller and Prokhorov3, Eqn. (3.22)], one has

(1.61)\begin{align} \log \left(D_\varkappa(r)\right)=-\frac{\varkappa}{4}r +\frac{\varkappa-\varkappa^2}{32}r^2+O(r^3),\quad r\to 0, \end{align}

which implies that $R(z)=-2{\mathrm{i}}+4{\mathrm{i}}\varkappa + 64(\varkappa-\varkappa^2)z + O(z^2)$ and $R'(z)=64(\varkappa-\varkappa^2)+O(z)$ as z → 0. Hence $R'(0)=64(\varkappa-\varkappa^2)$ and $R(0)^2+4=16(\varkappa-\varkappa^2)$. Since $\varkappa=a^2/(|a|^2+|b|^2)\in (0,1)$, we obtain (using z = 0 implies X = 0)

(1.62)\begin{align} \Psi(0,0;\mathbf{G})=\pm 8{\mathrm{i}}\sqrt{\varkappa-\varkappa^2}=\pm\frac{8{\mathrm{i}} a|b|}{a^2+|b|^2}. \end{align}

Comparing this with Theorem 1.6 and recalling that a > 0 while b is purely imaginary shows that we should choose the + (resp., −) sign in (1.60) when b is negative (resp., positive) imaginary, taking the square root as positive when X = 0.

The representation of $\Psi(X,0;\mathbf{G})$ in terms of a Fredholm determinant is easily generalized to arbitrary $(a,b)\in\mathbb{C}^2$ with ab ≠ 0 using Proposition 1.5, since

(1.63)\begin{align}\begin{aligned} \Psi(X,T;\mathbf{G}(a,b))={\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\Psi(X,T;\mathbf{G}(\mathfrak{a},\mathfrak{b})) \;\text{and}\; \Psi(X,T;\mathbf{G}(\mathfrak{a},{\mathrm{i}}\mathfrak{b}))=-{\mathrm{i}}\Psi(X,T;\mathbf{G}(\mathfrak{a},\mathfrak{b}))\\ \implies \Psi(X,T;\mathbf{G}(a,b))={\mathrm{i}}{\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}\Psi(X,T;\mathbf{G}(\mathfrak{a},{\mathrm{i}}\mathfrak{b})), \end{aligned} \end{align}

and the latter has parameters $a=\mathfrak{a}$ positive and $b={\mathrm{i}}\mathfrak{b}$ positive imaginary.

This proves the following.

Theorem 1.16 (Painlevé-III (D₈) and Bessel kernel determinant formula for $\Psi(X,0)$)

Let $(a,b)\in\mathbb{C}^2$ with $ab\neq 0$ correspond to normalized parameters $(\mathfrak{a},\mathfrak{b})$ by (1.11). Then the function $\Psi(X,0;\mathbf{G}(\mathfrak{a},{\mathrm{i}}\mathfrak{b}))=-{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}\arg(ab)}\Psi(X,0;\mathbf{G}(a,b))$ is expressible by (1.54) in terms of a solution $U(z)$ of the Painlevé-III (D₈) equation (1.52) having unit modulus for $z\in{\mathrm{i}}\mathbb{R}$, and moreover for $X$ in a neighborhood of the origin,

(1.64)\begin{align} \Psi(X,0;\mathbf{G}(a,b))=\left.\frac{{\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}R'(z)}{2\sqrt{R(z)^2+4}}\right|_{z={\mathrm{i}} X}, \end{align}

where

(1.65)\begin{align} R(z):=-2{\mathrm{i}} -\frac{1}{2}\frac{ \mathrm{d}}{ \mathrm{d} z}z\frac{ \mathrm{d}}{ \mathrm{d} z}\log \left(D_{\mathfrak{a}^2}(32{\mathrm{i}} z)\right) \end{align}

and $D_\varkappa(r)$ denotes the Fredholm determinant (1.57) of the Bessel kernel (1.58). Here the square root is taken to be positive when $X=0$ and the formula (1.64) admits real analytic continuation to $X\in\mathbb{R}$.

1.3.1. Behaviour of $\Psi(X,T;\mathbf{G})$ for $X$ large

Our first result concerns the behaviour of $\Psi(X,T;\mathbf{G})$ as $X\to\pm\infty$. Let $v_\mathrm{c}:=54^{-\frac{1}{2}}$, and define two functions of v on $(-v_\mathrm{c},v_\mathrm{c})$ by

(1.66)\begin{align} z_1(v):=\begin{cases}\displaystyle \frac{1}{6v}\left(-1+2\cos\left(\frac{1}{3}\arccos\left( 2\frac{v^2}{v_\mathrm{c}^2} -1\right)\right)\right),&\quad -v_\mathrm{c} \lt v \lt 0,\\\\ \displaystyle \frac{1}{6v}\left(-1+2\cos\left(\frac{1}{3}\arccos\left( 2\frac{v^2}{v_\mathrm{c}^2} -1\right)-\frac{2}{3}\pi\right)\right),&\quad 0 \lt v \lt v_\mathrm{c}, \end{cases} \end{align}

and

(1.67)\begin{align} z_2(v):=\begin{cases}\displaystyle \frac{1}{6v}\left(-1+2\cos\left(\frac{1}{3}\arccos\left( 2\frac{v^2}{v_\mathrm{c}^2}-1\right)-\frac{2}{3}\pi\right)\right),&\quad -v_\mathrm{c} \lt v \lt 0,\\\\ \displaystyle \frac{1}{6v}\left(-1+2\cos\left(\frac{1}{3}\arccos\left( 2\frac{v^2}{v_\mathrm{c}^2}-1\right)\right)\right),&\quad 0 \lt v \lt v_\mathrm{c}, \end{cases} \end{align}

which are equivalent for $|v| \lt v_\mathrm{c}/\sqrt{2}$ to

(1.68)\begin{align} \begin{split} z_1(v)&=\frac{1}{6v}\left(-1+2\cos\left(-\frac{1}{3}\arcsin\left(v\sqrt{216\left(1- \frac{v^2}{v_\mathrm{c}^2}\right)}\right)-\frac{1}{3}\pi\right)\right)\\ z_2(v)&=\frac{1}{6v}\left(-1+2\cos\left(-\frac{1}{3}\arcsin\left(v\sqrt{216\left(1- \frac{v^2}{v_\mathrm{c}^2}\right)}\right)+\frac{1}{3}\pi\right)\right). \end{split} \end{align}

In each case, the singularity at v = 0 is removable, and with the definition $z_1(0)=-\sqrt{2}$ and $z_2(0)=\sqrt{2}$, we see that $z_j(v)$, $j=1,2$, are both analytic functionsFootnote ³ of $v\in (-v_\mathrm{c},v_\mathrm{c})$ satisfying $z_1(v) \lt 0 \lt z_2(v)$. Introducing the function

(1.69)\begin{align} \vartheta(z;v):=z+vz^2+2z^{-1}, \end{align}

one can check that $\vartheta'(z_j(v);v)=0$ holds identically for $-v_\mathrm{c} \lt v \lt v_\mathrm{c}$, and that $\vartheta''(z_1(v);v) \lt 0$ while $\vartheta''(z_2(v);v) \gt 0$ (here prime means differentiation with respect to z for fixed v).

Theorem 1.18 (Large- $X$ regime)

Let $\tau:=|b/a|$ and $p:=\frac{1}{2\pi}\ln(1+\tau^2)$, and set $v:=TX^{-\frac{3}{2}}\in\mathbb{R}$. Then for each $\delta \gt 0$,

(1.70)\begin{align}\begin{aligned} \Psi(X, T;\mathbf{G}) & =\frac{{\mathrm{e}}^{-{\mathrm{i}} \arg(ab)}}{X^{\frac{3}{4}}} \left[ \frac{\sqrt{2p}}{\sqrt{-\vartheta''(z_1(v);v)}} {\mathrm{e}}^{-2 {\mathrm{i}} X^{1/2}\vartheta(z_1(v);v)} {\mathrm{e}}^{- {\mathrm{i}} \frac{1}{2} p \ln(X) } {\mathrm{e}}^{-{\mathrm{i}}( \phi_{z_1}(v) + \phi_{0}(v))} \right.\\ & \quad \left.+ \frac{\sqrt{2p}}{\sqrt{\vartheta''(z_2(v);v)}}{\mathrm{e}}^{-2 {\mathrm{i}} X^{1/2}\vartheta(z_2(v);v)} {\mathrm{e}}^{{\mathrm{i}} \frac{1}{2} p \ln(X) } {\mathrm{e}}^{{\mathrm{i}}( \phi_{z_2}(v) + \phi_{0}(v))} \right] + O(X^{-\frac{5}{4}}), \quad X\to+\infty \end{aligned} \end{align}

holds uniformly for $|v|\le v_\mathrm{c}-\delta$ and $\tau=O(1)$, where real phases $\phi_0(v)$, $\phi_{z_1}(v)$, and $\phi_{z_2}(v)$ are defined by

(1.71)\begin{align} \phi_0(v) := \frac{1}{4}\pi + p \ln( 2 (z_2(v)-z_1(v))^2 ) - \arg(\Gamma({\mathrm{i}} p)), \end{align}

and

(1.72)\begin{align} \begin{aligned} \phi_{z_1}(v)&:= p \ln(-\vartheta''(z_1(v);v ) ),\\ \phi_{z_2}(v)&:= p \ln(\vartheta''(z_2(v);v ) ). \end{aligned} \end{align}

These results generalize a theorem [Reference Bilman, Ling and Miller8, Theorem 4] of the authors with L. Ling to the general family of solutions parametrized by the $2\times 2$ matrix $\mathbf{G}(a,b)$, and they also sharpen the error estimate (see Remark 2.2 below). We prove Theorem 1.18 in Section 2. Note that the leading term of (1.70) vanishes as $p\to 0^+$, which is expected in light of Proposition 1.8 since this corresponds to b → 0. Similarly, the leading term of the asymptotic formula valid as $X\to-\infty$ vanishes as $\bar{p}\to 0^+$, which corresponds to a → 0. More generally, aside from the phase factor of ${\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}$, the dependence on (a, b) enters only via the value of p, which appears in the approximate formula (1.70) as an overall multiplier $\sqrt{2p}$ and in smaller corrections to the p-independent dominant phase terms proportional to $X^\frac{1}{2}$. In Figures 2–3, the accuracy of the approximation (1.70) is illustrated by comparing with numerical computations of $\Psi(X,T;\mathbf{G})$ achieved using the software package described in Section 1.4.

Figure 2.

Left: The real part (navy) and imaginary part (yellow) of the explicit terms on the right-hand side of (1.70) (curves) compared with numerical evaluation of $\Psi(X,T;\mathbf{G})$ (points) for $v=0.5v_\mathrm{c}$ and parameters $a=b=1$. Right: the logarithm of the absolute difference $E$ between $\Psi(X,T;\mathbf{G})$ and its explicit approximation in (1.70) for $v=0.5v_\mathrm{c}$ and parameters $a=b=1$ plotted against $\ln(X)$. The purple line is a least-squares best fit, and it has a slope of $-1.22883$, matching well with the predicted exponent of $-\frac{5}{4}$.

Figure 3.

Same as in Figure 2 but for $a=\frac{1}{2}{\mathrm{e}}^{{\mathrm{i}}\pi/4}$ and $b=1$. The best fit line in the right-hand plot has slope $-1.26571$, again matching well with the predicted exponent of $-\frac{5}{4}$.

Corollary 1.20 (Large- $X$ approximation of the squared modulus)

Under the hypotheses of Theorem 1.18, $ |\Psi(X,T;\mathbf{G}) |^2$ with $T=X^{\frac{3}{2}}v$ has the behaviour

(1.73)\begin{align}\begin{aligned} |\Psi(X,T;\mathbf{G})|^2 = \frac{2p}{X^\frac{3}{2}\sqrt{-\vartheta''(z_1(v);v)\vartheta''(z_2(v);v)}}\left[\sqrt{-\frac{\vartheta''(z_2(v);v)}{\vartheta''(z_1(v);v)}} +\sqrt{-\frac{\vartheta''(z_1(v);v)}{\vartheta''(z_2(v);v)}}\right.\\ \left.{}\vphantom{\sqrt{-\frac{\vartheta''(z_1(v);v)}{\vartheta''(z_2(v);v)}}}+2\cos\left( \Omega(X,v)\right)\right] +O(X^{-2}),\quad X\to+\infty, \end{aligned} \end{align}

where the phase is given by

(1.74)\begin{align} \Omega(X,v):=2\varrho(v)X^{\frac{1}{2}}+p \ln(X) +2\varsigma(v), \end{align}

where $\varrho(v):= \vartheta(z_1(v);v) - \vartheta(z_2(v);v) \lt 0$ and $\varsigma(v):=\frac{1}{2}\phi_{z_1}(v)+\frac{1}{2}\phi_{z_2}(v)+\phi_0(v)\in\mathbb{R}$.

Aside from an overall factor of $X^{-\frac{3}{2}}$, the only dependence of $|\Psi(X,T;\mathbf{G})|^2$ on $X\gg 1$ in the leading terms for fixed v appears in the phase $\Omega(X,v)$, making the leading contribution to $|\Psi(X,T;\mathbf{G})|^2$ highly oscillatory. It is therefore reasonable to assert that $|\Psi(X,T;\mathbf{G})|^2$ is approximately maximized along curves in the (X, T)-plane where $\cos(\Omega(X,TX^{-\frac{3}{2}}))=1$. Thus, we set $\Omega(X,v)=2\pi n$ for $n\in\mathbb{Z}$ and solve for X. Since $\varrho(v) \lt 0$ is bounded away from zero, X > 0 large implies that also $n=-N$ is a large negative integer. The equation $\Omega(X,v)=-2\pi N$ can then be rearranged as

(1.75)\begin{align} \frac{1}{p}\varrho(v)X^\frac{1}{2}\exp\left(\frac{1}{p}\varrho(v)X^\frac{1}{2}\right) = -{\mathrm{e}}^{-\eta},\quad\eta:=\frac{\pi N}{p}+\nu(v),\quad \nu(v):=\frac{\varsigma(v)}{p}+\ln\left(-\frac{p}{\varrho(v)}\right), \end{align}

which is solved using the Lambert W-function [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain27, Section 4.13]:

(1.76)\begin{align} X=\frac{p^2}{\varrho(v)^2}W_{\pm 1}(-{\mathrm{e}}^{-\eta}\mp {\mathrm{i}} 0)^2. \end{align}

Using the asymptotic formula $W_{\pm 1}(-{\mathrm{e}}^{-\eta}\mp{\mathrm{i}} 0)=-\eta -\ln(\eta)-\ln(\eta)/\eta + O(\ln(\eta)^2/\eta^2)$ as $\eta\to+\infty$ (see [Reference Olver, Daalhuis, Lozier, Schneider, Boisvert, Clark, Miller, Saunders, Cohl and McClain27, Eqn. 4.13.11]), we then obtain for each large positive integer N a solution $X=X_N(v)$ that is accurate up to a small absolute error:

(1.77)\begin{align} \begin{split} X_N(v)&=\frac{p^2}{\varrho(v)^2}\left[\eta^2 + 2\eta\ln(\eta) + 2\ln(\eta) + \ln(\eta)^2\right] + O\left(\frac{\ln(\eta)^2}{\eta}\right),\quad\eta\to+\infty\\ &=\frac{\pi^2}{\varrho(v)^2}N^2 + \frac{2\pi p}{\varrho(v)^2}N\ln(N) + \frac{2\pi p}{\varrho(v)^2}\left( \nu(v)+\ln\left(\frac{\pi}{p}\right) \right)N\\ &\quad +\frac{p^2}{\varrho(v)^2}\ln(N)^2+\frac{2p^2}{\varrho(v)^2}\left(1+ \nu(v)+\ln\left(\frac{\pi}{p}\right) \right)\ln(N)\\ &\quad +\frac{p^2}{\varrho(v)^2}\left[ 2\nu(v)+\nu(v)^2 + 2\left(1+ \nu(v) \right)\ln\left(\frac{\pi}{p}\right) +\log\left(\frac{\pi}{p}\right)^2\right] \\ &\quad +O(N^{-1}\ln(N)^2),\quad N\to+\infty. \end{split} \end{align}

This expansion is valid uniformly for $|v|\le v_\mathrm{c}-\delta$ for any δ > 0 however small, but it fails for $v\approx \pm v_\mathrm{c}$, where $\phi_{z_1}(v)$ and hence also $\varsigma(v)$ blows up. The curves $X=X_N(TX^{-\frac{3}{2}})$ are superimposed on a density plot of the square modulus of $\Psi(X,T;\mathbf{G})$ in Figure 6. This shows that these curves actually approximate the peaks of the modulus in the region $|TX^{-\frac{3}{2}}| \lt v_\mathrm{c}$ quite accurately even when X is not very large.

After some preparation in Sections 2.1–2.2, the proof of Theorem 1.18 is given in Section 2.3 below.

1.3.2. Behaviour of $\Psi(X,T;\mathbf{G})$ for T large

Next, we describe the behaviour of $\Psi(X,T;\mathbf{G})$ in the limit as $T\to\pm \infty$. Let $w_\mathrm{c}:=54^\frac{1}{3} \gt 0$. Define for $|w| \lt w_\mathrm{c}$ the real quantities:

(1.78)\begin{align} Z_1(w):=\frac{1}{12}\left(-w-\sqrt{w^2+8w_\mathrm{c}^2}\right) \lt 0\quad\text{and}\quad Z_2(w):=\frac{1}{12}\left(-w+\sqrt{w^2+8w_\mathrm{c}^2}\right) \gt 0, \end{align}

and the complex quantity

(1.79)\begin{align} Z_0(w):=\frac{1}{3}\left(-w +{\mathrm{i}}\sqrt{w_\mathrm{c}^2-w^2}\right)\in\mathbb{C}_+, \end{align}

where in each case the positive square root is taken. For convenience, we write

(1.80)\begin{align} V(w):=\mathrm{Im}(Z_0(w)). \end{align}

Related amplitudes are then defined by

(1.81)\begin{align} \begin{split} m_{Z_1}^\pm(w)&:=\frac{1}{2}(1\pm\cos(\arg(Z_1(w)-Z_0(w)))) \gt 0,\\ m_{Z_2}^\pm(w)&:=\frac{1}{2}(1\pm\cos(\arg(Z_2(w)-Z_0(w)))) \gt 0, \end{split} \end{align}

which of course satisfy $m_{Z_j}^+(w)+m_{Z_j}^-(w)=1$ for $j=1,2$. Now define some real phases by:

(1.82)\begin{align} \kappa(w):=-\frac{1}{3}(w^2+w_\mathrm{c}^2), \end{align}

(1.83)\begin{align}\begin{aligned} \Phi(w):= & 2\bar{p}\ln\left(\frac{Z_1(w)-\mathrm{Re}(Z_0(w))+|Z_1(w)-Z_0(w)|}{V(w)}\right)\\ & \qquad \qquad -2p\ln\left(\frac{Z_2(w)-\mathrm{Re}(Z_0(w)) + |Z_2(w)-Z_0(w)|}{V(w)}\right) -\frac{\pi}{2}, \end{aligned} \end{align}

(1.84)\begin{align}\begin{aligned} \Phi^0_{Z_1}(w): & =2p\ln\left(\frac{V(w)^2-2(V(w)-|Z_1(w)-Z_0(w)|)(V(w)-|Z_2(w)-Z_0(w)|)}{V(w)^2-2(V(w)+|Z_1(w)-Z_0(w)|)(V(w)-|Z_2(w)-Z_0(w)|)}\right)\\ & \quad +2p\ln\left(\frac{\mathrm{Re}(Z_0(w))-Z_1(w)}{|Z_1(w)-Z_0(w)|-V(w)}\right) + 2\bar{p}\ln\left(\frac{-Z_1(w)V(w)}{4|Z_1(w)-Z_0(w)|^\frac{5}{2}(Z_2(w)-Z_1(w))^\frac{1}{2}}\right)\\ & \quad +\frac{\pi}{4}+\arg(\Gamma({\mathrm{i}}\bar{p})), \end{aligned} \end{align}

(1.85)\begin{align}\begin{aligned} \Phi^0_{Z_2}(w): & =2\bar{p}\ln\left(\frac{V(w)^2-2(V(w)-|Z_2(w)-Z_0(w)|)(V(w)-|Z_1(w)-Z_0(w)|)}{V(w)^2-2(V(w)+|Z_2(w)-Z_0(w)|)(V(w)-|Z_1(w)-Z_0(w)|)}\right)\\ & \quad +2\bar{p}\ln\left(\frac{Z_2(w)-\mathrm{Re}(Z_0(w))}{|Z_2(w)-Z_0(w)|-V(w)}\right) +2p\ln\left(\frac{Z_2(w)V(w)}{4|Z_2(w)-Z_0(w)|^\frac{5}{2}(Z_2(w)-Z_1(w))^\frac{1}{2}}\right)\\ & \quad +\frac{\pi}{4}+\arg(\Gamma({\mathrm{i}} p)), \end{aligned} \end{align}

(1.86)\begin{align} \Phi_{Z_1}(T,w):=2\frac{|Z_1(w)-Z_0(w)|^3}{-Z_1(w)}T^\frac{1}{3} -\frac{1}{3}\bar{p}\ln(T) + \Phi^0_{Z_1}(w), \end{align}

and

(1.87)\begin{align} \Phi_{Z_2}(T,w):=2\frac{|Z_2(w)-Z_0(w)|^3}{Z_2(w)}T^\frac{1}{3} -\frac{1}{3}p\ln(T) + \Phi_{Z_2}^0(w). \end{align}

Theorem 1.22 (Large- $T$ regime)

Let $\tau:=|b/a|$ and $p:=\frac{1}{2\pi}\ln(1+\tau^2)$, $\bar{p}:=\frac{1}{2\pi}\ln(1+\tau^{-2})$, and set $w:=XT^{-\frac{2}{3}}\in\mathbb{R}$. Then, for each $\delta \gt 0$,

(1.88)\begin{align}\begin{aligned} \Psi(X,T;\mathbf{G}) & ={\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}{\mathrm{e}}^{-{\mathrm{i}} T^{1/3}\kappa(w)}{\mathrm{e}}^{{\mathrm{i}}\Phi(w)}\left[\frac{1}{3}\sqrt{w_\mathrm{c}^2-w^2}T^{-\frac{1}{3}}\right.\\ & \quad -\frac{1}{\sqrt{Z_2(w)-Z_1(w)}}\left\{ \frac{\sqrt{\bar{p}}|Z_1(w)|\left(m_{Z_1}^+(w){\mathrm{e}}^{{\mathrm{i}}\Phi_{Z_1}(T,w)}+m_{Z_1}^-(w){\mathrm{e}}^{-{\mathrm{i}}\Phi_{Z_1}(T,w)}\right)}{|Z_1(w)-Z_0(w)|^\frac{1}{2}}\right.\\ & \quad \left.\left.+\frac{\sqrt{p}Z_2(w)\left(m_{Z_2}^-(w){\mathrm{e}}^{{\mathrm{i}}\Phi_{Z_2}(T,w)}+m_{Z_2}^+(w){\mathrm{e}}^{-{\mathrm{i}}\Phi_{Z_2}(T,w)}\right)}{|Z_2(w)-Z_0(w)|^\frac{1}{2}}\right\}T^{-\frac{1}{2}} + O(T^{-\frac{2}{3}})\right],\quad T\to+\infty \end{aligned} \end{align}

holds uniformly for $|w|\le w_\mathrm{c}-\delta$, $\tau=O(1)$, and $\tau^{-1}=O(1)$.

These results generalize the long-time asymptotic theorem [Reference Bilman, Ling and Miller8, Theorem 5] of the authors with L. Ling to the general family of solutions parametrized by the $2\times 2$ matrix $\mathbf{G}(a,b)$. They also provide an explicit correction term proportional to $T^{-\frac{1}{2}}$ not previously obtained for any parameters. After some preliminary definitions in Sections 3.1–3.3, Theorem 1.22 is proved in Section 3.4 below. One can see from (1.88) that aside from the factor of ${\mathrm{e}}^{-{\mathrm{i}}\arg(ab)}$, the asymptotic formula depends on (a, b) only via the values of p and $\bar{p}$, which enter via $\sqrt{p}$ and $\sqrt{\bar{p}}$ as multipliers of the two correction terms, and which also appear as multipliers in the phase $\Phi(w)$ and in subdominant corrections to the leading terms proportional to $T^\frac{1}{3}$ in the phases $\Phi_{Z_1}(T,w)$ and $\Phi_{Z_2}(T,w)$. The accuracy of the large-T approximation afforded by Theorem 1.22 is illustrated in Figures 4–5.

Figure 4.

Left: The real part (navy) and imaginary part (yellow) of the explicit terms on the right-hand side of (1.88) (curves) compared with numerical evaluation of $\Psi(X,T;\mathbf{G})$ (points) for $w=0.85w_\mathrm{c}$ and parameters $a=b=1$. Right: the logarithm of the absolute difference $E$ between $\Psi(X,T;\mathbf{G})$ and its explicit approximation in (1.88) for $w=0.85w_\mathrm{c}$ and parameters $a=b=1$ plotted against $\ln(T)$. The purple line is a least-squares best fit, and it has a slope of $-0.66042$, matching well with the predicted exponent of $-\frac{2}{3}$.

Figure 5.

Same as in Figure 4 but for $a=\frac{1}{2}{\mathrm{e}}^{{\mathrm{i}}\pi/4}$ and $b=1$. The best fit line in the right-hand plot has slope $-0.69432$, again matching well with the predicted exponent of $-\frac{2}{3}$.

Corollary 1.24 (Large- $T$ approximation of the squared modulus)

Under the hypotheses of Theorem 1.22, $ |\Psi(X,T;\mathbf{G})|^2$ with $X=T^\frac{2}{3}w$ has the behaviour

(1.89)\begin{align}\begin{aligned} |\Psi(X,T;\mathbf{G})|^2 & = \frac{w_\mathrm{c}^2-w^2}{9}T^{-\frac{2}{3}}\\ & \quad -\frac{2}{3}\sqrt{\frac{w_\mathrm{c}^2-w^2}{Z_2(w)-Z_1(w)}}\left\{\frac{\sqrt{\bar{p}}|Z_1(w)|\cos(\Phi_{Z_1}(T,w))}{|Z_1(w)-Z_0(w)|^\frac{1}{2}} + \frac{\sqrt{p}Z_2(w)\cos(\Phi_{Z_2}(T,w))}{|Z_2(w)-Z_0(w)|^\frac{1}{2}}\right\}T^{-\frac{5}{6}}\\ & \quad +O(T^{-1}),\quad T\to+\infty \end{aligned} \end{align}

and for the limit $T\to-\infty$ we simply replace $T$ with $|T|$ on the right-hand side.

Comparing with the interpretation of Corollary 1.20 that the modulus of the solution is maximized along certain curves $X=X_N(TX^{-\frac{3}{2}})$ in the relevant part of the (X, T)-plane, here we see that instead the fluctuation proportional to $T^{-\frac{5}{6}}$ is a combination of two sinusoids with different phases, so that one term is approximately maximized along curves satisfying $\Phi_{Z_1}(T,w)=(2M+1)\pi$ while the other is approximately maximized along curves satisfying $\Phi_{Z_2}(T,w)=(2N+1)\pi$, for independent integers $(M,N)\in\mathbb{Z}^2$. These two families of curves are plotted over the region $|XT^{-\frac{2}{3}}| \lt w_\mathrm{c}$ for different values of the parameters (a, b) in Figures 6 and 7, and indeed one can see that even in the part of this region where T is not very large, still the peaks of the fluctuation appear to be localized near the intersections of one curve from each family, so that both terms of the fluctuation are simultaneously maximized.

Figure 6.

Density plot of $|\Psi(X,T)|^2$ with $a=b=1$ and the boundary curve (purple) $X/T^{\frac{2}{3}}=54^{\frac{1}{3}}$, overlaid with the level curves (green) on which the cosine in (1.73) in Corollary 1.20 is maximal and the level curves (red and mustard) on which $\cos(\Phi_{Z_1}(T,w))$ and $\cos(\Phi_{Z_2}(T,w))$ in (1.89) from Corollary 1.24 are minimal (respectively).

Figure 7.

As in Figure 6, but for parameters $a=\frac{1}{4}$ and $b=1$.

Like the curves $X=X_N(TX^{-\frac{3}{2}})$, the curves in the family $\Phi_{Z_2}(T,w)=(2N+1)\pi$ shown with mustard colour in Figures 6 and 7 appear to approach the common boundary in the first quadrant (X > 0 and T > 0) of the regions of validity of Corollaries 1.20 and 1.24 (shown in purple). It seems from the plot as though these two families of curves could be compared near the boundary curve. The equation $\Phi_{Z_2}(T,w)=(2N+1)\pi$ can be rewritten in the form

(1.90)\begin{align}\begin{aligned} -\frac{2|Z_2(w)-Z_0(w)|^3}{pZ_2(w)}T^\frac{1}{3}\exp\left(-\frac{2|Z_2(w)-Z_0(w)|^3}{pZ_2(w)}T^\frac{1}{3}\right)=-{\mathrm{e}}^{-\kappa},\quad\kappa:=\frac{2\pi N}{p} + \varpi(w),\\ \varpi(w):=\frac{\pi}{p}-\frac{1}{p}\Phi^0_{Z_2}(w)-\ln\left(\frac{2|Z_2(w)-Z_0(w)|^3}{pZ_2(w)}\right), \end{aligned} \end{align}

which is solved in terms of the Lambert W-function similarly to (1.76):

(1.91)\begin{align} T=-\frac{p^3Z_2(w)^3}{8|Z_2(w)-Z_0(w)|^9}W_\pm(-{\mathrm{e}}^{-\kappa}\mp{\mathrm{i}} 0)^3. \end{align}

Since a cube must be calculated in the limit $\kappa\to +\infty$, we need the more accurate asymptotic formula

(1.92)\begin{align}\begin{aligned} W_{\pm 1}(-{\mathrm{e}}^{-\kappa}\mp{\mathrm{i}} 0)=-\kappa-\ln(\kappa)-\kappa^{-1}\ln(\kappa)+\frac{1}{2}\kappa^{-2}\ln(\kappa)^2 -\kappa^{-2}\ln(\kappa) \\+ O(\kappa^{-3}\ln(\kappa)^3),\quad\kappa\to+\infty, \end{aligned} \end{align}

and therefore for each large positive integer N we obtain a solution $T=T_N(w)$ with the approximation

(1.93)\begin{align} \begin{split} T_N(w)&=\frac{p^3Z_2(w)^3}{8|Z_2(w)-Z_0(w)|^9}\left[\kappa^3+3\kappa^2\ln(\kappa)+3\kappa\ln(\kappa)^2+3\kappa\ln(\kappa)+\ln(\kappa)^3+\frac{9}{2}\ln(\kappa)^2 + 3\ln(\kappa)\right]\\ &\quad+O\left(\frac{\ln(\kappa)^3}{\kappa}\right),\quad\kappa\to+\infty\\ &=\frac{p^3Z_2(w)^3}{8|Z_2(w)-Z_0(w)|^9}\left[\left(\frac{2\pi}{p}\right)^3N^3+3\left(\frac{2\pi}{p}\right)^2N^2\ln(N)+3\left(\frac{2\pi}{p}\right)^2\left(\varpi(w)+\ln\left(\frac{2\pi}{p}\right)\right)N^2\right.\\ &\quad+\frac{6\pi}{p}N\ln(N)^2+\frac{6\pi}{p}\left(1+2\varpi(w)+2\ln\left(\frac{2\pi}{p}\right)\right)N\ln(N)\\ &\quad+\frac{6\pi}{p}\left(\varpi(w)^2+\varpi(w)+\left(2\varpi(w)+1\right)\ln\left(\frac{2\pi}{p}\right)+\ln\left(\frac{2\pi}{p}\right)^2\right)N + \ln(N)^3\\ &\quad+\left(3\varpi(w)+\frac{9}{2}+3\ln\left(\frac{2\pi}{p}\right)\right)\ln(N)^2\\ &\quad+\left(3\varpi(w)^2+9\varpi(w) + 3+ 3(2\varpi(w)+3)\ln\left(\frac{2\pi}{p}\right)+3\ln\left(\frac{2\pi}{p}\right)^2 \right)\ln(N)\\ &\quad+\varpi(w)^3+\frac{9}{2}\varpi(w)^2+3\varpi(w) +3\left(\varpi(w)^2+3\varpi(w)+1\right)\ln\left(\frac{2\pi}{p}\right) \\ &\quad\left.+ 3\left(\varpi(w)+\frac{3}{2}\right)\ln\left(\frac{2\pi}{p}\right)^2 + \ln\left(\frac{2\pi}{p}\right)^3\right] +O\left(\frac{\ln(N)^3}{N}\right),\quad N\to+\infty. \end{split} \end{align}

One can check that the first two terms of $54T_N(w)^2$, proportional to N ⁶ and $N^5\ln(N)$ respectively, have finite values in the limit $w\uparrow w_\mathrm{c}$. Exactly the same is true of the two leading terms of $X_N(v)^3$ in the limit $v\uparrow v_\mathrm{c}$. However, only the leading term of each expansion matches, and a discrepancy appears at the order $N^5\ln(N)$. Moreover, it is easy to see that the coefficient of $N^5\ln(N)$ cannot be changed in one of the expansions by adding any fixed integer to N (amounting to re-indexing the curves of the relevant family). Subsequent terms in each expansion actually blow up as $w\uparrow w_\mathrm{c}$ and $v\uparrow v_\mathrm{c}$ respectively. This computation shows that it is just an illusion that the families of curves appear to match along the boundary curve $54T^2=X^3$ in the first quadrant in Figures 6 and 7. Moreover, zooming in near the boundary curve shows that the curves actually turn sharply as they approach the boundary, becoming tangent to it as shown in Figure 8.

Figure 8.

Zoom-in plots on two different scales near the boundary curve corresponding to the parameters $a=b=1$ as in Figure 6, showing the mismatches of the amplitude-maximizing curves near the boundary.

1.3.3. Transitional behaviour of $\Psi(X,T;\mathbf{G})$

The domains of validity of Theorems 1.18 and 1.22 and their corollaries cover all asymptotic directions in the (X, T)-plane, except for those near the common boundary curves $|T||X|^{-\frac{3}{2}}=v_\mathrm{c}$ which is equivalent to $|X||T|^\frac{2}{3}=w_\mathrm{c}$. Our final asymptotic results concern the behaviour of $\Psi(X,T,\mathbf{G})$ in the transitional region for large (X, T) near these curves. To formulate them, we need to first recall the second Painlevé equation with parameter $\alpha\in\mathbb{C}$:

(1.94)\begin{align} \mathcal{U}''(x)=x\mathcal{U}(x)+2\mathcal{U}(x)^3-\alpha, \end{align}

every solution of which is a meromorphic function of $x\in\mathbb{C}$ all of whose poles are simple with residue ±1. There is a unique solution $\mathcal{U}(x)$ of (1.94) with asymptotic behaviour $\mathcal{U}(x)=-{\mathrm{i}} (\frac{1}{2}x)^\frac{1}{2}-\frac{1}{2}\alpha x^{-1}+O(|x|^{-\frac{5}{2}})$ as $x\to\infty$ with $|\arg(x)| \lt \frac{2}{3}\pi$. This is one of the so-called increasing tritronquée solutions of (1.94). We require this solution in the situation that $\alpha=\frac{1}{2}+{\mathrm{i}} p$, where $p=\frac{1}{2\pi}\ln(1+\tau^2)$ with $\tau=|b/a|$. Given the increasing tritronquée solution $\mathcal{U}(x)$ for such α determined by τ > 0, a related meromorphic function $\mathcal{V}(y;\tau)$ is uniquely defined by the conditions

(1.95)\begin{align} \frac{\mathcal{V}'(y;\tau)}{\mathcal{V}(y;\tau)}=-(\tfrac{2}{3})^\frac{1}{3}\mathcal{U}(-(\tfrac{2}{3})^\frac{1}{3}y),\quad \mathcal{V}(y;\tau)=-\left(\frac{y}{6}\right)^\alpha(1+O(y^{-\frac{3}{4}})),\quad y\to+\infty. \end{align}

Then it is shown in [Reference Miller25, Theorem 1.4] that $\mathcal{V}(y;\tau)$ is analytic and non-vanishing for all real y, and it has the complementary asymptotic behaviour

(1.96)\begin{align} \mathcal{V}(y;\tau)=\frac{\tau p\Gamma({\mathrm{i}} p)}{2\sqrt{\pi}}{\mathrm{e}}^{-3\pi{\mathrm{i}}/4}{\mathrm{e}}^{-\pi p/2}2^{-{\mathrm{i}} p}{\mathrm{e}}^{-2{\mathrm{i}} (-y/3)^{3/2}}(-3y)^{-\frac{1}{2}(\frac{1}{2}+{\mathrm{i}} p)}(1+O(|y|^{-\frac{5}{4}})),\quad y\to -\infty. \end{align}

See [Reference Miller25, Corollary 1.5], which is valid for all τ > 0 and $p=\frac{1}{2\pi}\ln(1+\tau^2)$.

Theorem 1.25 (Transition regime)

Let $\tau:=|b/a|$ and $p:=\frac{1}{2\pi}\ln(1+\tau^2)$, and set $v:=TX^{-\frac{3}{2}}$ and $v_\mathrm{c}:= 54^{-\frac{1}{2}}$. In the limit $X\to+\infty$,

(1.97)\begin{align} \Psi(X,T;\mathbf{G})=2\cdot 3^{\frac{2}{3}}X^{-\frac{2}{3}}\mathcal{V}(2^{\frac{5}{2}}3^{\frac{7}{6}}X^{\frac{1}{3}}(v-v_\mathrm{c});\tau){\mathrm{e}}^{{\mathrm{i}}\Omega_\mathrm{c}(X,v)} + 2^\frac{1}{4}3^{-\frac{1}{4}}p^\frac{1}{2}X^{-\frac{3}{4}} {\mathrm{e}}^{{\mathrm{i}}\Omega_2(X,v)} +O(X^{-\frac{5}{6}}) \end{align}

holds uniformly for $v-v_\mathrm{c}=O(X^{-\frac{1}{3}})$ and $\tau=O(1)$, where phases $\Omega_\mathrm{c}(X,v)$ and $\Omega_2(X,v)$ are defined by

(1.98)\begin{align} \Omega_\mathrm{c}(X,v):=24^{\frac{1}{2}}X^{\frac{1}{2}}- 12 X^{\frac{1}{2}}(v-v_\mathrm{c})-\frac{1}{3}p\ln(X) +\frac{\pi}{2}-\arg(ab) + p\ln(2)-\frac{5}{3}p\ln(3), \end{align}

and

(1.99)\begin{align}\begin{aligned} \Omega_2(X,v):& =-\left(\frac{75}{2}\right)^\frac{1}{2}X^{\frac{1}{2}}-3X^\frac{1}{2}(v-v_\mathrm{c}) +\frac{1}{2}p\ln(X)\\ & \qquad +\frac{\pi}{4}-\arg(\Gamma({\mathrm{i}} p))-\arg(ab)+\frac{1}{2}p\ln(2)+\frac{7}{2}p\ln(3). \end{aligned} \end{align}

Theorem 1.25 is proved in Section 4. The leading term in (1.97) was obtainedFootnote ⁴ in the special case of a = b in [Reference Bilman, Ling and Miller8, Theorem 6]. In the general case, we see that the limiting Painlevé-II function $\mathcal{V}(\diamond;\tau)$ corresponds to a variable parameter $\alpha=\frac{1}{2}+{\mathrm{i}} p$ depending on (a, b) via $\tau=|b/a|$ and $p=\frac{1}{2\pi}\ln(1+\tau^2)$.

The relative size of the correction term in (1.97) compared to the leading term is $O(X^{-\frac{1}{12}})$, which decays very slowly as $X\to\infty$. This observation motivates keeping the correction term although it is asymptotically negligible compared to the leading term. We observe that the correction term is essentially the same as the contribution to $\Psi(X,T;\mathbf{G})$ of the explicit term on the second line of (1.70), approximated in the limit $v\uparrow v_\mathrm{c}$. Theorem 1.25 shows that as $v\uparrow v_\mathrm{c}$, the contribution from the critical point $z_2(v)$ persists at the same order while that from $z_1(v)$ becomes larger by a factor proportional to $X^\frac{1}{12}$ and takes on a universal form expressed in terms of the Painlevé-II special function $\mathcal{V}(\diamond;\tau)$.

To illustrate the validity of Theorem 1.25, we took $a=b=1$ and applied the numerical method described in Appendix B to solve Riemann–Hilbert Problem 2.1 from [Reference Miller25] and hence obtain $\mathcal{V}(y;\tau=1)$ for a dense grid of y-values in the interval $[-0.2,0.2]$. Given such a value of y, and a large value of X > 0, a corresponding value of T is defined by $T=X^\frac{3}{2}v=X^\frac{3}{2}(v_\mathrm{c}+2^{-\frac{5}{2}}3^{-\frac{7}{6}}X^{-\frac{1}{3}}y)$. Then we numerically computed $\Psi(X,T;\mathbf{G}(1,1))$ using the method described in Section 5, and used the result to compare with the prediction of Theorem 1.25 as shown in Figure 9 and also to calculate the pointwise renormalized error

(1.100)\begin{align}\begin{aligned} E(y):=\left| 2^{-1}3^{-\frac{2}{3}}X^\frac{2}{3}\Psi(X,T;\mathbf{G}(1,1))-\mathcal{V}(y;1){\mathrm{e}}^{{\mathrm{i}}\Omega_\mathrm{c}(X,v)}-2^{-\frac{3}{4}}3^{-\frac{11}{12}}X^{-\frac{1}{12}}{\mathrm{e}}^{{\mathrm{i}}\Omega_2(X,v)}\right|,\\ v=v_\mathrm{c}+2^{-\frac{5}{2}}3^{-\frac{7}{6}}X^{-\frac{1}{3}}y,\quad T=X^\frac{3}{2}v=X^\frac{3}{2}(v_\mathrm{c}+2^{-\frac{5}{2}}3^{-\frac{7}{6}}X^{-\frac{1}{3}}y), \end{aligned} \end{align}

for $y\in [-1,1]$. We next set $E:=\max_{|y|\le 1}E(y)$ and plotted $\ln(E)$ against $\ln(X)$. See Figure 10.

Figure 9.

Numerical evaluation of $\Psi(X,T;\mathbf{G}(1,1))$ as a function of $y$ for fixed $X$ (points) compared with the approximation of the two explicit terms in Theorem 1.25 (solid curves). First row: real and imaginary parts; second row: modulus. Left-to-right: $X=4000$, $X=40,000$, $X=400,000$. The shaded region in each plot corresponds to an error bar proportional by a fixed constant to $X^{-5/6}$.

Figure 10.

The uniform renormalized error $E$ over $y\in [-1,1]$ as a function of $X$. Black points are numerical computations and the purple line is a least-squares best fit line with slope $-0.162334$. This matches very well the prediction of Theorem 1.25, namely $E=O(X^{-\frac{1}{6}})$, suggesting that the error term in (1.97) is sharp.