# The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone

Anne Boutet de Monvel, Jonatan Lenells, Dmitry Shepelsky

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Anne Boutet de Monvel, Jonatan Lenells, Dmitry Shepelsky. The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone. Journal of Differential Equations, 2025, 429, pp.747-801. 10.1016/j.jde.2025.02.016. hal-05114838

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Submitted on 16 Jun 2025

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# The focusing NLS equation with step-like oscillating background: Asymptotics in a transition zone

Anne Boutet de Monvel<sup>a</sup>, Jonatan Lenells<sup>b,*</sup>, Dmitry Shepelsky<sup>c</sup>

Received 6 June 2024; accepted 4 February 2025

Available online 26 February 2025

Dedicated to the memory of Henry McKean

## Abstract

In a recent paper, we presented scenarios of long-time asymptotics for the solution of the focusing non-linear Schrödinger equation with initial data approaching plane waves of the form $A_1e^{i\phi_1}e^{-2iB_1x}$ and $A_2e^{i\phi_2}e^{-2iB_2x}$ at minus and plus infinity, respectively. In the shock case $B_1 < B_2$ some scenarios include sectors of genus 3, that is, sectors $\xi_1 < \xi < \xi_2$, $\xi := x/t$, where the leading term of the asymptotics is expressed in terms of hyperelliptic functions attached to a Riemann surface of genus 3. The present paper deals with the asymptotic analysis in a transition zone between two genus 3 sectors. The leading term is expressed in terms of elliptic functions attached to a Riemann surface of genus 1. A central step in the derivation is the construction of a local parametrix in a neighborhood of two merging branch points.

© 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

We consider the focusing nonlinear Schrödinger (NLS) equation

$$iq_t + q_{xx} + 2|q|^2q = 0, \quad x \in \mathbb{R}, \quad t \ge 0, \quad (1.1)$$

<footnote><sup>a</sup> Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, 75205 Paris Cedex 13, France</footnote>

<footnote><sup>b</sup> Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden</footnote>

<footnote><sup>c</sup> B. Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkiv, Ukraine</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: anne.boutet-de-monvel@imj-prg.fr (A. Boutet de Monvel), jlenells@kth.se (J. Lenells), shepelsky@yahoo.com (D. Shepelsky).</footnote>
and a solution $q(x, t)$ of (1.1) such that

$$q(x, 0) \sim \begin{cases} A_1 e^{i\phi_1} e^{-2iB_1x}, & x \to -\infty, \\ A_2 e^{i\phi_2} e^{-2iB_2x}, & x \to +\infty, \end{cases} \quad (1.2)$$

where {$A_j, B_j, \phi_j$}$_1^2$ are real constants, $A_j > 0$. Our general objective is the study of the long-time behavior of the solution $q(x, t)$ of (1.1)–(1.2).

Equation (1.1) is a nonlinear PDE integrable by the Inverse Scattering Transform method [1]. The Riemann–Hilbert (RH) problem formalism of this method has proved to be highly efficient in studying various properties of solutions of integrable equations, particularly, the long time asymptotics of the solution of the Cauchy problem [21,23] and the small dispersion limit [22,33]. The nonlinear steepest descent method, put into a rigorous shape by Deift and Zhou in [23], was further developed in [21,22] with the introduction of the so-called “g-function mechanism”. This mechanism helps in the realization of the main idea of the method, which is to find a series of transformations of the original RH problem representation of the solution of the nonlinear equation in question leading ultimately to a model RH problem that can be solved explicitly. Moreover, the method makes it possible to obtain not only the main asymptotic term but also provides a way to derive rigorous error estimates, by solving appropriate local (“parametrix”) RH problems. The nonlinear steepest descent method has been successfully applied to study long-time asymptotics for the focusing nonlinear Schrödinger equation (see e.g. [7,9,11,12,32,39]) and other integrable equations (see e.g. [15,18,19,30,36,37,40,41]).

Expressing the $(x, t)$ dependence of the data for the associated RH problem (jump matrices and, if appropriate, residue conditions) in terms of $\xi := \frac{x}{t}$ and the large parameter $t$, it is natural to obtain large $t$ asymptotics for a fixed $\xi$ or for $\xi$ varying in certain intervals. These intervals are characterized by the fact that the asymptotics of the solution $q$ has the same qualitative form in any compact subset of each interval. In terms of the model RH problem, the intervals are such that the jump contour and the jump condition have the same structure within each interval, with only some parameters (as functions of $\xi$) being varying. On the other hand, the end points of the intervals correspond to changes in this structure: either some parts of the contour shrink to single points or new parts emerge from specific points. Accordingly, uniform error estimates break down when approaching the end points.

It turns out that already for problems with “zero boundary conditions”, i.e., when the solution is assumed to decay to 0 as $|x| \to \infty$, there exist various narrow transition regions in the $(x, t)$ plane, where the asymptotics has a qualitatively different form [21]. So it is natural to expect transition regions also in the more complicated situation of “non-zero boundary conditions” described by (1.2) (for other types of non-zero boundary conditions, see e.g. [2,8–10,24–26,28,29,31]).

Scenarios of long-time asymptotics of $q(x, t)$, where the $(x, t)$ half-plane is divided into sectors $\xi_1 < \xi < \xi_2$ with qualitatively different asymptotics, were presented in [13]. That work was motivated by [17], where the asymptotic scenario was obtained for a particular range of parameters in (1.2). In the “shock case” $B_1 < B_2$ some scenarios include genus 3 sectors where the leading term of the asymptotics of $q(x, t)$ is given in terms of theta functions attached to a hyperelliptic Riemann surface $M(\xi)$ of genus 3.

Of particular interest is the “symmetric shock case” in which

$$A := A_1 = A_2 > 0 \quad \text{and} \quad B := B_2 = -B_1 > 0. \quad (1.3)$$
<figure><img src="image_2.png" /><figcaption>Fig. 1.1. The asymptotic scenario in the symmetric shock case (1.3) for $1 < \frac{A}{B} < \frac{2}{7}(2 + 3\sqrt{2})$ (top), $\frac{A}{B} = \frac{2}{7}(2 + 3\sqrt{2})$ (middle), and $\frac{A}{B} > \frac{2}{7}(2 + 3\sqrt{2})$ (bottom).</figcaption></figure>

In this case, it was shown in [13] that there are five possible asymptotic scenarios depending on the value of the ratio $A/B$. In the third, fourth, and fifth of the scenarios presented in [13] (corresponding to $1 < \frac{A}{B} < \frac{2}{7}(2 + 3\sqrt{2})$, $\frac{A}{B} = \frac{2}{7}(2 + 3\sqrt{2})$, and $\frac{A}{B} > \frac{2}{7}(2 + 3\sqrt{2})$, respectively), the asymptotic picture involves two genus 3 sectors separated by the line $\xi = 0$, see Fig. 1.1. The asymptotic analysis in such genus 3 sectors was performed in [14]. In particular, [14, Theorem 2.1] implies that, if $A/B > 1$ (so that we are in the third, fourth, or fifth scenario), then there are intervals $[-\xi_2, 0)$ and $(0, \xi_2]$ with $\xi_2 > 0$ such that for any fixed $\xi \in [-\xi_2, 0) \cup (0, \xi_2]$ the leading behavior of $q(x, t)$ as $t \to +\infty$ can be expressed in terms of theta functions associated with a genus 3 Riemann surface $M(\xi)$. Moreover, the convergence is uniform for $\xi$ in compact subsets of $[-\xi_2, 0) \cup (0, \xi_2]$. However, the genus 3 theorem in [14] does not give the asymptotics in a narrow transition zone surrounding the line $\xi = 0$. The purpose of this paper is to derive the asymptotic behavior of the solution $q(x, t)$ of (1.1) on the line $\xi = 0$ and in a narrow region containing this line.

Although the behavior of the solution $q(x, t)$ in the transition zone around $\xi = 0$ is of interest in its own right, our original motivation for considering the behavior on and near the line $\xi = 0$
was to reach a better understanding of the initial-boundary value problem for (1.1) posed on the half-line $x \ge 0$. The most challenging step in the analysis of (1.1) on the half-line is to construct the generalized Dirichlet-to-Neumann map, which expresses the unknown boundary values in terms of the given initial and boundary data. For the purposes of calculating the long-time behavior of the solution, it is often sufficient to know the asymptotics of the boundary values. An important source of explicit examples of half-line solutions is obtained by restricting solutions on the line to the half-line $x \ge 0$. In this context, it is of interest to compute the asymptotics of the Dirichlet and Neumann values, $q(0, t)$ and $q_x(0, t)$, of the restriction of the solution $q(x, t)$ of (1.1)–(1.2) to the half-line $x \ge 0$. We therefore seek a formula which is valid on the line $\xi = 0$ and close to this line, so that both $q(0, t)$ and the $x$-derivative $q_x(0, t)$ can be retrieved.

As $\xi \to 0$, the genus 3 Riemann surface $M(\xi)$ degenerates to a genus 1 Riemann surface $\tilde{M}$. Our main result expresses the asymptotics of $q(x, t)$ in a sector $S$ containing the line $\xi = 0$ in terms of elliptic theta functions associated with $\tilde{M}$. In fact, since it entails no additional work, we will treat the more general case of two genus 3 sectors $\xi_1 < \xi < \xi_0$ and $\xi_0 < \xi < \xi_2$ separated by a point $\xi_0 \in \mathbb{R}$ at which the genus 3 Riemann surface $M(\xi)$ degenerates to a genus 1 Riemann surface $\tilde{M}$. In the symmetric case (1.3), it is always the case that $\xi_0 = 0$, but by allowing for nonzero values of $\xi_0$, our result applies also to non-symmetric situations.

## 1.1. Organization of the paper

The main theorem is stated in Section 2. Our proof of this theorem is based on a steepest descent analysis of a RH problem which is described in Section 3. In Section 3, we also introduce some necessary notation. In Section 4, we implement a number of transformations of the RH problem that are required for the steepest descent analysis. In Section 5, we construct a global parametrix by solving a model RH problem in terms of theta functions associated to a genus 1 Riemann surface. The global parametrix eventually gives rise to the leading term of $O(1)$ in the final asymptotic formula for $q(x, t)$. In Section 6, we construct local parametrices; in particular, we construct a local parametrix in a neighborhood of two merging branch points $\alpha$ and $\beta$. This is achieved by relating the original RH problem to an exactly solvable RH problem whose solution is presented in the appendix. The local parametrices eventually give rise to the subleading terms in the final asymptotic formula for $q(x, t)$. The proof of the main theorem is finalized in Section 7.

## 2. Main result

Consider the Cauchy problem (1.1)–(1.2) in the shock case $B_1 < B_2$. For simplicity, we will assume that (1.2) holds not only asymptotically as $x \to \pm\infty$, but that $q(x, 0)$ is identically equal to a plane wave for $|x|$ large enough. More precisely, we suppose that $q: \mathbb{R} \times [0, \infty) \to \mathbb{C}$ is a smooth solution of (1.1) satisfying

$$ q(x, 0) = \begin{cases} q_{01}(x, 0) & \text{for } x < -C, \\ q_{02}(x, 0) & \text{for } x > C, \end{cases} \qquad (2.1) $$

where $q_{0j}(x, 0) = A_j e^{i\phi_j} e^{-2iB_j x}$, $j = 1, 2$, for some real constants $C > 0$ and $\{A_j, B_j, \phi_j\}_1^2$ with $A_1 > 0$, $A_2 > 0$, and $B_1 < B_2$.
<figure><img src="image_3.png" /><figcaption>Fig. 2.1. The vertical segments $\Sigma_1 = [\bar{E}_1, E_1]$ and $\Sigma_2 = [\bar{E}_2, E_2]$ together with the real axis.</figcaption></figure>

The conditions (2.1) at time $t=0$ must be supplemented with some boundary conditions for $t > 0$. The natural assumption is that $q(x, t) \sim q_{0j}(x, t)$ as $x \to (-1)^j \infty$ for $j = 1, 2$, where $q_{0j}(x, t)$ are the plane wave solutions of the NLS equation with initial data $q_{0j}(x, 0)$, that is,

$$q_{0j}(x, t) = A_j e^{i\phi_j} e^{-2iB_j x + 2i\omega_j t}, \quad \omega_j := A_j^2 - 2B_j^2. \qquad (2.2)$$

More precisely, we assume that $q(x, t)$ satisfies

$$\int_{0}^{(-1)^j \infty} |q(x, t) - q_{0j}(x, t)| dx < \infty \quad \text{for } t \ge 0 \text{ and } j = 1, 2. \qquad (2.3)$$

## 2.1. Spectral functions

The formulation of our main result involves two spectral functions $a(k)$ and $b(k)$ defined as follows, cf. [13, Section 2]. For $j = 1, 2$, let $E_j := B_j + iA_j$ and let $\Sigma_j := [\bar{E}_j, E_j]$ be the vertical segment $[\bar{E}_j, E_j]$ oriented upward, see Fig. 2.1. Define

$$X_j(k) := \sqrt{(k - E_j)(k - \bar{E}_j)},$$

$$v_j(k) := \left( \frac{k - E_j}{k - \bar{E}_j} \right)^{\frac{1}{4}}, \quad \mathcal{E}_j(k) := \frac{1}{2} \begin{pmatrix} v_j(k) + v_j(k)^{-1} & v_j(k) - v_j(k)^{-1} \\ v_j(k) - v_j(k)^{-1} & v_j(k) + v_j(k)^{-1} \end{pmatrix},$$

where the branches are such that these functions are analytic in $C \setminus \Sigma_j$ and satisfy

$$X_j(k) = k - B_j + O(k^{-1}), \quad v_j(k) = 1 + O(k^{-1}), \quad \mathcal{E}_j(k) = I + O(k^{-1})$$

as $k \to \infty$. Let $\sigma_j, j = 1, 2, 3$, denote the three Pauli matrices. Let

$$Q_0(x) := \begin{pmatrix} 0 & q(x, 0) \\ -q(x, 0) & 0 \end{pmatrix}, \quad Q_{0j}(x) := \begin{pmatrix} 0 & q_{0j}(x, 0) \\ -q_{0j}(x, 0) & 0 \end{pmatrix},$$

and define the background solutions $\Phi_{0j}(x, k)$, $j = 1, 2$, by
$$ \Phi_{0j}(x, k) := e^{-iB_j x \sigma_3} N_j(k) e^{-iX_j(k)x \sigma_3}, \quad \text{where } N_j(k) := e^{\frac{i\phi_j}{2} \sigma_3} \mathcal{E}_j(k) e^{-\frac{i\phi_j}{2} \sigma_3}. $$

Let $\mu_j(x, k)$, $j = 1, 2$, be the unique solutions of the linear Volterra integral equations

$$ \mu_j(x, k) = e^{-iB_j x \sigma_3} N_j(k) + \int_{(-1)^j \infty}^{x} \Phi_{0j}(x, k) \Phi_{0j}^{-1}(y, k) [Q_0(y) - Q_{0j}(y)] \mu_j(y, k) e^{-iX_j(k)(y-x)\sigma_3} dy. $$

The functions $a(k)$ and $b(k)$ are defined by the relation

$$ \Phi_2(x, k) = \Phi_1(x, k) \begin{pmatrix} a^*(k) & b(k) \\ -b^*(k) & a(k) \end{pmatrix}, \quad k \in \mathbb{R} \setminus \{B_1, B_2\}, $$

where $\Phi_j(x, k) := \mu_j(x, k)e^{-iX_j(k)x \sigma_3}$ and we use the notation $f^*(k) := \overline{f(\bar{k})}$ for the Schwarz conjugate of a complex-valued function $f(k)$. Under the conditions (2.1), the spectral functions $a(k)$ and $b(k)$ extend to analytic functions of $k \in \mathbb{C} \setminus (\Sigma_1 \cup \Sigma_2)$ with jumps across $\Sigma_1 \cup \Sigma_2$. Furthermore, $a(k) = 1 + O(1/k)$ as $k \to \infty$ in $\mathbb{C}^+$ and $b(k) = O(1/k)$ as $k \to \infty$ for $k \in \mathbb{R}$, where we write $\mathbb{C}^{\pm} := \{\pm \operatorname{Im} k > 0\}$ for the open upper and lower halves of the complex plane.

**Remark.** More generally, if $q(x, 0)$ is not necessarily identically equal to $q_{0j}(x, 0)$ for large $|x|$ but approaches $q_{0j}(x, 0)$ sufficiently fast as $x \to (-1)^{j\infty}$, then $a(k)$ is an analytic function of $k \in \mathbb{C}^+ \setminus (\Sigma_1 \cup \Sigma_2)$ and $b(k)$ is a continuous function of $k \in \mathbb{R} \setminus (\Sigma_1 \cup \Sigma_2)$, see [13, Proposition 2.1]. By making the stronger assumption (2.1), we ensure that $a(k)$ and $b(k)$ extend to analytic functions of $k \in \mathbb{C} \setminus (\Sigma_1 \cup \Sigma_2)$. This allows us to work directly with these functions thus avoiding analytical approximations, which would make the realization of the main ideas of the asymptotic analysis less transparent.

We make the following assumption, which in particular implies that the solution is solitonless.

**Assumption.** We assume that $a(k) \neq 0$ and $b(k) \neq 0$ for $k \in \mathbb{C}^+ \cup \mathbb{R}$.

We let $r(k)$ be the reflection coefficient defined by (see [13, (2.36)])

$$ r(k) := \frac{b^*(k)}{a(k)}. $$

## 2.2. The transition zone

Our main result gives the long-time behavior of $q(x, t)$ in an asymptotic sector $\mathcal{S}$. To define $\mathcal{S}$, we need to introduce the genus 3 Riemann surface $M(\xi)$, which is defined by the equation (see [14, Section 3.2])

$$ w^2 = (k - E_1)(k - \bar{E}_1)(k - E_2)(k - \bar{E}_2)(k - \alpha)(k - \bar{\alpha})(k - \beta)(k - \bar{\beta}) \quad (2.4) $$
<figure><img src="image_4.png" /><figcaption>Fig. 2.2. The set of branch cuts $C = \Sigma_1 \cup \Sigma_2 \cup \gamma_{(\beta, \alpha)} \cup \gamma_{(\bar{\alpha}, \bar{\beta})}$ (red) together with the regions where $\operatorname{Im} g > 0$ (white) and $\operatorname{Im} g < 0$ (shaded). (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)</figcaption></figure>

where $\alpha \equiv \alpha(\xi)$ and $\beta \equiv \beta(\xi)$ are critical points of the $g$-function $g(\xi, k)$, that is, zeros of $dg/dk$. More precisely, $\alpha$ and $\beta$ are defined as follows. Introduce the set of branch cuts

$$C := \Sigma_1 \cup \Sigma_2 \cup \gamma_{(\beta, \alpha)} \cup \gamma_{(\bar{\alpha}, \bar{\beta})}, \qquad (2.5)$$

where $\gamma_{(\beta, \alpha)}$ is a contour from $\beta$ to $\alpha$, and $\gamma_{(\bar{\alpha}, \bar{\beta})}$ is a contour from $\bar{\alpha}$ to $\bar{\beta}$, see Fig. 2.2. Let $dg(k) \equiv dg(\xi, k)$ denote the differential on $\mathbb{C} \setminus C$ defined by

$$dg := 4 \frac{(k - \mu)(k - \alpha)(k - \bar{\alpha})(k - \bar{\beta})(k - \bar{\beta})}{w(k)} dk, \quad k \in \mathbb{C} \setminus C, \qquad (2.6)$$

where $\mu \equiv \mu(\xi)$ is a real number and

$$w(k) := \sqrt{(k - E_1)(k - \bar{E}_1)(k - E_2)(k - \bar{E}_2)(k - \alpha)(k - \bar{\alpha})(k - \beta)(k - \bar{\beta})}.$$

We view $M$ as a two-sheeted cover of the complex $k$-plane and let $\hat{dg}$ denote the differential on $M$ given by $dg$ on the upper sheet and by $-dg$ on the lower sheet. Let $\alpha_1, \beta_1$ and $\alpha_2, \beta_2$ denote the real and imaginary parts of $\alpha$ and $\beta$:

$$\alpha = \alpha_1 + i\alpha_2, \quad \beta = \beta_1 + i\beta_2.$$
<figure><img src="image_5.png" /><figcaption>Fig. 2.3. The canonical homology basis {$a_j, b_j$}$_{j=1}^3$ for the genus three surface M.</figcaption></figure>

The five real numbers $\alpha_1, \beta_1, \alpha_2, \beta_2$, and $\mu$ are determined by the five conditions

$$ \int_{a_1} \hat{dg} = \int_{a_2} \hat{dg} = \int_{a_3} \hat{dg} = 0, \qquad (2.7a) $$

$$ \lim_{k \to \infty} \left( \frac{dg}{dk} - 4k \right) = \xi, \quad \lim_{k \to \infty} k \left( \frac{dg}{dk} - 4k - \xi \right) = 0, \qquad (2.7b) $$

where {$a_j, b_j$}$_{j=1}^3$ is the canonical homology basis for the genus three surface M displayed in Fig. 2.3. The g-function $g(k) \equiv g(\xi, k)$ is defined for $k \in \mathbb{C} \setminus \mathcal{C}$ by

$$ g(k) := \int_{\tilde{E}_2}^{k} dg, \qquad (2.8) $$

where the contour lies in $\mathbb{C} \setminus \mathcal{C}$. As in [14], we henceforth choose the branch cuts so that $\operatorname{Im} g = 0$ on $\gamma(\beta, \alpha) \cup \gamma(\tilde{\alpha}, \tilde{\beta})$. We also let $\gamma(\mu, \beta)$ be a curve from $\mu$ to $\beta$ on which $\operatorname{Im} g = 0$, see dashed curve in Fig. 2.2. We define the complex-valued function $\tilde{v}(\xi)$ by

$$ \tilde{v}(\xi) = \frac{1}{2\pi} \ln(1 + r(\beta(\xi))r^*(\beta(\xi))), \qquad (2.9) $$

where the branch of $\ln(1 + rr^*)$ is fixed by requiring that $\ln(1 + r(k)r^*(k))$ is a continuous function of $k \in \gamma(\mu, \beta)$ whose value at $k = \mu$ is strictly positive.
In the symmetric shock case (1.3) with $A/B > 1$, the branch points $\alpha(\xi)$ and $\beta(\xi)$ merge at the point $\alpha(0) = \beta(0) = i\sqrt{A^2 - B^2}$ on the imaginary axis as $\xi \to 0$ (see Fig. 2.2 for the location of $\alpha$ and $\beta$ when $\xi$ is small but positive) and there exists a genus 3 sector on either side of the line $\xi = 0$ (see Fig. 1.1). More generally, we assume that $\alpha(\xi)$ and $\beta(\xi)$ merge at the point $\alpha(\xi_0) = \beta(\xi_0)$ for some $\xi_0 \in \mathbb{R}$. Let $\tilde{v}_0 := \tilde{v}(\xi_0)$ denote the value of $\tilde{v}$ at $\xi = \xi_0$. We will consider the asymptotics in the sector $\mathcal{S}$ defined by

$$ \mathcal{S} := \left\{ (x, t) \in \mathbb{R}^2 \mid \xi \ge \xi_0, t \ge T, |\alpha(\xi) - \alpha(\xi_0)| + |\beta(\xi) - \beta(\xi_0)| < Mt^{-\frac{1+|\mathrm{Im}\,\tilde{v}_0|}{2}-\delta} \right\}, \quad (2.10) $$

where the constants $\delta, M, T$ are strictly positive. The sector $\mathcal{S}$ is characterized by the fact that $\alpha(\xi)$ and $\beta(\xi)$ approach $\alpha(\xi_0) = \beta(\xi_0)$ faster than $t^{-\frac{1+|\mathrm{Im}\,\tilde{v}_0|}{2}}$ as $t \to +\infty$.

**Remark.** For definiteness, we assume that $\xi \ge \xi_0$ in $\mathcal{S}$, i.e., $\mathcal{S}$ contains the line $\xi = \xi_0$ and extends to the right of this line in the $(x, t)$-plane. A similar analysis can be used to obtain the asymptotics in the analogous sector which contains the line $\xi = \xi_0$ and extends to the left it.

Since our main result gives no information on the asymptotics for $|\mathrm{Im}\,\tilde{v}_0| \ge \frac{1}{2}$, we make the following assumption.

**Assumption.** We suppose that

$$ \mathrm{Im} \, \tilde{v}_0 \in \left(-\frac{1}{2}, \frac{1}{2}\right), \quad (2.11) $$

i.e. $\arg(1 + r(\beta(\xi_0))r^*(\beta(\xi_0))) \in (-\pi, \pi)$.

See Remarks 2.7 and 2.8 for classes of initial data for which this assumption is fulfilled.

The asymptotics of $q$ in $\mathcal{S}$ can be expressed in terms of quantities defined on the genus 1 Riemann surface $\tilde{M}$ with branch cuts along $\Sigma_1$ and $\Sigma_2$ defined by the equation

$$ \tilde{w}^2 = (k - E_1)(k - \tilde{E}_1)(k - E_2)(k - \tilde{E}_2). \quad (2.12) $$

The genus 3 surface $M(\xi)$ degenerates to $\tilde{M}$ in the limit $\xi \to \xi_0$.

The following is our main result. The theorem is formulated under the assumption that $B_1 = -1$, $B_2 = 1$, and $\phi_2 = 0$; a simple scaling argument shows that this implies no loss of generality, see [13, Section 2.2].

**Theorem 2.1.** The asymptotics in the sector $\mathcal{S}$ is given by

$$ q(x,t) = Q_0(\xi, t) + \frac{Q_1(\xi, t)}{\sqrt{t}} + O(F(\xi, t)^2 + t|\alpha - \beta|^2), \quad t \to +\infty, (x,t) \in \mathcal{S}. \quad (2.13) $$

• The first term $Q_0$ is the leading order term. It is given by

$$ Q_0(\xi, t) = e^{2i(tg^{(0)}(\xi) + \tilde{h}(\xi, \infty))} (A_1 + A_2) \frac{\tilde{\Theta}(\tilde{\varphi}(\infty^+) + \tilde{d})\tilde{\Theta}(\tilde{\varphi}(\infty^+) - \tilde{v}(\xi, t) - \tilde{d})}{\tilde{\Theta}(\tilde{\varphi}(\infty^+) + \tilde{v}(\xi, t) + \tilde{d})\tilde{\Theta}(\tilde{\varphi}(\infty^+) - \tilde{d})} $$
• The coefficient $Q_1$ of the second term is given by

$$Q_1(\xi, t) = 2ie^{2i(tg^{(0)}(\xi) + \tilde{h}(\xi, \infty))} \left( (T_{\mu}(x, t))_{12} + (T_{\beta}(x, t))_{12} - \overline{(T_{\beta}(x, t))_{21}} \right).$$

• Regarding the last term, the estimate is uniform with respect to $x$ and the function $F$ is given by

$$F(\xi, t) := t^{\text{Im}\,\tilde{v}|} (\ln t)^{|\text{Im}\,\tilde{v}|+2} \left( t|\alpha - \beta|^2 |\ln|\alpha - \beta|| + t^{-\frac{1}{2}} \right). \quad (2.14)$$

$\tilde{\Theta}$ is the Riemann theta function associated with the genus 1 Riemann surface $\tilde{M}$, the Abel map $\tilde{\phi}$ is defined in (3.6). The functions $\tilde{v}(t) \equiv \tilde{v}(\xi, t)$ and $\tilde{h}(k) \equiv \tilde{h}(\xi, k)$ are defined in (5.5) and (4.10), respectively. The constants $\tilde{v} \equiv \tilde{v}(\xi)$, $\tilde{d} \equiv \tilde{d}(\xi)$, and $g^{(0)} \equiv g^{(0)}(\xi)$ are defined in (2.9), (5.4), and [14, (3.28)], respectively. The functions $T_{\mu}(x, t)$ and $T_{\beta}(x, t)$ are defined in (7.10) and (7.11), respectively.

**Remark 2.2.** For any $\epsilon > 0$, the functions $T_{\mu}(x, t)$ and $T_{\beta}(x, t)$ satisfy the uniform estimates

$$\begin{aligned} |T_{\mu}(x, t)| &\le C, & \text{for } (x, t) \in \mathcal{S}. \quad (2.15) \\ |T_{\beta}(x, t)| &\le C t^{|\text{Im}\,\tilde{v}_0|+\epsilon}, & \end{aligned}$$

Indeed, all factors involved in (7.10) and (7.11) are bounded, with the exception of the factors $t^{\pm \frac{\tilde{v}}{2}\sigma_3}$. Moreover, by (2.10), $|\text{Im}\,\tilde{v}| = |\text{Im}\,\tilde{v}_0| + O(t^{-\frac{1+|\text{Im}\,\tilde{v}_0|}{2}-\delta})$ which implies $t^{|\text{Im}\,\tilde{v}|} = O(t^{|\text{Im}\,\tilde{v}_0|+\epsilon})$ for any $\epsilon > 0$. It follows from (2.11) and (2.15) that the second term on the right-hand side of (2.13), $Q_1/\sqrt{t}$, decays for large $t$, even though the coefficient $Q_1$ itself may grow with $t$.

**Remark 2.3.** The first term on the right-hand side of (2.13), $Q_0(\xi, t)$, is a bounded and oscillating term obtained by solving the “model” RH problem (see Section 5). It dominates the other two terms, $Q_1/\sqrt{t}$ and $O(F(\xi, t)^2 + t|\alpha - \beta|^2)$, which decay.

**Remark 2.4.** The region $\mathcal{S}$ depends on 4 parameters: $|\text{Im}\,\tilde{v}_0|$, $\delta$, $M$, and $T$. The first one, $|\text{Im}\,\tilde{v}_0|$, reflects the influence (via the reflection coefficient) of the initial data on the Cauchy problem while the others are at our choice, $\delta$ being the most important one. A smaller choice of $\delta > 0$ increases the size of the asymptotic sector $\mathcal{S}$, but it also makes the asymptotic formula less precise.

On the line $\xi = \xi_0$, we can take $\delta$ arbitrarily large and Theorem 2.1 reduces to the following.

**Corollary 2.5.** The asymptotics on the line $\xi = \xi_0$ is given by

$$q(x, t) = Q_0(\xi_0, t) + \frac{Q_1(\xi_0, t)}{\sqrt{t}} + O\left(t^{2|\text{Im}\,\tilde{v}_0|-1} (\ln t)^{2|\text{Im}\,\tilde{v}_0|+4}\right), \quad t \to +\infty, x = \xi_0 t. \quad (2.16)$$

On the other hand, keeping only the leading order term, Theorem 2.1 reduces to the following.
**Corollary 2.6 (Leading order asymptotics).** There exists a $c > 0$ such that

$$q(x, t) = Q_0(\xi, t) + O(t^{-c}), \quad t \to +\infty, \quad (x, t) \in \mathbb{S},$$

uniformly with respect to $x$.

**Remark 2.7.** The assumption (2.11) is at least satisfied in the symmetric case (1.3) whenever $A/B > 1$ is sufficiently close to 1. Indeed, in the symmetric case, we have $\xi_0 = 0$ and $\alpha(0) = \beta(0) = i\sqrt{A^2 - B^2} \to 0$ as $A/B \downarrow 1$. Moreover, if $\beta(0)$ is real, then $\mathrm{Im} \tilde{v}_0 = \mathrm{Im} \tilde{v}(0) = 0$, so by continuity $|\mathrm{Im} \tilde{v}_0| < 1/2$ whenever $A/B > 1$ is sufficiently close to 1.

**Remark 2.8.** The assumption (2.11) is also satisfied for any value of $A/B > 1$ in the symmetric case (1.3) provided that the initial data is a sufficiently small perturbation of the pure step-like profile

$$q_{0, \text{pure step}}(x) = \begin{cases} q_{01}(x, 0) & \text{for } x < 0, \\ q_{02}(x, 0) & \text{for } x > 0. \end{cases}$$

Indeed, if $q(x, 0) = q_{0, \text{pure step}}(x)$, then

$$\begin{pmatrix} a^*(k) & b(k) \\ -b^*(k) & a(k) \end{pmatrix} = e^{\frac{i\phi_1}{2}\sigma_3} \mathcal{E}_1(k)^{-1} e^{-\frac{i\phi_1}{2}\sigma_3} e^{\frac{i\phi_2}{2}\sigma_3} \mathcal{E}_2(k) e^{-\frac{i\phi_2}{2}\sigma_3},$$

and hence a calculation yields

$$1 + r(iy)r^*(iy) = f(A/B, y/B, \phi_1 - \phi_2) \quad \text{for all } y \ge 0,$$

where

$$f(A, y, \phi) := \frac{2((A+y)^2+1)\sqrt{1-\frac{4Ay}{(A+y)^2+1}}}{((A+y)^2+1)\sqrt{1-\frac{4Ay}{(A+y)^2+1}} + y^2 + 1 - A^2 \cos \phi}.$$

Observing that $f(A, y, \phi) > 0$ for all $A > 1$, $y \ge 0$, and $\phi \in \mathbb{R}$, and using that $\beta(0) = i\sqrt{A^2 - B^2}$, we conclude that

$$\tilde{v}_0 = \frac{1}{2\pi} \ln(1 + r(\beta(0))r^*(\beta(0))) = \frac{1}{2\pi} \ln f(A/B, \sqrt{(A/B)^2 - 1}, \phi_1 - \phi_2) \in \mathbb{R}.$$

This shows that $\mathrm{Im} \tilde{v}_0 = 0$ for the initial data $q_{0, \text{pure step}}(x)$, and hence (2.11) holds whenever $q(x, 0)$ is a sufficiently small perturbation of $q_{0, \text{pure step}}(x)$ by continuity.

**Remark 2.9.** Let us describe how Theorem 2.1 complements the asymptotic result of [14]. The genus 3 theorem [14, Theorem 2.1] gives information on the asymptotics of $q(x, t)$ in the genus 3 sectors $\xi \in (\xi_1 + \delta, \xi_0 - \delta)$ and $\xi \in (\xi_0 + \delta, \xi_2 - \delta)$ for any $\delta > 0$. In fact, by choosing the $\epsilon$-disks centered at $\alpha, \bar{\alpha}, \beta, \bar{\beta}$ to have $\xi$-dependent radius $\epsilon = \epsilon(\xi)$ in the proof of [14, Theorem 2.1], it can be seen that the genus 3 formula is valid uniformly for all $\xi_0 < \xi \le \xi_2$ if we include the
additional error term $O((t\epsilon^{3/2})^{-N})$. By choosing $\epsilon = \frac{|\alpha - \beta|}{3}$, we infer that the genus 3 theorem [14, Theorem 2.1] gives information on the solution whenever $t^{2/3}|\alpha - \beta| \to \infty$; however, it does not give any information on the asymptotics in the narrow region around the line $\xi = \xi_0$ in which $t^{2/3}|\alpha - \beta|$ remains bounded. Theorem 2.1 complements [14, Theorem 2.1] in that it gives the asymptotics in a region containing the line $\xi = \xi_0$ in which $t^{2/3}|\alpha - \beta|$ is allowed to stay bounded. In particular, Theorem 2.1 meets our objective mentioned in the introduction of yielding the large time behavior of $q$ and $q_x$ on the line $\xi = \xi_0$.

**Remark 2.10.** Numerical computations suggest that $|\alpha(\xi) - \alpha(\xi_0)| \sim |\xi - \xi_0|^{1/2}$ and $|\beta(\xi) - \beta(\xi_0)| \sim |\xi - \xi_0|^{1/2}$ as $\xi \downarrow \xi_0$ (at least in the symmetric case). If this is correct, then

$$ S = \{ (x, t) \in \mathbb{R}^2 \mid t \ge T, 0 \le \xi - \xi_0 < Mt^{-|\operatorname{Im} \tilde{v}_0|-1-2\delta} \}. \quad (2.17) $$

**Remark 2.11.** It is an interesting open question to understand how the asymptotic formula of Theorem 2.1 transitions into the genus 3 formula of [14, Theorem 2.1] as one moves further away from the line $\xi = \xi_0$. It is likely that this transition, which is expected to involve new finite-amplitude oscillations, can be understood by using the ideas of [4–6,16,20].

**Remark 2.12.** A degeneration of genus 2 (two-phase) solutions of (1.1) corresponding to the merging of two pairs of spectral points is considered in [3], where it is shown that it is possible to present the limiting behavior of the solution directly in terms of the solution of the degenerated RH problem, without resorting to the genus 2 solution before degeneration.

# 3. Preliminaries

## 3.1. Notations

The Riemann sphere will be denoted by $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$. We write $\ln k$ for the logarithm with the principal branch, that is, $\ln k = \ln|k| + i \arg k$ where $\arg k \in (-\pi, \pi]$. Unless specified otherwise, all complex powers will be defined using the principal branch, i.e., $z^\alpha = e^{\alpha \ln z}$. We let $f^*(k) := f(\bar{k})$ denote the Schwarz conjugate of a complex-valued function $f(k)$.

Given an open subset $D \subset \hat{\mathbb{C}}$ bounded by a piecewise smooth contour $\Sigma$, we let $\dot{E}_2(D)$ denote the Smirnoff class consisting of all functions $f(k)$ analytic in $D$ with the property that for each connected component $D_j$ of $D$ there exist curves $\{C_n\}_{n=1}^\infty$ in $D_j$ such that the $C_n$ eventually surround each compact subset of $D_j$ and $\sup_{n \ge 1} \|f\|_{L^2(C_n)} < \infty$. We let $E^\infty(D)$ denote the space of bounded analytic functions $D \to \mathbb{C}$. RH problems in the paper are generally 2 × 2 matrix-valued. They are formulated in the $L^2$-sense using Smirnoff classes (see [34,35]):

$$ \begin{cases} m \in I + \dot{E}^2(\mathbb{C} \setminus \Sigma), \\ m_+(k) = m_-(k) J(k) & \text{for a.e. } k \in \Sigma, \end{cases} \quad (3.1) $$

where $m_+$ and $m_-$ denote the boundary values of $m$ from the left and right sides of the contour $\Sigma$.
## 3.2. Set-up

Assume without loss of generality that $B_2 = -B_1 = 1$, $\phi_2 = 0$, and $\phi_1 = \phi$, see [13, Section 2.2]. According to [14, Proposition 3.1], the solution $q(x, t)$ of the Cauchy problem (1.1)-(1.2) is given by

$$q(x, t) = 2i \lim_{k \to \infty} k (\hat{m}(x, t, k))_{12}, \quad x \in \mathbb{R}, t \in [0, \infty), \qquad (3.2)$$

where $\hat{m}(x, t, \cdot)$ is the unique solution of the basic RH problem (see [14, (3.12)])

$$\begin{cases} \hat{m}(x, t, \cdot) \in I + \dot{E}^2 (\mathbb{C} \setminus (\mathbb{R} \cup \Sigma_1 \cup \Sigma_2)), \\ \hat{m}_+(x, t, k) = \hat{m}_-(x, t, k) \hat{J}(x, t, k) & \text{for a.e. } k \in \mathbb{R} \cup \Sigma_1 \cup \Sigma_2, \end{cases} \qquad (3.3)$$

with $\hat{J}$ defined as in [14, (3.11)]:

$$\hat{J}(x, t, k) := e^{-it\theta(k)\sigma_3} \hat{J}_0(k) e^{it\theta(k)\sigma_3}, \quad k \in \Sigma := \mathbb{R} \cup \Sigma_1 \cup \Sigma_2, \qquad (3.4a)$$

where the phase $\theta(k)$ is

$$\theta(k) \equiv \theta(\xi, k) := 2k^2 + \xi k, \quad \xi := \frac{x}{t}, \qquad (3.4b)$$

and

$$\hat{J}_0(k) = \begin{cases} \begin{pmatrix} 1 & \hat{r}^* \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \hat{r} & 1 \end{pmatrix} = \begin{pmatrix} 1 + \hat{r}\hat{r}^* & \hat{r}^* \\ \hat{r} & 1 \end{pmatrix}, & k \in \mathbb{R}, \\ \begin{pmatrix} -i & 0 \\ \frac{ie^{-i\phi}}{\hat{a}_+\hat{a}_-} & i \end{pmatrix}, & k \in \Sigma_1 \cap \mathbb{C}^+, \\ \begin{pmatrix} \frac{\hat{a}_-}{\hat{a}_+} & i\nu_1^2 \\ 0 & \frac{\hat{a}_+}{\hat{a}_-} \end{pmatrix}, & k \in \Sigma_2 \cap \mathbb{C}^+, \\ \begin{pmatrix} -i & \frac{ie^{i\phi}}{\hat{a}_+^*\hat{a}_-^*} \\ 0 & i \end{pmatrix}, & k \in \Sigma_1 \cap \mathbb{C}^-, \\ \begin{pmatrix} \frac{\hat{a}_+^*}{\hat{a}_-^*} & 0 \\ i\nu_1^{-2} & -\frac{\hat{a}_-^*}{\hat{a}_+^*} \end{pmatrix}, & k \in \Sigma_2 \cap \mathbb{C}^- \end{cases} \qquad (3.4c)$$

Recall [14, (3.10)] that $\hat{a}$, $\hat{b}$, and $\hat{r}$ are slight modifications of the scattering data $a$, $b$, and $r$:

$$\hat{a} := av_1, \quad \hat{b} := bv_1, \quad \hat{r} := \frac{\hat{b}^*}{\hat{a}} = rv_1^{-2}, \qquad (3.5)$$

where $v_1(k) := \left(\frac{k-E_1}{k-E_1}\right)^{\frac{1}{4}}$ with $v_1(\infty) = 1$, as in [14, (3.7)]. Note that $\hat{a}\hat{a}^* = aa^*$, $\hat{b}\hat{b}^* = bb^*$, and $\hat{r}\hat{r}^* = rr^*$.
We introduce a canonical homology basis {$a_1, b_1$} on the genus 1 Riemann surface $\tilde{M}$ defined in (2.12) by letting the cycles $a_1$ and $b_1$ be as in Fig. 2.3. We let $\tilde{\phi}: \tilde{M} \to \mathbb{C}$ be the limit of the Abel map $\varphi$ (see [14, (3.21)]) as $\xi \to \xi_0$, i.e.,

$$ \tilde{\phi}(P) = \int_{\tilde{E}_2}^{P} \tilde{\xi}, \quad P \in \tilde{M}, \qquad (3.6) $$

where {$\tilde{\xi}$} is the normalized basis of $\mathcal{H}^1(\tilde{M})$ dual to the canonical basis {$a_1, b_1$}, i.e., $\tilde{\xi}$ is a holomorphic differential form such that

$$ \int_{a_1} \tilde{\xi} = 1. $$

Let $\tau$ denote the period $\int_{b_1} \tilde{\xi}$ and let $\tilde{\Theta}$ denote the associated theta function which is defined by

$$ \tilde{\Theta}(z) := \sum_{n \in \mathbb{Z}} e^{2\pi i (\frac{1}{2} n^2 \tau + n z)}, \quad z \in \mathbb{C}. \qquad (3.7) $$

As in [14, (3.24)], we let $g(k) \equiv g(\xi, k)$ denote the $g$-function on $M(\xi)$ with derivative given by (2.6). We define $\tilde{w}(k)$ by

$$ \tilde{w}(k) := \sqrt{(k - E_1)(k - \tilde{E}_1)(k - E_2)(k - \tilde{E}_2)}, \quad k \in \mathbb{C} \setminus (\Sigma_1 \cup \Sigma_2), \qquad (3.8) $$

where the branch of the square root is such that $\tilde{w}(k) > 0$ for $k \gg 0$.

Throughout the paper $C, c > 0$ will denote generic constants independent of $k$ and of $(x, t) \in S$, which may change within a computation.

## 4. Transformations of the RH problem

In order to determine the long-time asymptotics of the solution $\hat{m}$ of the RH problem (3.3), we will perform a series of transformations of the RH problem. More precisely, starting with $\hat{m}$, we define functions $\hat{m}^{(j)}(x, t, k)$, $j = 1, \dots, 6$, such that each $\hat{m}^{(j)}$ satisfies an RH problem which is equivalent to the original RH problem (3.3). The RH problem for $\hat{m}^{(j)}$ has the form

$$ \left\{ \begin{array}{l} \hat{m}^{(j)}(x, t, \cdot) \in I + \dot{E}^2(\mathbb{C} \setminus \Sigma^{(j)}), \\ \hat{m}_+^{(j)}(x, t, k) = \hat{m}_-^{(j)}(x, t, k) \hat{v}^{(j)}(x, t, k) \quad \text{for a.e. } k \in \Sigma^{(j)}, \end{array} \right. \qquad (4.1) $$

where the contours $\Sigma^{(j)}$ and jump matrices $\hat{v}^{(j)}$ are specified below.

At each stage of the transformations, $\hat{m}^{(j)}$ and $\hat{v}^{(j)}$ will satisfy the symmetries

$$ \hat{v}^{(j)}(x, t, k) = \begin{cases} \sigma_3 \sigma_1 \overline{\hat{v}^{(j)}(x, t, \bar{k})} \sigma_1 \sigma_3, & k \in \Sigma^{(j)} \setminus \mathbb{R}, \\ \sigma_3 \sigma_1 \overline{\hat{v}^{(j)}(x, t, \bar{k})}^{-1} \sigma_1 \sigma_3, & k \in \Sigma^{(j)} \cap \mathbb{R}, \end{cases} \quad j = 1, \dots, 6, \qquad (4.2) $$
<figure><img src="image_6.png" /><figcaption>Fig. 4.1. The jump contour $\Sigma^{(3)}$ (red with arrows indicating orientation) together with the set where $\operatorname{Im} g = 0$ (dashed black). The region where $\operatorname{Im} g < 0$ is shaded and the region where $\operatorname{Im} g > 0$ is white.</figcaption></figure>

and

$$ \hat{m}^{(j)}(x, t, k) = \sigma_3 \sigma_1 \overline{\hat{m}^{(j)}(x, t, \bar{k})} \sigma_1 \sigma_3, \quad k \in \mathbb{C} \setminus \Sigma^{(j)}, \quad j = 1, \dots, 6. \quad (4.3) $$

### 4.1. First three transformations

The first three transformations are the same as in [14, Sections 4.1-4.3]. The first transformation $\hat{m} \mapsto \hat{m}^{(1)}$ implements the $g$-function: the key idea is to replace the phase function $\theta(\xi, k) = 2k^2 + \xi k$ in the original jump matrix (3.4) with the $g$-function $g(\xi, k)$ in such a way that after a series of transformations of the original RH problem we get a RH problem with jumps that are constant or decay to the identity as $t \to +\infty$. The second transformation $\hat{m}^{(1)} \mapsto \hat{m}^{(2)}$ makes sure that the jump matrix for $k \in (-\infty, \mu)$ has an appropriate triangular factorization. The third transformation $\hat{m}^{(2)} \mapsto \hat{m}^{(3)}$ is a first opening of lenses. This leads to a function $\hat{m}^{(3)}$ which satisfies the RH problem (4.1) for $j = 3$ with jump contour $\Sigma^{(3)}$ displayed in Fig. 4.1 and jump matrix $\hat{v}^{(3)}$ given by the following formulas [14, Section 4.3] ($\hat{v}_l^{(3)}$ denotes the restriction of $\hat{v}^{(3)}$ to the contour labeled by $l$ in Fig. 4.1):

$$ \begin{aligned} \hat{v}_1^{(3)} &= \begin{pmatrix} 1 & 0 \\ \hat{r}\delta^{-2}e^{2itg} & 1 \end{pmatrix}, & \hat{v}_2^{(3)} &= \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, \\ \hat{v}_3^{(3)} &= \begin{pmatrix} 0 & i\hat{a}_+\hat{a}_-e^{i\phi}\delta^2 e^{-it(g_++g_-)} \\ \frac{i\hat{a}_-\hat{a}_+}{\hat{a}_+\hat{a}_-}\delta^{-2}e^{it(g_++g_-)} & 0 \end{pmatrix}, & & \end{aligned} $$
$$ \hat{v}_4^{(3)} = \begin{pmatrix} 0 & i\nu_1^2 \delta^2 e^{-it(g_++g_-)} \\ i\nu_1^{-2} \delta^{-2} e^{it(g_++g_-)} & 0 \end{pmatrix}, $$

$$ \hat{v}_6^{(3)} = e^{-itg_--\sigma_3} \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 \\ \hat{r}\delta^{-2} & \hat{a}\hat{a}^* \end{pmatrix} e^{itg_+\sigma_3}, $$

$$ \hat{v}_8^{(3)} = e^{-itg_--\sigma_3} \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 \\ 0 & 1 \end{pmatrix} e^{itg_+\sigma_3}, $$

and extended to the lower half-plane by means of the symmetry (4.2). Recall [14, Section 4.2] that the complex-valued function $\delta(k) \equiv \delta(\xi, k)$ is defined by

$$ \delta(k) := e^{\frac{1}{2\pi i} \int_{-\infty}^{\mu} \frac{\ln(1+|r(s)|^2)}{s-k} ds}, \quad k \in \mathbb{C} \setminus (-\infty, \mu], $$

where $\mu$ is the only real critical point of the $g$-function $g(k)$.

**Remark 4.1.** $(\hat{v}_2^{(3)})_{12}$ has the opposite sign compared to that in [14] because the orientation of the contour 2 is opposite. The path labeled by 8 in Fig. 4.1 is labeled by 5 in [14, Figure 10].

## 4.2. Fourth transformation

The fourth transformation is also similar to the analogous transformation in [14, Section 4.4]. However, since $\alpha$ and $\beta$ are now merging, we do not need to factor the jump across $\gamma(\beta, \alpha)$, where $\gamma(\beta, \alpha)$ is the contour from $\beta$ to $\alpha$ in Fig. 4.1. As in [14, Figure 11], we let $V_3$ and $V_4$ be open sets that form a lens around $\gamma(\mu, \beta)$. Thus we define $\hat{m}^{(4)}$ for $k$ in the upper-half plane by

$$ \hat{m}^{(4)} := \hat{m}^{(3)} \times \left\{ \begin{array}{ll} \begin{pmatrix} 1 & 0 \\ -\frac{\hat{b}^*}{\hat{a}^2\hat{a}^*} \delta^{-2} e^{2itg} & 1 \\ 1 & -\frac{\hat{b}}{\hat{a}^*} \delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, & k \in V_3, \\ \\ \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, & k \in V_4, \\ \\ I, & \text{elsewhere in } \mathbb{C}^+ \end{array} \right. $$

and extend the definition to the lower half-plane by means of the symmetry (4.3). We note that the function $\delta(\xi, k)$ obeys the uniform bound (see [14, Lemma 4.1])

$$ |\delta(\xi, k)^{\pm 1}| \le C, \quad k \in \mathbb{C} \setminus (-\infty, \mu], \quad (x, t) \in \mathcal{S}. $$

Then $\hat{m}$ satisfies the RH problem (3.3) iff $\hat{m}^{(4)}$ satisfies the RH problem (4.1) for $j = 4$, where $\Sigma^{(4)}$ is the jump contour displayed in Fig. 4.2 and the jump matrix $\hat{v}^{(4)}$ is given by $(\hat{v}_l^{(4)})$ denotes the restriction of $\hat{v}^{(4)}$ to the contour labeled by $l$ in Fig. 4.2):

$$ \hat{v}_1^{(4)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}} \delta^{-2} e^{2itg} & 1 \end{pmatrix}, \qquad \hat{v}_2^{(4)} = \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, $$

$$ \hat{v}_3^{(4)} = \begin{pmatrix} 0 & i\hat{a} + \hat{a}e^{i\phi} \delta^2 e^{-it(g_++g_-)} \\ i\hat{a}e^{-i\phi} \delta^{-2} e^{it(g_++g_-)} & 0 \end{pmatrix}, $$

$$ \hat{v}_4^{(4)} = \begin{pmatrix} 0 & i\nu_1^2 \delta^2 e^{-it(g_++g_-)} \\ i\nu_1^{-2} \delta^{-2} e^{it(g_++g_-)} & 0 \end{pmatrix}, \qquad \hat{v}_5^{(4)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2 \hat{a}^*} \delta^{-2} e^{2itg} & 1 \end{pmatrix}, $$
<figure><img src="image_7.png" /><figcaption>Fig. 4.2. The jump contour $\Sigma^{(4)}$ (red with arrows indicating orientation) together with the set where $\operatorname{Im} g = 0$ (dashed black). The region where $\operatorname{Im} g < 0$ is shaded and the region where $\operatorname{Im} g > 0$ is white.</figcaption></figure>

$$ \hat{v}_6^{(4)} = \begin{pmatrix} \frac{e^{it(g_+ - g_-)}}{\hat{a}\hat{a}^*} & 0 \\ 0 & \hat{a}\hat{a}^* e^{-it(g_+ - g_-)} \end{pmatrix}, $$

$$ \hat{v}_8^{(4)} = e^{-itg_--\sigma_3} \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 \\ 0 & 1 \end{pmatrix} e^{itg_+ + \sigma_3}, $$

$$ \hat{v}_7^{(4)} = \begin{pmatrix} 1 & -\frac{\hat{b}}{\hat{a}^*} \delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, $$

and is extended to the part of $\Sigma^{(4)}$ that lies in the lower half-plane by means of the symmetry (4.2).

**Remark.** The paths labeled by 5, 6, 7, and 8 in Fig. 4.2 are labeled by 12, 11, 10 in [14, Figure 12], and 5 in [14, Figure 10]. Moreover, $(\hat{v}_2^{(4)})_{12}$ has the opposite sign compared to that in [14].

### 4.3. Fifth transformation

The purpose of the fifth transformation is to remove the factors $\hat{a}\hat{a}^* = aa^*$ from the jump across $\gamma_{(\tilde{\beta}, \beta)}$, i.e., from $v_6^{(4)}$ in the upper half-plane.

As we did with $\delta$ in [14, Section 4.2], we introduce a complex-valued function $\tilde{\delta}(k) \equiv \tilde{\delta}(\xi, k)$ by

$$ \tilde{\delta}(\xi, k) = e^{-\frac{1}{2\pi i} \int_{\gamma_{(\mu, \beta)}} \frac{\ln(a(s)a^*(s))}{s-k} ds} e^{\frac{1}{2\pi i} \int_{\gamma_{(\tilde{\beta}, \mu)}} \frac{\ln(a(s)a^*(s))}{s-k} ds}, \quad k \in \mathbb{C} \setminus \gamma_{(\tilde{\beta}, \beta)}, \quad (x, t) \in \mathbb{S}, \quad (4.4) $$

where the branch of the logarithm is chosen so that $\ln(a(s)a^*(s))$ is continuous on each contour and strictly positive for $s = \mu$. Since $a(k)$ has no zeros or poles, the function $aa^*$ is nonzero and
finite everywhere on the contours. The function $\tilde{\delta}$ is in general singular at $\beta$ and $\tilde{\beta}$. Hence we cut out small neighborhoods of these points. Thus let

$$R \equiv R(\xi, t) := c(\xi)t^{-1/2}, \qquad (4.5)$$

where $c(\xi) > c > 0$ is independent of $t$ (we will fix $c(\xi)$ below).

For $r > 0$ and $z \in \mathbb{C}$, we let $D_r(z)$ denote the open disk of radius $r$ centered at $z$. Since $|\alpha - \beta| = O(t^{-1/2-\delta})$, by increasing $T$ in the definition (2.10) of $\mathbb{S}$ if necessary, we may assume that $\alpha \in D_{R/2}(\beta)$ for all $(x, t) \in \mathbb{S}$.

The following lemma shows that $\tilde{\delta}$ is nonsingular in the complement of $D_R(\beta) \cup D_R(\tilde{\beta})$.

**Lemma 4.2.** For each $(x, t) \in \mathbb{S}$, the function $\tilde{\delta}(\xi, k)$ has the following properties:

(a) $\tilde{\delta}(\xi, k)$ and $(\tilde{\delta}(\xi, k))^{-1}$ are bounded and analytic functions of $k \in \mathbb{C} \setminus (D_R(\beta) \cup D_R(\tilde{\beta}) \cup \gamma(\tilde{\beta}, \beta))$.

(b) $\tilde{\delta}(\xi, k)$ obeys the symmetry

$$\tilde{\delta} = (\tilde{\delta}^*)^{-1}, \quad k \in \mathbb{C} \setminus \gamma(\tilde{\beta}, \beta).$$

(c) Across $\gamma(\tilde{\beta}, \beta)$, $\tilde{\delta}(\xi, k)$ satisfies the jump condition

$$\tilde{\delta}_+ = \tilde{\delta}_- \times \begin{cases} \frac{1}{aa^*}, & k \in \gamma(\mu, \beta), \\ aa^*, & k \in \gamma(\tilde{\beta}, \mu). \end{cases} \qquad (4.6)$$

**Proof.** Since $\ln(|a(\mu)|^2) > 0$, [14, Lemma C.1] shows that $\tilde{\delta}(\xi, k)$ is bounded as $k$ approaches $\mu$. The other properties follow easily from the definition (4.4). We can indeed write $\tilde{\delta} = e^{-f_{\beta}+f_{\tilde{\beta}}}$ with

$$f_{\beta}(k) := \frac{1}{2\pi i} \int_{\gamma(\mu, \beta)} \frac{\ln(a(s)a^*(s))}{s-k} ds, \qquad f_{\tilde{\beta}}(k) := \frac{1}{2\pi i} \int_{\gamma(\tilde{\beta}, \mu)} \frac{\ln(a(s)a^*(s))}{s-k} ds,$$

and $f_{\beta} = f_{\tilde{\beta}}^*$. $\square$

Let $\epsilon \in (0, 1/2)$ be such that the $\xi$-dependent disks

$$D_{\epsilon}(\beta), \quad D_{\epsilon}(\mu), \quad D_{\epsilon}(\tilde{\beta}) \qquad (4.7)$$

are disjoint from each other and from the cuts $\Sigma_1$ and $\Sigma_2$ for all $(x, t) \in \mathbb{S}$. Increasing $T$ in (2.10) if necessary, we may assume that $R(\xi, t) < \epsilon/2$ for all $(x, t) \in \mathbb{S}$.

We define $\hat{m}^{(5)} \equiv \hat{m}^{(5)}(x, t, k)$ by

$$\hat{m}^{(5)} = \hat{m}^{(4)} \times \begin{cases} \tilde{\delta}^{-\sigma_3}, & k \in \mathbb{C} \setminus (D_R(\beta) \cup D_R(\tilde{\beta})), \\ I, & k \in D_R(\beta) \cup D_R(\tilde{\beta}). \end{cases}$$

We define the complex-valued function $\delta_2(k) \equiv \delta_2(\xi, k)$ by
<figure><img src="image_8.png" /><figcaption>Fig. 4.3. The jump contour $\Sigma^{(5)} = \Sigma^{(6)}$ (red with arrows indicating orientation) together with the set where $\operatorname{Im} g = 0$ (dashed black). The region where $\operatorname{Im} g < 0$ is shaded and the region where $\operatorname{Im} g > 0$ is white.</figcaption></figure>

$$ \delta_2 := \delta \tilde{\delta}. $$

Using Lemma 4.2 we see that $\hat{m}$ satisfies the RH problem (3.3) iff $\hat{m}^{(5)}$ satisfies the RH problem (4.1) with $j = 5$, where $\Sigma^{(5)} := \Sigma^{(4)} \cup \partial D_R(\beta) \cup \partial D_R(\tilde{\beta})$ and the jump matrix $\hat{v}^{(5)}$ is given in $\mathbb{C}^+ \setminus \overline{D_R(\beta)}$ (see Fig. 4.3) by

$$ \hat{v}_1^{(5)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}} \delta_2^{-2} e^{2itg} & 1 \end{pmatrix}, $$

$$ \hat{v}_2^{(5)} = \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta_2^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, $$

$$ \hat{v}_3^{(5)} = \begin{pmatrix} 0 & i\hat{a}_+\hat{a}_-e^{i\phi}\delta_2^2e^{-it(g_++g_-)} \\ \frac{ie^{-i\phi}}{\hat{a}_+\hat{a}_-}\delta_2^{-2}e^{it(g_++g_-)} & 0 \end{pmatrix}, $$

$$ \hat{v}_4^{(5)} = \begin{pmatrix} 0 & i\nu_1^2\delta_2^2e^{-it(g_++g_-)} \\ i\nu_1^{-2}\delta_2^{-2}e^{it(g_++g_-)} & 0 \end{pmatrix}, $$

$$ \hat{v}_5^{(5)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2\hat{a}^*}\delta_2^{-2}e^{2itg} & 1 \end{pmatrix}, $$

$$ \hat{v}_6^{(5)} = \begin{pmatrix} e^{it(g_+-g_-)} & 0 \\ 0 & e^{-it(g_+-g_-)} \end{pmatrix}, $$

$$ \hat{v}_7^{(5)} = \begin{pmatrix} 1 & -\frac{\hat{b}}{\hat{a}^*}\delta_2^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, $$

where $\hat{v}_l^{(5)}$ denotes the restriction of $\hat{v}^{(5)}$ to the contour labeled by $l$ in Fig. 4.3, outside of $\overline{D_R(\beta)}$.

In order to give an expression for $\hat{v}^{(5)}$ on the part of the contour that lies in $\overline{D_R(\beta)}$, we let $\mathcal{Y} := \Sigma^{(5)} \cap D_\epsilon(\beta)$ denote the restriction of $\Sigma^{(5)}$ to $D_\epsilon(\beta)$ and write $\mathcal{Y} = \cup_{l=1}^{16} \mathcal{Y}_l$ where the curves $\{\mathcal{Y}_l\}_{16}$ are as in Fig. 4.4.
<figure><img src="image_9.png" /><figcaption>Fig. 4.4. The contour $\mathcal{Y} = \Sigma^{(5)} \cap D_{\epsilon}(\beta) = \cup_{j=1}^{16} \mathcal{Y}_j$ in the disk of radius $\epsilon$ centered at $\beta$ and the domains $\{S_j\}_1^{10}$.</figcaption></figure>

Then $\hat{v}^{(5)}$ is given in $D_{\epsilon}(\beta)$ by

$$
\begin{align*}
\hat{v}_{\mathcal{Y}_1}^{(5)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{a} \delta_2^{-2} e^{2itg} & 1 \end{pmatrix}, & \hat{v}_{\mathcal{Y}_2}^{(5)} &= \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta_2^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_3}^{(5)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2\hat{a}^*} \delta_2^{-2} e^{2itg} & 1 \end{pmatrix}, & \hat{v}_{\mathcal{Y}_4}^{(5)} &= \begin{pmatrix} 1 & -\frac{\hat{b}^*}{\hat{a}^*} \delta_2^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_5}^{(5)} &= \hat{v}_{\mathcal{Y}_6}^{(5)} = \hat{v}_{\mathcal{Y}_7}^{(5)} = \hat{v}_{\mathcal{Y}_8}^{(5)} = \hat{v}_{\mathcal{Y}_9}^{(5)} = \delta\sigma_3, \\
\hat{v}_{\mathcal{Y}_{10}}^{(5)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{a} \delta^{-2} e^{2itg} & 1 \end{pmatrix}, & \hat{v}_{\mathcal{Y}_{11}}^{(5)} &= \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_{12}}^{(5)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2\hat{a}^*} \delta^{-2} e^{2itg} & 1 \end{pmatrix}, & \hat{v}_{\mathcal{Y}_{13}}^{(5)} &= \begin{pmatrix} 1 & -\frac{\hat{b}^*}{\hat{a}^*} \delta^2 e^{-2itg} \\ 0 & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_{14}}^{(5)} &= e^{-itg - \sigma_3} \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 \\ 0 & 1 \end{pmatrix} e^{itg + \sigma_3}, \\
\hat{v}_{\mathcal{Y}_{15}}^{(5)} &= \begin{pmatrix} \frac{e^{it(g_+-g_-)}}{\hat{a}\hat{a}^*} & 0 \\ 0 & \hat{a}\hat{a}^* e^{-it(g_+-g_-)} \end{pmatrix}, & \hat{v}_{\mathcal{Y}_{16}}^{(5)} &= \begin{pmatrix} e^{it(g_+-g_-)} & 0 \\ 0 & e^{-it(g_+-g_-)} \end{pmatrix},
\end{align*}
$$

where $\hat{v}_{\mathcal{Y}_l}^{(5)}$ denotes the restriction of $\hat{v}^{(5)}$ to $\mathcal{Y}_l$. We extend the definition of $\hat{v}^{(5)}$ to the lower half-plane by means of the symmetry (4.2).
### 4.4. Sixth transformation

The purpose of the sixth transformation $\hat{m}^{(5)} \to \hat{m}^{(6)}$ is to make the jumps across the branch cuts $\Sigma_1$ and $\Sigma_2$ constant in $k$. This transformation is similar to the fifth transformation in [14], except that the fifth transformation in [14] also addresses the jumps on $\gamma(\beta, \alpha)$, $\gamma(\tilde{\alpha}, \tilde{\beta})$, and $\gamma(\tilde{\beta}, \beta)$. In the present case, the contours $\gamma(\beta, \alpha)$ and $\gamma(\tilde{\alpha}, \tilde{\beta})$ are so small that they are included in the local parametrices, and the jump on $\gamma(\tilde{\beta}, \beta)$ has already been addressed in the transformation $\hat{m}^{(4)} \to \hat{m}^{(5)}$.

Let $\Sigma^{\text{mod}} := \Sigma_1 \cup \Sigma_2$ denote the union of the branch cuts $\Sigma_1$ and $\Sigma_2$ oriented upwards. Given a real number $\omega_1 \equiv \omega_1(\xi)$, we define the function $\mathcal{H} \equiv \mathcal{H}(\xi, k)$ for $k \in \Sigma^{\text{mod}} \cap \mathbb{C}^+$ by

$$ \mathcal{H} = \begin{cases} \frac{2\omega_1 - i \ln(\hat{a}_+ \hat{a}_- \delta_2^2 e^{i\phi})}{\tilde{w}_+}, & k \in \Sigma_1 \cap \mathbb{C}^+, \quad (x, t) \in \mathbb{S}, \\ -\frac{i \ln(v_1^2 \delta_2^2)}{\tilde{w}_+}, & k \in \Sigma_2 \cap \mathbb{C}^+, \quad (x, t) \in \mathbb{S}, \end{cases} \quad (4.8) $$

where $\tilde{w}$ is defined in (3.8), and extend it to $\Sigma^{\text{mod}} \cap \mathbb{C}^-$ by the symmetry

$$ \mathcal{H} = \mathcal{H}^*, \quad (4.9) $$

i.e., $\mathcal{H}(\xi, k) = \overline{\mathcal{H}(\xi, \bar{k})}$. The branches of the logarithms in (4.8) can be chosen arbitrarily as long as $\mathcal{H}(\xi, k)$ is a continuous function of $k$ on each of the three segments $\Sigma_1 \cap \mathbb{C}^+$, $\Sigma_1 \cap \mathbb{C}^-$, and $\Sigma_2$ (such a choice exists because $|v_1^2 \delta_2^2| = 1$ at the point where $\Sigma_2$ crosses the real line).

We define $\tilde{h} \equiv \tilde{h}(\xi, k)$ by

$$ \tilde{h}(\xi, k) = \frac{\tilde{w}(k)}{2\pi i} \int_{\Sigma^{\text{mod}}} \frac{\mathcal{H}(\xi, s)}{s-k} ds, \quad k \in \mathbb{C} \setminus \Sigma^{\text{mod}}. \quad (4.10) $$

In general, the function $\tilde{h}$ has a pole at $k = \infty$. The following lemma shows that by choosing $\omega_1(\xi)$ appropriately, the pole at $\infty$ can be removed.

**Lemma 4.3.** There is a unique choice of $\omega_1(\xi)$ such that the function $\tilde{h}(\xi, k)$ defined in (4.10) has the following properties:

(a) $\tilde{h}$ obeys the symmetry $\tilde{h} = \tilde{h}^*$, i.e.,

$$ \tilde{h}(\xi, k) = \overline{\tilde{h}(\xi, \bar{k})}, \quad k \in \mathbb{C} \setminus \Sigma^{\text{mod}}, \quad (x, t) \in \mathbb{S}, \quad (4.11) $$

(b) As $k$ goes to infinity,

$$ \tilde{h}(\xi, k) = \tilde{h}(\xi, \infty) + O(k^{-1}), \quad k \to \infty, \quad (x, t) \in \mathbb{S}, \quad (4.12) $$

where

$$ \tilde{h}(\xi, \infty) = -\frac{1}{2\pi i} \int_{\Sigma^{\text{mod}}} s \mathcal{H}(\xi, s) ds \quad (4.13) $$
is a real-valued bounded function of $\xi$.

(c) For each $(x, t) \in \mathbb{S}$, $e^{i\tilde{h}(\xi, k)\sigma_3}$ is an analytic function of $k \in \hat{\mathbb{C}} \setminus \Sigma^{\text{mod}}$.

(d) $e^{i\tilde{h}(\xi, k)\sigma_3}$ satisfies the uniform bound

$$|e^{i\tilde{h}(\xi, k)\sigma_3}| \le C, \quad k \in \hat{\mathbb{C}} \setminus \Sigma^{\text{mod}}, (x, t) \in \mathbb{S}.$$

(e) $\tilde{h}(\xi, k)$ satisfies the following jump conditions across $\Sigma^{\text{mod}}$:

$$\tilde{h}_+ + \tilde{h}_- = \begin{cases} 2\omega_1 - i \ln(\hat{a}_+ \hat{a}_- \delta_2^2 e^{i\phi}), & k \in \Sigma_1 \cap \mathbb{C}^+, \\ 2\omega_1 + i \ln(\hat{a}_+^* \hat{a}_-^* \delta_2^{-2} e^{-i\phi}), & k \in \Sigma_1 \cap \mathbb{C}^-, \\ -i \ln(v_1^2 \delta_2^2), & k \in \Sigma_2. \end{cases} \quad (4.14)$$

**Proof.** The proof is similar to the analogous proof in the genus 3 sector [14, Lemma 4.2]. □

We henceforth fix $\omega_1(\xi)$ to be the unique choice which ensures that $\tilde{h}(\xi, k)$ has the properties of Lemma 4.3. We define $\hat{m}^{(6)} \equiv \hat{m}^{(6)}(x, t, k)$ by

$$\hat{m}^{(6)}(x, t, k) := e^{-i\tilde{h}(\xi, \infty)\sigma_3} \hat{m}^{(5)}(x, t, k) e^{i\tilde{h}(\xi, k)\sigma_3}.$$

By Lemma 4.3, $\hat{m}$ satisfies the RH problem (3.3) iff $\hat{m}^{(6)}$ satisfies the RH problem (4.1) with $j=6$, where $\Sigma^{(6)} = \Sigma^{(5)}$ and

$$\hat{v}^{(6)} = e^{-i\tilde{h}_{-}\sigma_3} \hat{v}^{(5)} e^{i\tilde{h}_{+}\sigma_3}.$$

The jump matrix $\hat{v}^{(6)}$ is given explicitly in $\mathbb{C}^+ \setminus \overline{D_R(\beta)}$ by

$$\begin{align*}
\hat{v}_1^{(6)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}} \delta_2^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, & \hat{v}_2^{(6)} &= \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta_2^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix}, \\
\hat{v}_3^{(6)} &= \begin{pmatrix} 0 & ie^{-2i(t\Omega_1+\omega_1)} \\ ie^{2i(t\Omega_1+\omega_1)} & 0 \end{pmatrix}, & \hat{v}_4^{(6)} &= \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \\
\hat{v}_5^{(6)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2\hat{a}^*} \delta_2^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, & \hat{v}_6^{(6)} &= \begin{pmatrix} e^{it(g_+-g_-)} & 0 \\ 0 & e^{-it(g_+-g_-)} \end{pmatrix}, \\
\hat{v}_7^{(6)} &= \begin{pmatrix} 1 & -\frac{\hat{b}^*}{\hat{a}^*} \delta_2^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix}, & &
\end{align*}$$

where we used (4.14) and the relations $g_+ + g_- = 2\Omega_1$ along $\Sigma_1$ and $g_+ + g_- = 0$ along $\Sigma_2$ [14, Lemma 3.2 (c)]. Here, $\omega_1 \equiv \omega_1(\xi)$ and $\Omega_1 \equiv \Omega_1(\xi)$ are two real constants and $\hat{v}_l^{(6)}$ denotes the restriction of $\hat{v}^{(6)}$ to the contour labeled by $l$ in Fig. 4.3, outside of $\overline{D_R(\beta)}$.

On the part of the contour that lies in $D_\epsilon(\beta)$ the jump matrix $\hat{v}^{(6)}$ is given by

$$\hat{v}_{\gamma_l}^{(6)} = e^{-i\tilde{h}\sigma_3} \hat{v}_{\gamma_l}^{(5)} e^{i\tilde{h}\sigma_3}, \quad l = 1, \dots, 16.$$

That is
$$
\begin{align*}
\hat{v}_{\mathcal{Y}_1}^{(6)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^*} \delta_2^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_3}^{(6)} &= \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2 \hat{a}^*} \delta_2^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, \\
\hat{v}_{\mathcal{Y}_5}^{(6)} &= \hat{v}_{\mathcal{Y}_6}^{(6)} = \hat{v}_{\mathcal{Y}_7}^{(6)} = \hat{v}_{\mathcal{Y}_8}^{(6)} = \hat{v}_{\mathcal{Y}_9}^{(6)} = \delta\sigma_3,
\end{align*}
$$

$$
\hat{v}_{\mathcal{Y}_{10}}^{(6)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^*} \delta^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, \qquad \hat{v}_{\mathcal{Y}_{11}}^{(6)} = \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

$$
\hat{v}_{\mathcal{Y}_{12}}^{(6)} = \begin{pmatrix} 1 & 0 \\ \frac{\hat{b}^*}{\hat{a}^2 \hat{a}^*} \delta^{-2} e^{2itg} e^{2i\tilde{h}} & 1 \end{pmatrix}, \qquad \hat{v}_{\mathcal{Y}_{13}}^{(6)} = \begin{pmatrix} 1 & -\frac{\hat{b}}{\hat{a}^*} \delta^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

$$
\hat{v}_{\mathcal{Y}_{14}}^{(6)} = e^{-itg - \sigma_3} \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix} e^{itg + \sigma_3},
$$

$$
\hat{v}_{\mathcal{Y}_{15}}^{(6)} = \begin{pmatrix} \frac{e^{it(g_+ - g_-)}}{\hat{a}\hat{a}^*} & 0 \\ 0 & \hat{a}\hat{a}^* e^{-it(g_+ - g_-)} \end{pmatrix}, \qquad \hat{v}_{\mathcal{Y}_{16}}^{(6)} = \begin{pmatrix} e^{it(g_+ - g_-)} & 0 \\ 0 & e^{-it(g_+ - g_-)} \end{pmatrix}.
$$

$$
\hat{v}_{\mathcal{Y}_2}^{(6)} = \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta_2^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

$$
\hat{v}_{\mathcal{Y}_4}^{(6)} = \begin{pmatrix} 1 & -\frac{\hat{b}}{\hat{a}^*} \delta_2^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

$$
\hat{v}_{\mathcal{Y}_{11}}^{(6)} = \begin{pmatrix} 1 & -\hat{a}\hat{b}\delta^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

$$
\hat{v}_{\mathcal{Y}_{13}}^{(6)} = \begin{pmatrix} 1 & -\frac{\hat{b}}{\hat{a}^*} \delta^2 e^{-2itg} e^{-2i\tilde{h}} \\ 0 & 1 \end{pmatrix},
$$

5. Global parametrix

Away from $\Sigma^{\text{mod}}$ and the critical points, the jump matrix $\hat{v}^{(6)}$ approaches the identity matrix as $t \to +\infty$. This leads us to expect that in the limit $t \to +\infty$, the solution $\hat{m}^{(6)}$ approaches the solution $m^{\text{mod}}$ of the RH problem

$$
\left\{
\begin{array}{l}
m^{\text{mod}}(x, t, \cdot) \in I + \dot{E}^2(\hat{C} \setminus \Sigma^{\text{mod}}), \\
m_+^{\text{mod}}(x, t, k) = m_-^{\text{mod}}(x, t, k) v^{\text{mod}}(x, t, k) \quad \text{for a.e. } k \in \Sigma^{\text{mod}},
\end{array}
\right.
\qquad (5.1)
$$

where $v^{\text{mod}}$ denotes the restriction of $\hat{v}^{(6)}$ to $\Sigma^{\text{mod}}$, i.e.,

$$
v^{\text{mod}} = \left\{
\begin{array}{ll}
\begin{pmatrix} 0 & i e^{-2i(t\Omega_1(\xi) + \omega_1(\xi))} \\ i e^{2i(t\Omega_1(\xi) + \omega_1(\xi))} & 0 \end{pmatrix}, & k \in \Sigma_1, \\
\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, & k \in \Sigma_2.
\end{array}
\right.
\qquad (5.2)
$$

The jump matrix $v^{\text{mod}}$ is off-diagonal and independent of $k$. This implies that we can write down an explicit solution of the RH problem (5.1) in terms of theta functions.

Define the function $v \equiv v(k)$ for $k \in \mathbb{C} \setminus \Sigma^{\text{mod}}$ by

$$
\nu := \left( \frac{(k - E_1)(k - E_2)}{(k - \bar{E}_1)(k - \bar{E}_2)} \right)^{\frac{1}{4}}, \quad k \in \mathbb{C} \setminus \Sigma^{\text{mod}},
$$

where the branch of the fourth root is fixed by requiring that $\nu = 1 + O(k^{-1})$ as $k \to \infty$. Let
$\hat{v}: \tilde{M} \to \hat{C}$ denote the function which is given by $\nu^2$ on the upper sheet and by $-\nu^2$ on the lower
sheet of $\tilde{M}$, that is, $\hat{v}(k^{\pm}) = \pm v(k)^2$ for $k \in \mathbb{C} \setminus \Sigma^{\text{mod}}$. Then $\hat{v}$ is a meromorphic function on $\tilde{M}$. Noting, for example, that $\hat{v}$ has two simple zeros at $E_1, E_2$, we see that $\hat{v}: \tilde{M} \to \mathbb{C}$ has degree two. Hence the function $\hat{v} - 1$ has two zeros on $\tilde{M}$ counting multiplicity; we denote these zeros by $\infty^{+}$, $P_1 \in \tilde{M}$. We define the complex constant $\tilde{\mathcal{K}}$ by

$$ \tilde{\mathcal{K}} := \frac{1}{2}(1 + \tau), \qquad (5.3) $$

where $\tau$ denotes the period $\int_{b_1} \tilde{\zeta}$. We define the constant $\tilde{d} \in \mathbb{C}$ by

$$ \tilde{d} := \tilde{\varphi}(P_1) + \tilde{\mathcal{K}}, \qquad (5.4) $$

where $\tilde{\varphi}: \tilde{M} \to \mathbb{C}$ is the Abel map defined in (3.6); if $P_1$ has projection in $\Sigma^{\text{mod}}$, then we fix the value of $\tilde{d}$ by letting $\tilde{\varphi}(P_1)$ denote the boundary value $\tilde{\varphi}_+(P_1)$ from the left say. We also define the complex-valued function $\tilde{v} = \tilde{v}(\xi, t)$ by

$$ \tilde{v}(\xi, t) := -\frac{1}{\pi} (t\Omega_1(\xi) + \omega_1(\xi)). \qquad (5.5) $$

**Theorem 5.1 (Solution of the model RH problem).** For each choice of the real constants $\Omega_1$ and $\omega_1$ and for each $t \ge 0$, the RH problem (5.1) has a unique solution $m^{\text{mod}}(x, t, k)$. Moreover, this solution satisfies

$$ \lim_{k \to \infty} k(m^{\text{mod}}(x, t, k))_{12} = -\frac{i}{2} \operatorname{Im}(E_1 + E_2) \times \frac{\tilde{\Theta}(\tilde{\varphi}(\infty^+) + \tilde{d}) \tilde{\Theta}(\tilde{\varphi}(\infty^+) - \tilde{v}(\xi, t) - \tilde{d})}{\tilde{\Theta}(\tilde{\varphi}(\infty^+) + \tilde{v}(\xi, t) + \tilde{d}) \tilde{\Theta}(\tilde{\varphi}(\infty^+) - \tilde{d})}, \qquad (5.6) $$

where the limit is uniform with respect to $\arg k \in [0, 2\pi]$ and $\tilde{\Theta}$ is the Riemann theta function associated to the genus 1 Riemann surface $\tilde{M}$, see (3.7).

**Proof.** The proof is similar to the analogous proof in the genus 3 sector [14, Theorem 5.1]. $\square$

# 6. Local parametrices

The solution $m^{\text{mod}}$ is a good approximation of $\hat{m}^{(6)}$ as $t \to +\infty$ except for $k$ near the five critical points $\{\alpha, \beta, \mu, \tilde{\alpha}, \tilde{\beta}\}$. In this section, we introduce local model solutions which are good approximations of $\hat{m}^{(6)}$ near these critical points. More precisely, we define two local solutions, denoted by $m^{\beta}(x, t, k)$ and $m^{\mu}(x, t, k)$, which are good approximations of $\hat{m}^{(6)}$ for $k$ in the disks $D_{\epsilon}(\beta)$ and $D_{\epsilon}(\mu)$, respectively.

## 6.1. Local model near $\mu$

The local model $m^{\mu}$ near $\mu$ is defined as in the case of the genus 3 sector [14, Section 6.3].
## 6.2. Local model near $\beta$

**Remark 6.1 (Motivating remark).** The derivative $dg/dk$ of the $g$-function for the genus 3 sector behaves like $(k - \alpha)^{1/2}$ and $(k - \beta)^{1/2}$ as $k$ approaches the critical points $\alpha$ and $\beta$, respectively. If $\alpha$ and $\beta$ stay separated, we have $g(k) \sim (k - \alpha)^{3/2}$ as $k \to \alpha$ and $g(k) \sim (k - \beta)^{3/2}$ as $k \to \beta$. The contributions to the solution $q(x, t)$ from $\alpha$ and $\beta$ can then be computed using the Airy local model [14, Appendix B] and each contribution is of order $O(t^{-N})$ for any $N \ge 1$. However, if $\alpha$ approaches $\beta$, $dg/dk$ behaves roughly like $(k - \alpha)^{1/2}(k - \beta)^{1/2} \approx k - \beta$ for $k$ in $D_\epsilon(\beta) \setminus D_R(\beta)$, where $R$ is some radius such that $|\alpha - \beta| \ll R < \epsilon$. Hence $g(k)$ behaves as if it had a double zero at $\beta$. This suggests the construction of a local model near $\beta$ as follows.

Let $\{S_j\}_{j=1}^{10}$ denote the open subsets of $D_\epsilon(\beta)$ shown in Fig. 4.4 which separate the $Y_j$. Let $\gamma(\alpha, \infty)$ denote the contour from $\alpha$ to infinity labeled by 2 in Fig. 4.2. Let $\gamma_{\text{cut}} := \gamma(\beta, \alpha) \cup \gamma(\alpha, \infty)$. Let $S_+ := S_3 \cup S_4 \cup S_8 \cup S_9$ and $S_- := S_1 \cup S_2 \cup S_5 \cup S_6 \cup S_7 \cup S_{10}$ denote the parts of $D_\epsilon(\beta)$ that lie to the left and right of the curve $\gamma(\mu, \beta) \cup \gamma_{\text{cut}}$, respectively.

### 6.2.1. First transformation

We define the function $g_\beta(\xi, k)$ for $k$ near $\beta$ by

$$g_{\beta}(\xi, k) = \int_{\beta}^{k} dg = \begin{cases} g(\xi, k) - g_{+}(\xi, \beta), & k \in S_{+}, \\ g(\xi, k) - g_{-}(\xi, \beta), & k \in S_{-}, \end{cases} \quad (x, t) \in \mathbb{S}. $$

For each $\xi$, $g_\beta(\xi, k)$ is an analytic function of $k \in D_\epsilon(\beta) \setminus \gamma_{\text{cut}}$ with jump across $\gamma_{\text{cut}}$.

Let

$$m^{(\beta 0)}(x, t, k) := \hat{m}^{(6)}(x, t, k)e^{-i\hat{h}(\xi, k)\sigma_3} A(x, t, k), \quad k \in D_{\epsilon}(\beta) \setminus \mathcal{Y}, (x, t) \in \mathbb{S},$$

where $A(x, t, k)$ denotes the sectionally holomorphic function

$$A(x, t, k) := \begin{cases} (\delta^2 \hat{a} \hat{b})^{\frac{\sigma_3}{2}} e^{-i t g_+(\xi, \beta) \sigma_3}, & k \in S_+, \\ (\delta^2 \hat{a} \hat{b})^{\frac{\sigma_3}{2}} e^{-i t g_-(\xi, \beta) \sigma_3}, & k \in S_-, \end{cases} \quad (x, t) \in \mathbb{S}. \quad (6.1)$$

Since the function $\delta^2 \hat{a} \hat{b}$ is nonzero and analytic in $D_\epsilon(\beta)$, the square root $(\delta^2 \hat{a} \hat{b})^{\frac{1}{2}}$ is well-defined and analytic; the branch of $(\delta^2 \hat{a} \hat{b})^{\frac{1}{2}}$ can be chosen arbitrarily.

Then (recall that $\hat{a}\hat{a}^* = aa^*, \hat{b}\hat{b}^* = bb^*, \text{and } \hat{r}\hat{r}^* = rr^*$)

$$ \begin{align*} v_{\mathcal{Y}_1}^{(\beta 0)} &= \begin{pmatrix} 1 & 0 \\ bb^* \tilde{\delta}^{-2} e^{2itg_\beta} & 1 \end{pmatrix}, & v_{\mathcal{Y}_2}^{(\beta 0)} &= \begin{pmatrix} 1 & -\tilde{\delta}^2 e^{-it(g_\beta+g_{\beta-})} \\ 0 & 1 \end{pmatrix}, \\ v_{\mathcal{Y}_3}^{(\beta 0)} &= \begin{pmatrix} 1 & 0 \\ \frac{bb^*}{aa^*} \tilde{\delta}^{-2} e^{2itg_\beta} & 1 \end{pmatrix}, & v_{\mathcal{Y}_4}^{(\beta 0)} &= \begin{pmatrix} 1 & -\frac{1}{aa^*} \tilde{\delta}^2 e^{-2itg_\beta} \\ 0 & 1 \end{pmatrix}, \\ v_{\mathcal{Y}_5}^{(\beta 0)} &= v_{\mathcal{Y}_6}^{(\beta 0)} = v_{\mathcal{Y}_7}^{(\beta 0)} = v_{\mathcal{Y}_8}^{(\beta 0)} = v_{\mathcal{Y}_9}^{(\beta 0)} = \tilde{\delta}^{\sigma_3}, & v_{\mathcal{Y}_{10}}^{(\beta 0)} &= \begin{pmatrix} 1 & 0 \\ bb^* e^{2itg_\beta} & 1 \end{pmatrix}, & v_{\mathcal{Y}_{11}}^{(\beta 0)} &= \begin{pmatrix} 1 & -e^{-it(g_\beta+g_{\beta-})} \\ 0 & 1 \end{pmatrix}, \end{align*} $$
$$v_{\mathcal{Y}_{12}}^{(\beta 0)} = \begin{pmatrix} 1 & 0 \\ \frac{bb^*}{aa^*} e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{14}}^{(\beta 0)} = e^{-itg_{\beta}-\sigma_3} \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} e^{itg_{\beta}+\sigma_3},$$

$$v_{\mathcal{Y}_{16}}^{(\beta 0)} = I,$$

$$v_{\mathcal{Y}_{13}}^{(\beta 0)} = \begin{pmatrix} 1 & -\frac{1}{aa^*} e^{-2itg_{\beta}} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{15}}^{(\beta 0)} = \begin{pmatrix} \frac{1}{aa^*} & 0 \\ 0 & aa^* \end{pmatrix},$$

or, in terms of $r$,

$$v_{\mathcal{Y}_1}^{(\beta 0)} = \begin{pmatrix} 1 & 0 \\ \frac{rr^*}{1+rr^*} \tilde{\delta}^{-2} e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_3}^{(\beta 0)} = \begin{pmatrix} 1 & 0 \\ rr^* \tilde{\delta}^{-2} e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_5}^{(\beta 0)} = v_{\mathcal{Y}_6}^{(\beta 0)} = v_{\mathcal{Y}_7}^{(\beta 0)} = v_{\mathcal{Y}_8}^{(\beta 0)} = v_{\mathcal{Y}_9}^{(\beta 0)} = \tilde{\delta}\sigma_3,$$

$$v_{\mathcal{Y}_{10}}^{(\beta 0)} = \begin{pmatrix} 1 & 0 \\ \frac{rr^*}{1+rr^*} e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{12}}^{(\beta 0)} = \begin{pmatrix} 1 & 0 \\ rr^* e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{14}}^{(\beta 0)} = e^{-itg_{\beta}-\sigma_3} \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix} e^{itg_{\beta}+\sigma_3},$$

$$v_{\mathcal{Y}_{16}}^{(\beta 0)} = I.$$

$$v_{\mathcal{Y}_2}^{(\beta 0)} = \begin{pmatrix} 1 & -\tilde{\delta}^2 e^{-it(g_{\beta}+g_{\beta}-)} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_4}^{(\beta 0)} = \begin{pmatrix} 1 & -(1+rr^*)\tilde{\delta}^2 e^{-2itg_{\beta}} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{11}}^{(\beta 0)} = \begin{pmatrix} 1 & -e^{-it(g_{\beta}+g_{\beta}-)} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{13}}^{(\beta 0)} = \begin{pmatrix} 1 & -(1+rr^*)e^{-2itg_{\beta}} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{15}}^{(\beta 0)} = \begin{pmatrix} 1+rr^* & 0 \\ 0 & \frac{1}{1+rr^*} \end{pmatrix},$$

### 6.2.2. Second transformation

We would next like to remove all jumps within $D_R(\beta)$. This is not quite possible, because $g_\beta$ has a jump across $\mathcal{Y}_{11} \cup \mathcal{Y}_{14}$. However, the following transformation removes all jumps within $D_R(\beta)$ except those across $\mathcal{Y}_{11}$ and $\mathcal{Y}_{14}$. The remaining jumps across $\mathcal{Y}_{11}$ and $\mathcal{Y}_{14}$ go away in the limit $\alpha \to \beta$.

Let $m^{(\beta 1)} := m^{(\beta 0)} B$, where $B(x, t, k)$ denotes the sectionally holomorphic function

$$B(x, t, k) := \begin{cases} \begin{pmatrix} 1 & 0 \\ -\frac{rr^*}{1+rr^*} e^{2itg_{\beta}} & 1 \end{pmatrix}, & k \in S_7, \\ \begin{pmatrix} 1 & 0 \\ -rr^* e^{2itg_{\beta}} & 1 \end{pmatrix} \begin{pmatrix} \frac{1}{1+rr^*} & e^{-2itg_{\beta}} \\ 0 & 1+rr^* \end{pmatrix}, & k \in S_8, \\ \begin{pmatrix} \frac{1}{1+rr^*} & e^{-2itg_{\beta}} \\ 0 & 1+rr^* \end{pmatrix}, & k \in S_9, \\ \begin{pmatrix} 1 & (1+rr^*)e^{-2itg_{\beta}} \\ 0 & 1 \end{pmatrix}, & k \in S_{10}, \\ I, & \text{else,} \end{cases} \quad (x, t) \in \mathbb{S}. \quad (6.2)$$

Then
$$v_{\mathcal{Y}_1}^{(\beta 1)} = \begin{pmatrix} \frac{1}{rr^*} & 0 \\ \frac{\tilde{\delta}^2 e^{-2itg_{\beta}}}{1+rr^*} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_3}^{(\beta 1)} = \begin{pmatrix} 1 & 0 \\ rr^* \tilde{\delta}^2 e^{-2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_5}^{(\beta 1)} = \tilde{\delta}^{\sigma_3},$$

$$v_{\mathcal{Y}_7}^{(\beta 1)} = \tilde{\delta}^{\sigma_3} \begin{pmatrix} \frac{1}{1+rr^*} & e^{-2itg_{\beta}} \\ -\frac{rr^*}{1+rr^*} e^{2itg_{\beta}} & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_9}^{(\beta 1)} = \tilde{\delta}^{\sigma_3} \begin{pmatrix} 1 & (1+rr^*)e^{-2itg_{\beta}} \\ 0 & 1 \end{pmatrix},$$

$$v_{\mathcal{Y}_{11}}^{(\beta 1)} = \begin{pmatrix} \frac{1+rr^* e^{it(g_{\beta}+g_{\beta}-)}}{1+rr^*} & e^{-2itg_{\beta}+}(1-e^{it(g_{\beta}+g_{\beta}-)}) \\ rr^* e^{2itg_{\beta}+}(e^{-2it(g_{\beta}+g_{\beta}-)}-1+rr^*(e^{-it(g_{\beta}+g_{\beta}-)}-1)) & 1-\frac{rr^* e^{-it(g_{\beta}+g_{\beta}-)}(1-e^{-it(g_{\beta}+g_{\beta}-)})}{1+rr^*} \end{pmatrix},$$

$$v_{\mathcal{Y}_{14}}^{(\beta 1)} = \begin{pmatrix} e^{it(g_{\beta}+g_{\beta}-)} & 0 \\ 0 & e^{-it(g_{\beta}+g_{\beta}-)} \end{pmatrix}, \qquad v_{\mathcal{Y}_{10}}^{(\beta 1)} = v_{\mathcal{Y}_{12}}^{(\beta 1)} = v_{\mathcal{Y}_{13}}^{(\beta 1)} = v_{\mathcal{Y}_{15}}^{(\beta 1)} = v_{\mathcal{Y}_{16}}^{(\beta 1)} = I.$$

We note that in the limit as $\alpha$ approaches $\beta$, we have $g_{\beta+} - g_{\beta-} \to 0$ on $\mathcal{Y}_{11}$ and $\mathcal{Y}_{14}$. Hence the jumps across $\mathcal{Y}_{11}$ and $\mathcal{Y}_{14}$ vanish in this limit.

### 6.2.3. Analytic approximation of $g_{\beta}$

We want to map the RH problem for $m^{(\beta 1)}$ to an RH problem which, in the large $t$ limit, can be approximated by the exactly solvable RH problem of Appendix A. Thus, we want to introduce new variables $\lambda \equiv \lambda(x, t, k)$ and $y \equiv y(x, t)$ such that

$$2itg_{\beta}(k) \approx -\lambda^2 - 2y\lambda. \qquad (6.3)$$

For each $(x, t) \in \mathcal{S}$, we would like the map $k \mapsto \lambda$ to be a biholomorphism from $D_{\epsilon}(\beta)$ to a neighborhood of 0 in the complex $\lambda$-plane. However, $g_{\beta}$ is not analytic at $k = \beta$. Our next goal is to prove Lemma 6.2, which overcomes this problem by showing that $g_{\beta}$ can be well-approximated in $D_{\epsilon}(\beta) \setminus D_{R}(\beta)$ by an analytic function.

In this subsection, we will come across square roots of the form $\sqrt{s-k}$ with $k \in \gamma_{(\beta, \alpha)}$. For $k \in \gamma_{(\beta, \alpha)}$, let $\gamma_{(k, \infty)}$ denote the part of $\gamma_{\text{cut}}$ that stretches from $k$ to $\infty$. Given $k \in \gamma_{(\beta, \alpha)}$, we then fix the branch of the square root $\sqrt{s-k}$ (or, more generally, of the complex power $(s-k)^a$) so that $\sqrt{s-k}$ is an analytic function of $s \in \mathbb{C} \setminus \gamma_{(k, \infty)}$ and $\sqrt{s-k} \sim \sqrt{s}$ as $s \to +\infty$. In particular, the functions $\sqrt{s-\beta}$ and $\sqrt{s-\alpha}$ are analytic for $s \in \mathbb{C} \setminus \gamma_{\text{cut}}$ and $s \in \mathbb{C} \setminus \gamma_{(\alpha, \infty)}$, respectively.

We define the function $g_1(s) \equiv g_1(\xi, s)$ by

$$g_1(s) := \frac{g'(s)}{\sqrt{s-\beta}\sqrt{s-\alpha}}, \quad s \in D_{\epsilon}(\beta) \setminus \gamma_{\text{cut}}.$$

The definition of $g(s)$ implies that, for each $\xi$, $g_1(\xi, s)$ is an analytic function of $s \in D_{\epsilon}(\beta)$. Moreover,
$$ \sup_{(x,t) \in \mathbb{S}} \sup_{s \in D_{\epsilon}(\beta)} \left| \frac{\partial^n g_1}{\partial s^n} (\xi, s) \right| < \infty, \quad n = 0, 1, 2, \dots \qquad (6.4) $$

**Lemma 6.2.** We have

$$ g_{\beta}(k) = \int_{\beta}^{k} g_{1}(\xi, s) \left(s - \beta + \frac{\beta - \alpha}{2}\right) ds + O\left(|\alpha - \beta|^2 \ln|\alpha - \beta|\right), \quad t \to +\infty, \\ k \in D_{\epsilon}(\beta) \setminus (D_{R}(\beta) \cup \gamma_{\text{cut}}), \quad (x, t) \in \mathbb{S}, $$

where the error term is uniform with respect to $k$ and $x$ in the given ranges.

**Proof.** For each $s \in D_{\epsilon}(\beta) \setminus \gamma_{\text{cut}}$, $\sqrt{s-k}$ is a smooth function of $k \in \gamma_{(\beta,\alpha)}$. Hence integration by parts gives the following Taylor expansion of $\sqrt{s-\alpha}$ as $\alpha \to \beta$:

$$ \sqrt{s-\alpha} = \sqrt{s-\beta} + \frac{\beta-\alpha}{2\sqrt{s-\beta}} + \int_{\gamma_{(\beta,\alpha)}} \frac{u-\alpha}{4(s-u)^{3/2}} du, \quad s \in D_{\epsilon}(\beta) \setminus \gamma_{\text{cut}}. \qquad (6.5) $$

Using (6.5) we can write

$$ g_{\beta}(k) = \int_{\beta}^{k} dg = \int_{\beta}^{k} g_{1}(\xi, s) \sqrt{s-\alpha} \sqrt{s-\beta} \, ds \\ = \int_{\beta}^{k} g_{1}(\xi, s) \left(s - \beta + \frac{\beta - \alpha}{2}\right) ds + E(\xi, k), \quad k \in D_{\epsilon}(\beta) \setminus (D_{R}(\beta) \cup \gamma_{\text{cut}}), $$

where the error term $E(\xi, k)$ is given by

$$ E(\xi, k) = \int_{\beta}^{k} g_{1}(\xi, s) \left( \int_{\gamma_{(\beta, \alpha)}} \frac{u-\alpha}{4(s-u)^{3/2}} du \right) \sqrt{s-\beta} \, ds. \qquad (6.6) $$

Given $k \in D_{\epsilon}(\beta) \setminus (D_{R}(\beta) \cup \gamma_{\text{cut}})$, we let the contour from $\beta$ to $k$ in (6.6) consist of the straight segment from $\beta$ to $\beta + \frac{[k-\beta]}{|\alpha-\beta|}(\beta-\alpha)$ followed by an arc of the cut circle $\partial D_{|k-\beta|}(\beta) \setminus \gamma_{\text{cut}}$, see Fig. 6.1.

Increasing $T$ in (2.10) if necessary, we may assume that $t$ is so large that $\gamma_{(\beta,\alpha)}$ deviates only slightly from the straight line segment $[\alpha, \beta]$ from $\alpha$ to $\beta$. Then $|s-\beta| \le C|s-u|$ for all $s, u$ on the given contours and for all $(x, t) \in \mathbb{S}$. Hence, using (6.4), we can estimate

$$ |E(\xi, k)| \le C \int_{\beta}^{k} \int_{\gamma_{(\beta, \alpha)}} \frac{|u-\alpha|}{|s-u|} |du| |ds| $$
<figure><img src="image_10.png" /><figcaption>Fig. 6.1. The integration contour from $\beta$ to $k$ in equation (6.6).</figcaption></figure>

$$ 
\begin{aligned}
&\le C \int_{\beta}^{k} \int_{0}^{|\alpha-\beta|} \frac{|\alpha - \beta| - v}{v + |s - \beta|} dv |ds| \\
&\le C \int_{0}^{|k-\beta|} \int_{0}^{|\alpha-\beta|} \frac{|\alpha - \beta| - v}{v + r} dv dr + 2\pi C |k - \beta| \int_{0}^{|\alpha-\beta|} \frac{|\alpha - \beta| - v}{v + |k - \beta|} dv \\
&\le C \int_{0}^{|\alpha-\beta|} (|\alpha - \beta| - v) \ln\left(\frac{v + |k - \beta|}{v}\right) dv + C \int_{0}^{|\alpha-\beta|} (|\alpha - \beta| - v) dv \\
&\le C \int_{0}^{|\alpha-\beta|} (|\alpha - \beta| - v)(1 + \ln|v|) dv + C |\alpha - \beta|^2 \\
&\le C |\alpha - \beta|^2 |\ln|\alpha - \beta||, \quad k \in D_{\epsilon}(\beta) \setminus (D_{R}(\beta) \cup \gamma_{\text{cut}}), (x, t) \in \mathbb{S}.
\end{aligned}
$$

This completes the proof. □

### 6.2.4. A local change of variables

Taylor expanding $g_1(\xi, s)$ around $s = \beta$ and using (6.4), we find

$$ |g_1(\xi, s) - g_1(\xi, \beta) - g_{1s}(\xi, \beta)(s - \beta)| \le C|s - \beta|^2, \quad s \in D_{\epsilon}(\beta), (x, t) \in \mathbb{S}, $$

and hence

$$ \left| \int_{\beta}^{k} g_{1}(\xi, s) \left(s - \beta + \frac{\beta - \alpha}{2}\right) ds - \left\{ g_{1}(\xi, \beta) \frac{\beta - \alpha}{2} (k - \beta) \right\} \right| \le C |s - \beta|^2, \quad (x, t) \in \mathbb{S}. $$
$$ \left. + \left( g_1(\xi, \beta) + g_{1s}(\xi, \beta) \frac{\beta - \alpha}{2} \right) \frac{(k - \beta)^2}{2} \right\} \le C(k - \beta)^3, \quad k \in D_{\epsilon}(\beta), (x, t) \in \mathbb{S}. $$

It follows that if we define the functions $\lambda \equiv \lambda(x, t, k)$ and $y \equiv y(x, t)$ by

$$ \lambda := \sqrt{t} \psi_{\beta}(\xi)(k - \beta), \quad (6.7) $$

$$ y := -\sqrt{t} \frac{ig_1(\xi, \beta)(\beta - \alpha)}{2\psi_{\beta}(\xi)}, \quad (6.8) $$

where

$$ \psi_{\beta}(\xi) := e^{-\frac{i\pi}{4}} \sqrt{g_1(\xi, \beta) + g_{1s}(\xi, \beta) \frac{\beta - \alpha}{2}}, \quad (6.9) $$

then

$$ \left| 2it \int_{\beta}^{k} g_1(\xi, s) \left(s - \beta + \frac{\beta - \alpha}{2}\right) ds - (-\lambda^2 - 2y\lambda) \right| \le Ct|k - \beta|^3, \quad k \in D_{\epsilon}(\beta), (x, t) \in \mathbb{S}. \quad (6.10) $$

For each $(x, t) \in \mathbb{S}$, the map $k \mapsto \lambda$ is a biholomorphism of the disk $D_{\epsilon}(\beta)$ onto the open disk of radius $\sqrt{t}|\psi_{\beta}(\xi)|\epsilon$ centered at the origin. By deforming the contour $\Sigma^{(6)}$ slightly and fixing the branch of the square root in (6.9) appropriately, we may assume that $\mathcal{Y}_j$ is mapped into the ray $Y_j$ defined in (A.1) for $j = 1, \dots, 4$. Furthermore, by choosing

$$ c(\xi) = |\psi_{\beta}(\xi)|^{-1} $$

in the definition (4.5) of $R(\xi, t)$, we can arrange so that the circle $\partial D_R(\beta)$ is mapped onto the unit circle. Finally, we choose the branch cut $C$ in Fig. A.1 so that $\mathcal{Y}_{15}$ and $\mathcal{Y}_{16}$ are mapped into $C$.

**Remark 6.3.** For $\xi = \xi_0 = 0$ we have

$$ \psi_{\beta}(0) = e^{-\frac{i\pi}{4}} \sqrt{g_1(0, \beta(0))}, $$

and, by symmetry, $g_1(0, \beta(0)) > 0$. So in this case we can fix the branch of the square root in (6.9) by requiring that

$$ \arg \psi_{\beta}(\xi) \in \left(-\frac{\pi}{2}, 0\right) $$

for all small enough $\xi$.
**Lemma 6.4.** We have

$$2itg_{\beta}(\xi, k) = -\lambda^2 - 2y\lambda + O(t|\alpha - \beta|^2|\ln|\alpha - \beta||) + O(t|k - \beta|^3), \\ t \to +\infty, k \in D_{\epsilon}(\beta) \setminus (D_{R}(\beta) \cup \gamma_{\text{cut}}), (x, t) \in \mathbb{S}, \quad (6.11)$$

and

$$|g_{\beta+}(\xi, k) - g_{\beta-}(\xi, k)| = O(|\alpha - \beta|^2), \quad t \to +\infty, k \in \gamma_{\text{cut}}, (x, t) \in \mathbb{S}, \quad (6.12)$$

where the error terms are uniform with respect to $k$ and $x$ in the given ranges.

**Proof.** The estimate (6.11) follows immediately from Lemma 6.2 and (6.10).

In order to prove (6.12), we note that

$$\begin{align*} g_{\beta+}(\xi, k) - g_{\beta-}(\xi, k) &= \int_{\beta}^{k} \left( \left( \frac{dg}{dk} \right)_{+}(\xi, s) - \left( \frac{dg}{dk} \right)_{-}(\xi, s) \right) ds \\ &= \int_{\beta}^{k} g_{1}(\xi, s) \left( \left( \sqrt{s-\alpha}\sqrt{s-\beta} \right)_{+} - \left( \sqrt{s-\alpha}\sqrt{s-\beta} \right)_{-} \right) ds \\ &= 2 \int_{\beta}^{k} g_{1}(\xi, s) \left( \sqrt{s-\alpha}\sqrt{s-\beta} \right)_{+} ds, \quad k \in \gamma_{(\beta, \alpha)}, (x, t) \in \mathbb{S}, \end{align*}$$

where the integration contour follows the curve $\gamma_{(\beta, \alpha)}$ from $\beta$ to $k$. Hence, by (6.4),

$$\begin{align*} |g_{\beta+}(\xi, k) - g_{\beta-}(\xi, k)| &\le C \int_{0}^{|k-\beta|} \sqrt{|\alpha-\beta|-u} \sqrt{u} \, du \\ &\le C \int_{0}^{|\alpha-\beta|} \sqrt{|\alpha-\beta|-u} \sqrt{u} \, du \\ &= C \frac{\pi |\alpha-\beta|^2}{8}, \quad k \in \gamma_{(\beta, \alpha)}, (x, t) \in \mathbb{S}. \end{align*}$$

This establishes (6.12) for $k \in \gamma_{(\beta, \alpha)}$. Since

$$g_{\beta+}(\xi, k) - g_{\beta-}(\xi, k) = g_{\beta+}(\xi, \alpha) - g_{\beta-}(\xi, \alpha), \quad k \in \gamma_{(\alpha, \infty)}, (x, t) \in \mathbb{S},$$

the estimate (6.12) holds also for $k \in \gamma_{(\alpha, \infty)}$. $\square$
### 6.2.5. Behavior of $\tilde{\delta}$ as $k \to \beta$

We next consider the behavior of $\tilde{\delta}$ as $k$ approaches $\beta$. Let $\ln_{\mu}(k - \beta)$ denote the function $\ln(k - \beta)$ with branch cut along $\gamma_{(\mu, \beta)}$. More precisely,

$$ \ln_{\mu}(k - \beta) = \ln(k - \beta), \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}, $$

where the branch is fixed by requiring that

$$ \ln_C(\lambda) = \ln_{\mu}(k - \beta) + \ln(\sqrt{t}\psi_{\beta}(\xi)), \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}, \quad (x, t) \in \mathbb{S}, \quad (6.13) $$

where $\ln_C(\lambda)$ denotes the logarithm of $\lambda$ with cut along $C$, see Appendix A.

We also define the function $L_{\mu}$ by

$$ L_{\mu}(s, k) = \ln(k - s), \quad s \in \gamma_{(\mu, \beta)}, \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}, $$

where the branch is fixed by requiring that:

(i) $L_{\mu}(s, k)$ is a continuous function of $s \in \gamma_{(\mu, \beta)}$ for each $k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}$.

(ii) For $s = \beta$, we have $L_{\mu}(\beta, k) = \ln_{\mu}(k - \beta)$.

Integrating by parts, we find

$$ \int_{\gamma_{(\mu, \beta)}} \frac{\ln(1+r(s)r^*(s))}{s-k} ds = \ln_{\mu}(k-\beta) \ln(1+r(\beta)r^*(\beta)) - L_{\mu}(\mu, k) \ln(1+|r(\mu)|^2) \\ - \int_{\gamma_{(\mu, \beta)}} L_{\mu}(s, k) d\ln(1+r(s)r^*(s)). $$

Hence we can write

$$ \tilde{\delta}(\xi, k) = e^{-i\tilde{v}(\xi) \ln_{\mu}(k-\beta) + \tilde{\chi}(\xi, k)}, \quad k \in \mathbb{C} \setminus \gamma_{(\tilde{\beta}, \beta)}, \quad (x, t) \in \mathbb{S}, \quad (6.14) $$

where $\tilde{v}(\xi)$ is defined in (2.9) and the function $\tilde{\chi}(\xi, k)$ is defined by

$$ \tilde{\chi}(\xi, k) := -\frac{1}{2\pi i} L_{\mu}(\mu, k) \ln(1 + |r(\mu)|^2) - \frac{1}{2\pi i} \int_{\gamma_{(\mu, \beta)}} L_{\mu}(s, k) d\ln(1 + r(s)r^*(s)) \\ - \frac{1}{2\pi i} \int_{\gamma_{(\tilde{\beta}, \mu)}} \frac{\ln(1 + r(s)r^*(s))}{s - k} ds, \quad k \in \mathbb{C} \setminus \gamma_{(\tilde{\beta}, \beta)}, \quad (x, t) \in \mathbb{S}. \quad (6.15) $$

**Lemma 6.5.** We have

$$ \tilde{\delta}(\xi, k) = \lambda^{-i\tilde{v}(\xi)} t^{\frac{i\tilde{v}(\xi)}{2}} \delta_0(\xi) \delta_1(\xi, k), \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}, \quad (x, t) \in \mathbb{S}, $$

where the functions $\delta_0$ and $\delta_1$ are defined by
$$ \delta_0(\xi) := e^{i\tilde{v}(\xi) \ln \psi_{\beta}(\xi)} e^{\tilde{\chi}(\xi, \beta)}, \quad (x, t) \in \mathbb{S}, $$

$$ \delta_1(\xi, k) := e^{\tilde{\chi}(\xi, k) - \tilde{\chi}(\xi, \beta)}, \quad (x, t) \in \mathbb{S}, \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}. $$

**Proof.** Immediate from (6.13) and (6.14). $\square$

We also note that $\tilde{\chi}(\xi, k)$ is uniformly bounded, i.e.,

$$ |\tilde{\chi}(\xi, k)| \le C, \quad k \in D_{\epsilon}(\beta) \setminus \gamma_{(\mu, \beta)}, \quad (x, t) \in \mathbb{S}. $$

### 6.2.6. Third transformation

Let $m^{(\beta2)}(x, t, k) = m^{(\beta1)}(x, t, k)D(x, t, k)$, where $D$ is defined by

$$ D(x, t, k) = \begin{cases} t^{\frac{i\tilde{v}(\xi)\sigma_3}{2}} \delta_0(\xi)^{\sigma_3}, & k \in S_1 \cup S_2 \cup S_3 \cup S_4 \cup S_5, \\ I, & \text{else,} \end{cases} \quad (x, t) \in \mathbb{S}. \quad (6.16) $$

Then

$$ v_{\mathcal{Y}_1}^{(\beta2)} = \begin{pmatrix} \frac{rr^*}{1+rr^*} \lambda 2i\tilde{v} \delta_1^{-2} e^{2itg_\beta} & 0 \\ 0 & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_2}^{(\beta2)} = \begin{pmatrix} 1 & -\lambda^{-2i\tilde{v}} \delta_1^2 e^{-it(g_{\beta+} + g_{\beta-})} \\ 0 & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_3}^{(\beta2)} = \begin{pmatrix} 1 & 0 \\ rr^* \lambda 2i\tilde{v} \delta_1^{-2} e^{2itg_\beta} & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_4}^{(\beta2)} = \begin{pmatrix} 1 & -(1+rr^*)\lambda^{-2i\tilde{v}}\delta_1^2 e^{-2itg_\beta} \\ 0 & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_5}^{(\beta2)} = \lambda^{-i\tilde{v}\sigma_3}\delta_1^{\sigma_3}, $$

$$ v_{\mathcal{Y}_6}^{(\beta2)} = \lambda^{-i\tilde{v}\sigma_3}\delta_1^{\sigma_3} \begin{pmatrix} 1 & 0 \\ -\frac{rr^*}{1+rr^*}e^{2itg_\beta} & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_7}^{(\beta2)} = \lambda^{-i\tilde{v}\sigma_3}\delta_1^{\sigma_3} \begin{pmatrix} \frac{1}{1+rr^*} & e^{-2itg_\beta} \\ -\frac{rr^*}{1+rr^*}e^{2itg_\beta} & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_8}^{(\beta2)} = \lambda^{-i\tilde{v}\sigma_3}\delta_1^{\sigma_3} \begin{pmatrix} \frac{1}{1+rr^*} & e^{-2itg_\beta} \\ 0 & 1+rr^* \end{pmatrix}, $$

$$ v_{\mathcal{Y}_9}^{(\beta2)} = \lambda^{-i\tilde{v}\sigma_3}\delta_1^{\sigma_3} \begin{pmatrix} 1 & (1+rr^*)e^{-2itg_\beta} \\ 0 & 1 \end{pmatrix}, $$

$$ v_{\mathcal{Y}_{11}}^{(\beta2)} = \begin{pmatrix} \frac{1+rr^*e^{it(g_\beta+g_{\beta-})}}{1+rr^*} & e^{-2itg_\beta+}(1-e^{it(g_\beta+g_{\beta-})}) \\ rr^*e^{2itg_\beta+}(e^{-2it(g_\beta+g_{\beta-})}-1+rr^*e^{-it(g_\beta+g_{\beta-})}-1) & 1-\frac{rr^*e^{-it(g_\beta+g_{\beta-})}(1-e^{-it(g_\beta+g_{\beta-})})}{1+rr^*} \end{pmatrix}, $$

$$ v_{\mathcal{Y}_{14}}^{(\beta2)} = \begin{pmatrix} e^{it(g_\beta+g_{\beta-})} & 0 \\ 0 & e^{-it(g_\beta+g_{\beta-})} \end{pmatrix}, \qquad v_{\mathcal{Y}_{10}}^{(\beta2)} = v_{\mathcal{Y}_{12}}^{(\beta2)} = v_{\mathcal{Y}_{13}}^{(\beta2)} = v_{\mathcal{Y}_{15}}^{(\beta2)} = v_{\mathcal{Y}_{16}}^{(\beta2)} = I. \quad (6.17) $$

Let $\tilde{v}^Y$ denote the jump matrix $v^Y$ defined in (A.2) pulled back to $\mathcal{Y}$ via the map $k \mapsto \lambda$, that is,

$$ \tilde{v}^Y(x, t, k) = v^Y(y(x, t), \tilde{v}(\xi), \lambda(x, t, k)), \quad k \in \mathcal{Y}, (x, t) \in \mathbb{S}, \quad (6.18) $$

where we assume that $v^Y(y, \tilde{v}, \lambda)$ is the identity matrix whenever $\lambda \notin \mathcal{Y}$ and where $\lambda(x, t, k)$ and $y(x, t)$ are defined in (6.7) and (6.8), respectively. Define $\rho \equiv \rho(\xi)$ and $\rho^* \equiv \rho^*(\xi)$ by
$$ \rho = r(\beta) \quad \text{and} \quad \rho^* = r^*(\beta). $$

For a fixed $\lambda$, $r(k) \to \rho$ and $\delta_1(\xi, k) \to 1$ as $t \to +\infty$. This suggests that $v^{(\beta 2)}$ tends to $\tilde{v}^Y$ for large $t$. In the following subsection, we make this statement precise.

### 6.2.7. Estimates of $v^{(\beta 2)} - \tilde{v}^Y$

First an auxiliary lemma.

**Lemma 6.6.** Shrinking $\epsilon > 0$ if necessary, we have

$$ \left\{ \begin{array}{ll} |e^{2itg_{\beta}(k)}| \le C e^{-ct|k-\beta|^2}, & k \in \mathcal{Y}_1 \cup \mathcal{Y}_3, \\ |e^{-2itg_{\beta}(k)}| \le C e^{-ct|k-\beta|^2}, & k \in \mathcal{Y}_2 \cup \mathcal{Y}_4, \end{array} \right. \quad (x,t) \in \mathbb{S}, \qquad (6.19) $$

for some constants $c$, $C > 0$ independent of $k$, $x$, $t$.

**Proof.** It follows from (6.11) that

$$ |e^{2itg_{\beta}}| \le |e^{-\lambda^2 - 2y\lambda}| |e^{Ct|\alpha - \beta|^2} |\ln|\alpha - \beta|| e^{Ct|k - \beta|^3}, \quad k \in \mathcal{Y}_1 \cup \mathcal{Y}_3, (x,t) \in \mathbb{S}. $$

Since $y = O(\sqrt{t}|\alpha - \beta|) = O(t^{-\delta})$ tends uniformly to zero as $t \to +\infty$, we have (increasing $T$ in (2.10) if necessary)

$$ \left\{ \begin{array}{ll} |e^{-\lambda^2 - 2y\lambda}| \le C e^{-\frac{|\lambda|^2}{2}}, & \lambda \in Y_1 \cup Y_3, \\ |e^{\lambda^2 + 2y\lambda}| \le C e^{-\frac{|\lambda|^2}{2}}, & \lambda \in Y_2 \cup Y_4, \end{array} \right. \quad (x,t) \in \mathbb{S}. \qquad (6.20) $$

Moreover, for every $d > 0$ sufficiently small, we have

$$ \begin{aligned} t|\alpha - \beta|^2 |\ln|\alpha - \beta|| &= t|\alpha - \beta|^{2-d} \times |\alpha - \beta|^d |\ln|\alpha - \beta|| \\ &\le t|\alpha - \beta|^{2-d} \le Ct^{-c}, \quad (x,t) \in \mathbb{S}. \end{aligned} \qquad (6.21) $$

Hence

$$ |e^{2itg_{\beta}}| \le C e^{-\frac{|\lambda|^2}{2}} e^{Ct|k-\beta|^3}, \quad k \in \mathcal{Y}_1 \cup \mathcal{Y}_3, (x,t) \in \mathbb{S}. $$

From the definition (6.7), we have

$$ |\lambda| \le C\sqrt{t}|k - \beta| \le C|\lambda|, \quad k \in D_{\epsilon}(\beta), (x, t) \in \mathbb{S}. $$

Hence there exists a $C_1 > 0$ such that

$$ |e^{2itg_{\beta}}| \le C e^{-C_1 t |k-\beta|^2} e^{Ct|k-\beta|^3} = C e^{-t|k-\beta|^2(C_1-C|k-\beta|)}, \quad k \in \mathcal{Y}_1 \cup \mathcal{Y}_3, (x,t) \in \mathbb{S}. $$

Shrinking $\epsilon > 0$ if necessary, we may assume that $C_1 - C\epsilon > 0$. This proves the lemma in the case of $k \in \mathcal{Y}_1 \cup \mathcal{Y}_3$; a similar proof applies when $k \in \mathcal{Y}_2 \cup \mathcal{Y}_4$. $\square$
**Lemma 6.7.** As $t \to +\infty$, $v^{(\beta2)}$ approaches the jump matrix $\tilde{v}^Y$ defined in (6.18) in the sense that

$$ \|v^{(\beta2)}(x, t, \cdot) - \tilde{v}^Y(x, t, \cdot)\|_{L^1(\mathcal{Y})} \le C e_1(x, t) \quad (x, t) \in \mathbb{S}, \quad (6.22) $$

$$ \|v^{(\beta2)}(x, t, \cdot) - \tilde{v}^Y(x, t, \cdot)\|_{L^\infty(\mathcal{Y})} \le C e_\infty(x, t) \quad (x, t) \in \mathbb{S}, \quad (6.23) $$

where $e_1$ and $e_\infty$ are given by

$$ \begin{aligned} e_\infty(x, t) &:= t(\ln t)^{|\ln \tilde{v}|} |\alpha - \beta|^2 |\ln|\alpha - \beta|| + t^{-1/2}(\ln t)^{|\ln \tilde{v}|+\frac{3}{2}}, \\ e_1(x, t) &:= e_\infty(x, t) \sqrt{\frac{\ln t}{t}}. \end{aligned} $$

**Proof.** We will show that the following estimates hold uniformly for $(x, t) \in \mathbb{S}$:

$$ \|v^{(\beta2)}(x, t, \cdot) - \tilde{v}^Y(x, t, \cdot)\|_{L^1(\mathcal{Y}_j)} \le C e_1(x, t), \quad j = 1, \dots, 9, \quad (6.24a) $$

$$ \|v^{(\beta2)}(x, t, \cdot) - I\|_{L^1(\mathcal{Y}_{11})} \le C\sqrt{t} |\alpha - \beta|^2, \quad (6.24b) $$

$$ \|v^{(\beta2)}(x, t, \cdot) - I\|_{L^1(\mathcal{Y}_{14})} \le Ct|\alpha - \beta|^3, \quad (6.24c) $$

and

$$ \|v^{(\beta2)}(x, t, \cdot) - \tilde{v}^Y(x, t, \cdot)\|_{L^\infty(\mathcal{Y}_j)} \le C e_\infty(x, t), \quad j = 1, \dots, 9, \quad (6.25a) $$

$$ \|v^{(\beta2)}(x, t, \cdot) - I\|_{L^\infty(\mathcal{Y}_{11} \cup \mathcal{Y}_{14})} \le Ct|\alpha - \beta|^2. \quad (6.25b) $$

This will complete the proof.

We begin by proving (6.25a) for $j = 1$. Only the (21) entry of $v^{(\beta2)} - \tilde{v}^Y$ is nonzero for $k \in \mathcal{Y}_1$, so we find

$$ \begin{aligned} |v^{(\beta2)} - \tilde{v}^Y| &= \left| \frac{rr^*}{1+rr^*} \lambda^{2i\tilde{v}} \delta_1^{-2} e^{2itg_\beta} - \frac{\rho\rho^*}{1+\rho\rho^*} \lambda^{2i\tilde{v}} e^{-\lambda^2-2y\lambda} \right| \\ &\le |\lambda^{2i\tilde{v}}| \left( \left| \frac{rr^*}{1+rr^*} - \frac{\rho\rho^*}{1+\rho\rho^*} \right| |\delta_1^{-2} e^{2itg_\beta}| + \left| \frac{\rho\rho^*}{1+\rho\rho^*} \right| |\delta_1^{-2} - 1| |e^{2itg_\beta}| \right. \\ & \qquad \left. + \left| \frac{\rho\rho^*}{1+\rho\rho^*} \right| |e^{2itg_\beta} - e^{-\lambda^2-2y\lambda}| \right), \quad k \in \mathcal{Y}_1, (x,t) \in \mathbb{S}. \end{aligned} \quad (6.26) $$

Standard estimates show that

$$ |\tilde{\chi}(\xi, k) - \tilde{\chi}(\xi, \beta)| \le C|k - \beta| |\ln|k - \beta||, \quad k \in D_\epsilon(\beta) \setminus \gamma(\mu, \beta), (x, t) \in \mathbb{S}. $$

Using the general inequality

$$ |e^w - 1| \le |w| \max(1, e^{\Re w}), \quad w \in \mathbb{C}, \quad (6.27) $$

this yields
$$|\delta_1(\xi, k) - 1| \le C|k - \beta| |\ln|k - \beta||, \quad k \in D_\epsilon(\beta) \setminus \gamma_{(\mu, \beta)}, (x, t) \in \mathbb{S}. \quad (6.28)$$

On the other hand, the smoothness of $r(k)$ implies

$$|r(k) - r(\beta)| \le C|k - \beta|, \quad k \in D_{\epsilon}(\beta). \quad (6.29)$$

Using these inequalities, we see that (6.26) implies

$$|v^{(\beta 2)} - \tilde{v}^Y| \le C(F_1 + F_2), \quad k \in \mathcal{Y}_1, (x, t) \in \mathbb{S}, \quad (6.30)$$

where the functions $F_j \equiv F_j(x, t, k)$, $j = 1, 2$, are defined by

$$\begin{aligned} F_1 &= |\lambda|^{-2 \operatorname{Im} \tilde{v}} |k - \beta| |\ln|k - \beta|| |e^{2 \operatorname{Im} g \beta}|, \\ F_2 &= |\lambda|^{-2 \operatorname{Im} \tilde{v}} |e^{2 \operatorname{Im} g \beta} - e^{-\lambda^2 - 2y \lambda}|. \end{aligned}$$

We first consider $F_1$. Employing (6.19) and using that $\sqrt{t}|k - \beta| \ge C > 0$ on $\mathcal{Y}_1$ we obtain

$$\begin{aligned} |F_1| &\le C (\sqrt{t}|k - \beta|)^{-2 \operatorname{Im} \tilde{v}} |k - \beta| |\ln|k - \beta|| e^{-ct|k-\beta|^2} \\ &\le \frac{C}{\sqrt{t}} (\sqrt{t}|k - \beta|)^{1-2 \operatorname{Im} \tilde{v}} |\ln|k - \beta|| e^{-ct|k-\beta|^2} \\ &\le \frac{C}{\sqrt{t}} |\ln|k - \beta|| e^{-\frac{c}{2}t|k-\beta|^2}, \quad k \in \mathcal{Y}_1, (x, t) \in \mathbb{S}. \end{aligned}$$

It follows that

$$\|F_1\|_{L^1(\mathcal{Y}_1)} \le \frac{C}{\sqrt{t}} \int_0^\epsilon |\ln u| e^{-\frac{c}{2}tu^2} du \le Ct^{-1} \ln t, \quad (x, t) \in \mathbb{S}, \quad (6.31a)$$

and

$$\|F_1\|_{L^\infty(\mathcal{Y}_1)} \le Ct^{-1/2} \ln t, \quad (x, t) \in \mathbb{S}, \quad (6.31b)$$

We now consider $F_2$. On the one hand, the estimates (6.19) and (6.20) give

$$|e^{2 \operatorname{Im} g \beta} - e^{-\lambda^2 - 2y \lambda}| \le |e^{2 \operatorname{Im} g \beta}| + |e^{-\lambda^2 - 2y \lambda}| \le Ce^{-ct|k-\beta|^2}, \quad k \in \mathcal{Y}_1, (x, t) \in \mathbb{S}. \quad (6.32)$$

On the other hand, (6.11), (6.21), and (6.27) yield

$$\begin{aligned} |e^{2 \operatorname{Im} g \beta} - e^{-\lambda^2 - 2y \lambda}| &\le |e^{-\lambda^2 - 2y \lambda}||e^{\lambda^2 + 2y \lambda + 2 \operatorname{Im} g \beta} - 1| \\ &\le Ct (|\alpha - \beta|^2 |\ln|\alpha - \beta|| + |k - \beta|^3) e^{Ct|k-\beta|^3}, \quad k \in \mathcal{Y}_1, (x, t) \in \mathbb{S}. \end{aligned} \quad (6.33)$$
Let $d > 0$. On the part of $\mathcal{Y}_1$ on which $|k - \beta| \ge \sqrt{\frac{d \ln t}{ct}}$, we use the estimate (6.32) to find

$$|F_2| \le C(\sqrt{t}|k - \beta|)^{-2\operatorname{Im}\tilde{v}} t^{-d} \le Ct^{\operatorname{Im}\tilde{v}-d}, \quad k \in \mathcal{Y}_1, |k - \beta| \ge \sqrt{\frac{d \ln t}{ct}}. \quad (6.34)$$

On the part of $\mathcal{Y}_1$ on which $|k - \beta| \le \sqrt{\frac{d \ln t}{ct}}$, we use the estimate (6.33) to find

$$\begin{aligned} |F_2| &\le C|\lambda|^{-2\operatorname{Im}\tilde{v}} t \left(|\alpha - \beta|^2 |\ln|\alpha - \beta|| + |k - \beta|^3\right) \\ &\le C(\ln t)^{\operatorname{Im}\tilde{v}} \left(t|\alpha - \beta|^2 |\ln|\alpha - \beta|| + |k - \beta|^3\right), \quad k \in \mathcal{Y}_1, |k - \beta| \le \sqrt{\frac{d \ln t}{ct}}. \end{aligned} \quad (6.35)$$

Choosing $d$ larger than $\sup_{(x,t) \in \mathcal{S}} |\operatorname{Im}\tilde{v}| + 1$, it follows from (6.34) and (6.35) that

$$\begin{aligned} \|F_2\|_{L^1(\mathcal{Y}_1)} &\le Ct^{\operatorname{Im}\tilde{v}-d} + C(\ln t)^{\operatorname{Im}\tilde{v}} t|\alpha - \beta|^2 |\ln|\alpha - \beta|| \left(\sqrt{\frac{d \ln t}{ct}} - R\right) + C(\ln t)^{\operatorname{Im}\tilde{v}} t \int_R^{\sqrt{\frac{d \ln t}{ct}}} u^3 du \\ &\le Ct^{-1} + C(\ln t)^{\operatorname{Im}\tilde{v}+\frac{1}{2}} \sqrt{t} |\alpha - \beta|^2 |\ln|\alpha - \beta|| + C(\ln t)^{\operatorname{Im}\tilde{v}} t^{-1} \left((\ln t)^2 + C\right) \\ &\le C\sqrt{t} (\ln t)^{\operatorname{Im}\tilde{v}+\frac{1}{2}} |\alpha - \beta|^2 |\ln|\alpha - \beta|| + Ct^{-1}(\ln t)^{\operatorname{Im}\tilde{v}+2}, \quad (x,t) \in \mathcal{S}, \end{aligned} \quad (6.36a)$$

and that

$$\|F_2\|_{L^\infty(\mathcal{Y}_1)} \le Ct(\ln t)^{\operatorname{Im}\tilde{v}} |\alpha - \beta|^2 |\ln|\alpha - \beta|| + Ct^{-1/2}(\ln t)^{\operatorname{Im}\tilde{v}+\frac{3}{2}}, \quad (x,t) \in \mathcal{S}. \quad (6.36b)$$

Combining equations (6.30), (6.31), and (6.36), we obtain (6.24a) and (6.25a) for $j = 1$; the proofs for $j = 2, 3, 4$ are similar.

We next prove (6.25a) and (6.24a) for $j = 6$. We have

$$v^{(\beta 2)} - \tilde{v}^Y = \lambda^{-i\tilde{v}\sigma_3} \left( \delta_1^{\sigma_3} \begin{pmatrix} 1 & 0 \\ -\frac{rr^*}{1+rr^*} e^{2itg\beta} & 1 \end{pmatrix} - \begin{pmatrix} 1 & 0 \\ -\frac{\rho\rho^*}{1+\rho\rho^*} e^{-\lambda^2 - 2y\lambda} & 1 \end{pmatrix} \right), \quad k \in \mathcal{Y}_6, (x,t) \in \mathcal{S}. \quad (6.37)$$

Using (6.28) and the facts that $|\lambda| = 1$ and $|k - \beta| = O(t^{-1/2})$ on $\mathcal{Y}_6$, we see that the absolute value of the (11) element in (6.37) is bounded above by

$$|\lambda^{-i\tilde{v}}||\delta_1 - 1| \le C|k - \beta||\ln|k - \beta|| \le Ct^{-1/2}\ln t, \quad k \in \mathcal{Y}_6, (x,t) \in \mathcal{S}.$$

The (22) element satisfies a similar estimate. On the other hand, the (21) element in (6.37) is bounded above by
$$ \begin{aligned} & |\lambda^{i\tilde{v}}| \left| \frac{rr^*}{1+rr^*} \delta_1^{-1} e^{2itg_\beta} - \frac{\rho\rho^*}{1+\rho\rho^*} e^{-\lambda^2-2y\lambda} \right| \\ & \le C \left| \frac{rr^*}{1+rr^*} - \frac{\rho\rho^*}{1+\rho\rho^*} \right| |\delta_1^{-1} e^{2itg_\beta}| + C \left| \frac{\rho\rho^*}{1+\rho\rho^*} \right| |\delta_1^{-1}-1| |e^{2itg_\beta}| \\ & \qquad + C \left| \frac{\rho\rho^*}{1+\rho\rho^*} \right| |e^{2itg_\beta} - e^{-\lambda^2-2y\lambda}|, \quad k \in \mathcal{Y}_6, (x,t) \in \mathbb{S}. \end{aligned} $$

Recall that $y = O(t^{-\delta})$ and note that $|\lambda| = 1$ for $\lambda \in \mathcal{Y}_6$. Hence we have the following analog of (6.33):

$$ \begin{aligned} & |e^{2itg_\beta} - e^{-\lambda^2-2y\lambda}| \le C|e^{\lambda^2+2y\lambda+2itg_\beta} - 1| \\ & \le Ct|\alpha - \beta|^2|\ln|\alpha - \beta|| + Ct^{-1/2}, \quad k \in \mathcal{Y}_6, (x,t) \in \mathbb{S}. \end{aligned} \quad (6.38) $$

Using (6.28), we conclude that the (21) element in (6.37) is bounded above by

$$ \begin{aligned} & C|k - \beta||\ln|k - \beta|| + Ct|\alpha - \beta|^2|\ln|\alpha - \beta|| + Ct^{-1/2} \\ & \le Ct^{-1/2}\ln t + Ct|\alpha - \beta|^2|\ln|\alpha - \beta||, \quad k \in \mathcal{Y}_6, (x,t) \in \mathbb{S}. \end{aligned} $$

Hence, since $\mathcal{Y}_6$ has length of order $O(t^{-1/2})$, we arrive at

$$ \begin{aligned} \|v^{(\beta2)} - \tilde{v}^Y\|_{L^1(\mathcal{Y}_6)} &\le Ct^{-1} \ln t + C\sqrt{t} |\alpha - \beta|^2 |\ln|\alpha - \beta||, \\ \|v^{(\beta2)} - \tilde{v}^Y\|_{L^\infty(\mathcal{Y}_6)} &\le Ct^{-1/2} \ln t + Ct |\alpha - \beta|^2 |\ln|\alpha - \beta||. \end{aligned} $$

This proves (6.24a) and (6.25a) for $j = 6$; the proofs for $j = 5, 7, 8, 9$ are similar.

Since $\mathcal{Y}_{11}$ has length of order $O(t^{-1/2})$ and $\mathcal{Y}_{14}$ has length of order $O(|\alpha - \beta|)$, equations (6.24b) and (6.24c) are a direct consequence of (6.25b). It therefore only remains to prove (6.25b).

According to (6.12) we have

$$ t|g_{\beta+}(k) - g_{\beta-}(k)| \le Ct|\alpha - \beta|^2 \le C, \quad k \in \mathcal{Y}_{11} \cup \mathcal{Y}_{14}, (x,t) \in \mathbb{S}. $$

Thus, by (6.27),

$$ |e^{it(g_{\beta+}(k)-g_{\beta-}(k))} - 1| \le Ct|\alpha - \beta|^2, \quad k \in \mathcal{Y}_{11} \cup \mathcal{Y}_{14}, (x,t) \in \mathbb{S}. $$

The estimate (6.25b) now follows from the expressions for $v_{\mathcal{Y}_{11}}^{(\beta2)}$ and $v_{\mathcal{Y}_{14}}^{(\beta2)}$ given in (6.17). $\square$

Let $\tilde{m}^Y$ denote the solution $m^Y$ of Appendix A pulled back to $D_\epsilon(\beta)$ via the map $k \mapsto \lambda$, that is,

$$ \tilde{m}^Y(x, t, k) = m^Y(y(x, t), \tilde{v}(\xi), \lambda(x, t, k)), \quad k \in D_\epsilon(\beta) \setminus \mathcal{Y}, (x, t) \in \mathbb{S}, \quad (6.39) $$

where $\tilde{v}(\xi)$, $\lambda(x, t, k)$, and $y(x, t)$ are defined in (2.9), (6.7), and (6.8), respectively.
We have $m^{(\beta 2)} = \hat{m}^{(6)}G$ where

$$G(x, t, k) := e^{-i\tilde{h}\sigma_3} ABD$$

and the matrix-valued functions A, B, D are defined in (6.1), (6.2), and (6.16), respectively. Lemma 6.7 shows that the jumps of $m^{(\beta 2)} = \hat{m}^{(6)}G$ across $\mathcal{Y}$ approach those of $\tilde{m}^Y$. In other words, as $t \to +\infty$, the jumps of $\hat{m}^{(6)}$ approach those of the function $\tilde{m}^Y G^{-1}$. This suggests that we approximate $\hat{m}^{(6)}$ in $D_\epsilon(\beta)$ by a $2 \times 2$-matrix valued function $m^\beta$ of the form

$$m^{\beta} := Y_{\beta} t^{\frac{i\tilde{h}}{2}\sigma_3} \tilde{m}^{Y} G^{-1}, \qquad (6.40)$$

where $Y_\beta(x, t, k)$ is a function which is analytic for $k \in D_\epsilon(\beta)$ and we have included the $k$-independent factor $t^{\frac{i\tilde{h}}{2}\sigma_3}$ in order to make $Y_\beta$ of order O(1). To ensure that $m^\beta$ is a good approximation of $\hat{m}^{(6)}$ for large $t$, we want to choose $Y_\beta(k)$ so that $m^\beta(m^{\text{mod}})^{-1} \to I$ on $\partial D_\epsilon(\beta)$ as $t \to +\infty$. Now

$$G = e^{-i\tilde{h}\sigma_3}(\delta^2\hat{a}\hat{b})^{\sigma_3/2}t^{\frac{i\tilde{h}}{2}\sigma_3}\delta_0^{\sigma_3} \times \begin{cases} e^{-itg_+(\beta)\sigma_3} & k \in (D_\epsilon(\beta) \setminus D_R(\beta)) \cap S_+, \\ e^{-itg_-(\beta)\sigma_3} & k \in (D_\epsilon(\beta) \setminus D_R(\beta)) \cap S_- \end{cases}$$

We therefore define (we arbitrarily choose to define $Y_\beta$ using the expression involving $g_+(\beta)$; we could equally well have used the expression involving $g_-(\beta)$)

$$Y_{\beta}(x, t, k) := m^{\text{mod}}(x, t, k)e^{-i\tilde{h}\sigma_3}(\delta^2\hat{a}\hat{b})^{\sigma_3/2}\delta_0^{\sigma_3}e^{-itg_+(\beta)\sigma_3}. \qquad (6.41)$$

**Lemma 6.8.** For each $(x, t) \in \mathbb{S}$, the function $m^\beta(x, t, k)$ defined in (6.40) is an analytic function of $k \in D_\epsilon(\beta) \setminus \mathcal{Y}$ and the function $Y_\beta(x, t, k)$ defined in (6.41) is an analytic function of $k \in D_\epsilon(\beta)$. Moreover, we have the uniform estimate

$$|Y_{\beta}(x, t, k)| \le C, \quad k \in D_{\epsilon}(\beta), \quad (x, t) \in \mathbb{S}. \qquad (6.42)$$

Across $\mathcal{Y}$, $m^\beta$ obeys the jump condition $m_+^\beta = m_-^\beta v^\beta$, where the jump matrix $v^\beta$ satisfies

$$\|m_-^\beta(\hat{v}^{(6)} - v^\beta)(m_+^\beta)^{-1}\|_{L^\infty(\mathcal{Y})} \le CE_\infty(x, t), \quad (x, t) \in \mathbb{S}, \qquad (6.43a)$$

$$\|m_-^\beta(\hat{v}^{(6)} - v^\beta)(m_+^\beta)^{-1}\|_{L^1(\mathcal{Y})} \le CE_1(x, t), \quad (x, t) \in \mathbb{S}, \qquad (6.43b)$$

with the functions $E_1$ and $E_\infty$ defined by

$$\begin{aligned} E_\infty(x, t) &:= t^{|\text{Im } \tilde{v}|} e_\infty(x, t) \\ &= t^{|\text{Im } \tilde{v}|+1} (\ln t)^{|\text{Im } \tilde{v}|} |\alpha - \beta|^2 |\ln|\alpha - \beta|| + t^{|\text{Im } \tilde{v}|-\frac{1}{2}|\ln t|} |\text{Im } \tilde{v}|+\frac{3}{2}, \end{aligned} \qquad (6.44)$$

$$E_1(x, t) := E_\infty(x, t) \sqrt{\frac{\ln t}{t}}. \qquad (6.45)$$

Furthermore, on $\partial D_\epsilon(\beta)$ the quotient $m^{\text{mod}}(m^\beta)^{-1}$ satisfies
$$
\left| \frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\beta)} \left( m^{\text{mod}}(m^{\beta})^{-1} - I \right) dk - \frac{Y_{\beta}(x, t, \beta) t^{\frac{i\hat{v}}{2}\sigma_3} m_1^Y(y, \tilde{v}) t^{-\frac{i\hat{v}}{2}\sigma_3} Y_{\beta}(x, t, \beta)^{-1}}{\sqrt{t} \psi_{\beta}(\xi)} \right| \\
\leq C t^{\left|\text{Im} \, \tilde{v}\right|} (t^{-1} + t^{|\alpha - \beta|^2}), \quad (x, t) \in \mathcal{S}, \qquad (6.46)
$$

and

$$
\|m^{\text{mod}}(m^{\beta})^{-1} - I\|_{L^{\infty}(\partial D_{\epsilon}(\beta))} \leq C t^{|\text{Im}\,\tilde{v}| - \frac{1}{2}} + C t^{|\text{Im}\,\tilde{v}| + 1} |\alpha - \beta|^2, \quad (x, t) \in \mathcal{S}, \quad (6.47)
$$

where $m_1^Y(y, \tilde{v})$ is defined in (A.6).

**Proof.** The analyticity properties of $m^\beta$ and $Y_\beta$ are immediate. Using that $\text{Im } g_+(\beta) = 0$ (see [14, Section 3]), the bound (6.42) on $Y_\beta$ follows from the definition (6.41).

We next establish the estimates (6.43). We have

$$
\hat{v}^{(6)} - v^{\beta} = G_{-} \left( v^{(\beta 2)} - \tilde{v}^{Y} \right) G_{+}^{-1}, \quad k \in \mathcal{Y}, (x, t) \in \mathcal{S},
$$

and so

$$
m_{-}^{\beta}(\hat{v}^{(6)} - v^{\beta})(m_{+}^{\beta})^{-1} = Y_{\beta} t^{\frac{i\hat{v}}{2}\sigma_3} \tilde{m}_{-}^{Y}(v^{(\beta 2)} - \tilde{v}^{Y})(\tilde{m}_{+}^{Y})^{-1} t^{-\frac{i\hat{v}}{2}\sigma_3} Y_{\beta}^{-1}.
$$

In view of the bounds (6.42) and (A.7) on $Y_\beta$ and $m^Y$, this gives

$$
|m_{-}^{\beta}(\hat{v}^{(6)} - v^{\beta})(m_{+}^{\beta})^{-1}| \leq C t^{|\text{Im}\,\tilde{v}|} |v^{(\beta 2)} - \tilde{v}^{Y}|, \quad k \in \mathcal{Y}, (x, t) \in \mathcal{S}.
$$

The estimates (6.43) now follow from Lemma 6.7.

It remains to prove (6.46) and (6.47). Note that

$$
m^{\text{mod}}(m^{\beta})^{-1} - I = Y_{\beta} t^{\frac{i\hat{v}}{2}\sigma_3} (\tilde{m}^{Y})^{-1} - I) t^{-\frac{i\hat{v}}{2}\sigma_3} Y_{\beta}^{-1} \\
+ \begin{cases} 0, & k \in \partial D_{\epsilon}(\beta) \cap S_{+}, \\ Y_{\beta} t^{\frac{i\hat{v}}{2}\sigma_3} (e^{i\hat{v}(g_{+}(\beta)-g_{-}(\beta))\sigma_3} - I) (\tilde{m}^{Y})^{-1} t^{-\frac{i\hat{v}}{2}\sigma_3} Y_{\beta}^{-1}, & k \in \partial D_{\epsilon}(\beta) \cap S_{-}. \end{cases}
$$

Since $\inf_{(x,t) \in \mathcal{S}} |\psi_{\beta}(\xi)| > 0$, the variable $\lambda = \sqrt{t(k-\beta)\psi_{\beta}(\xi)}$ goes to infinity as $t \to +\infty$ whenever $k \in \partial D_{\epsilon}(\beta)$. Thus equation (A.5) yields (see (6.39))

$$
\tilde{m}^Y(x, t, k) = I + \frac{m_1^Y(y, \tilde{v})}{\sqrt{t(k-\beta)\psi_{\beta}(\xi)}} + O(t^{-1}), \quad t \to +\infty, k \in \partial D_{\epsilon}(\beta), (x, t) \in \mathcal{S},
$$

uniformly with respect to $k \in \partial D_{\epsilon}(\beta)$ and $(x, t) \in \mathcal{S}$. Consequently, using (6.42) and (A.7),

$$
\left| m^{\text{mod}}(m^{\beta})^{-1} - I + \frac{Y_{\beta} t^{\frac{i\hat{v}}{2}\sigma_3} m_1^Y t^{-\frac{i\hat{v}}{2}\sigma_3} Y_{\beta}^{-1}}{\sqrt{t}(k-\beta)\psi_{\beta}(\xi)} \right| \le C t^{|\text{Im}\,\tilde{v}|-1} + C t^{|\text{Im}\,\tilde{v}|+1} |g_+(\beta) - g_-(\beta)|, \\
k \in \partial D_{\epsilon}(\beta), (x, t) \in \mathcal{S}. \quad (6.48)
$$
But, by (6.12),

$$|g_{+}(\beta) - g_{-}(\beta)| \leq C|\alpha - \beta|^2.$$

Thus, using that the functions $Y_{\bar{\beta}}^{\pm 1}$ are analytic in $D_{\epsilon}(\beta)$, equation (6.46) follows from (6.48) and Cauchy's formula. Since $|\psi_{\beta}(\xi)|^{-1} \leq C$ for $(x, t) \in \mathbb{S}$, (6.47) also follows from (6.48). $\square$

# 7. Final steps

## 7.1. The approximate solution

Define a local solution $m^{\bar{\beta}}$ for $k \in D_{\epsilon}(\bar{\beta})$ by

$$m^{\bar{\beta}}(x, t, k) = \sigma_3 \sigma_1 \overline{m^{\beta}(x, t, \bar{k})} \sigma_1 \sigma_3.$$

We define an approximate solution $m^{\text{app}}(x, t, k)$ by

$$m^{\text{app}} = \begin{cases} m^{\beta}, & k \in D_{\epsilon}(\beta), \\ m^{\mu}, & k \in D_{\epsilon}(\mu), \\ m^{\bar{\beta}}, & k \in D_{\epsilon}(\bar{\beta}), \\ m^{\text{mod}}, & \text{elsewhere.} \end{cases}$$

## 7.2. The solution $m^{\text{dif}}$

We will show that the function $m^{\text{dif}}(x, t, k)$ defined by

$$m^{\text{dif}} = \hat{m}^{(6)}(m^{\text{app}})^{-1}$$

is such that $m^{\text{dif}} - I$ is small for large $t$. Let $\mathcal{D} = D_{\epsilon}(\beta) \cup D_{\epsilon}(\mu) \cup D_{\epsilon}(\bar{\beta})$ denote the union of the three open disks in (4.7). The function $m^{\text{dif}}$ satisfies the RH problem

$$\begin{cases} m^{\text{dif}}(x, t, \cdot) \in I + \dot{E}^2(\mathbb{C} \setminus \Sigma^{\text{dif}}), \\ m_{+}^{\text{dif}}(x, t, k) = m_{-}^{\text{dif}}(x, t, k)v^{\text{dif}}(x, t, k) \quad \text{for a.e. } k \in \Sigma^{\text{dif}}, \end{cases} \quad (7.1)$$

where the contour $\Sigma^{\text{dif}} := (\Sigma^{(6)} \setminus \Sigma^{\text{mod}}) \cup \partial \mathcal{D}$ is displayed in Fig. 7.1 and the jump matrix $v^{\text{dif}}$ is given by

$$v^{\text{dif}} = \begin{cases} m^{\text{mod}}\hat{v}^{(6)}(m^{\text{mod}})^{-1}, & k \in \Sigma^{(6)} \setminus \bar{D}, \\ m^{\text{mod}}(m^{\beta})^{-1}, & k \in \partial D_{\epsilon}(\beta), \\ m^{\text{mod}}(m^{\mu})^{-1}, & k \in \partial D_{\epsilon}(\mu), \\ m^{\text{mod}}(m^{\bar{\beta}})^{-1}, & k \in \partial D_{\epsilon}(\bar{\beta}), \\ m_{-}^{\beta}\hat{v}^{(6)}(m_{+}^{\beta})^{-1}, & k \in \mathcal{Y}, \\ m_{-}^{\mu}\hat{v}^{(6)}(m_{+}^{\mu})^{-1}, & k \in \mathcal{X}, \\ m_{-}^{\bar{\beta}}\hat{v}^{(6)}(m_{+}^{\bar{\beta}})^{-1}, & k \in \Sigma^{(6)} \cap D_{\epsilon}(\bar{\beta}) \end{cases}$$
<figure><img src="image_11.png" /><figcaption>Fig. 7.1. The jump contour $\Sigma^{\text{dif}}$.</figcaption></figure>

with $\mathcal{Y} := \Sigma^{(6)} \cap D_{\epsilon}(\beta)$ (see Fig. 4.4) and $\mathcal{X} := \Sigma^{(6)} \cap D_{\epsilon}(\mu)$.

Let $\hat{w} := v^{\text{dif}} - I$. The matrix $\hat{v}^{(6)} - I$ is exponentially small on $\Sigma' := \Sigma^{(6)} \setminus (\bar{D} \cup \gamma_{(\tilde{\beta}, \beta)})$, uniformly with respect to $(x, t) \in \mathbb{S}$, that is,

$$ \|\hat{w}\|_{(L^1 \cap L^\infty)(\Sigma')} \le C e^{-ct}, \quad (x, t) \in \mathbb{S}. \qquad (7.2a) $$

On the other hand, equation (6.47) implies

$$ \|\hat{w}\|_{(L^1 \cap L^\infty)(\partial D_\epsilon(\beta) \cup \partial D_\epsilon(\tilde{\beta}))} \le C t^{|\text{Im}\,\tilde{v}|-\frac{1}{2}} + C t^{|\text{Im}\,\tilde{v}|+1} |\alpha - \beta|^2, \quad (x, t) \in \mathbb{S}. \qquad (7.2b) $$

As in [14, (7.2c)], we have

$$ \|\hat{w}\|_{(L^1 \cap L^\infty)(\partial D_\epsilon(\mu))} \le C t^{-1/2}, \quad (x, t) \in \mathbb{S}, \qquad (7.2c) $$

and

$$ \left\{ \begin{array}{l} \|\hat{w}\|_{L^1(\mathcal{X})} \le C t^{-1} \ln t, \\ \|\hat{w}\|_{L^2(\mathcal{X})} \le C t^{-3/4} \ln t, \quad (x, t) \in \mathbb{S}. \\ \|\hat{w}\|_{L^\infty(\mathcal{X})} \le C t^{-1/2} \ln t, \end{array} \right. \qquad (7.2d) $$

For $k \in \mathcal{Y}$, we have

$$ \hat{w} = m_{-}^{\beta}(\hat{v}^{(6)} - v^{\beta})(m_{+}^{\beta})^{-1}, $$
so Lemma 6.8 yields

$$ \left\{ \begin{array}{ll} \|\hat{w}\|_{L^1(\mathcal{Y})} \le C E_1(x, t), & (x, t) \in \mathbb{S}. \\ \|\hat{w}\|_{L^\infty(\mathcal{Y})} \le C E_\infty(x, t), & (x, t) \in \mathbb{S}. \end{array} \right. \qquad (7.2e) $$

For $k \in \gamma_{(\tilde{\beta}, \beta)} \setminus \bar{\mathcal{D}}$ we have $\hat{w} = m^{\text{mod}}(e^{i\tilde{t}(g_+ - g_-)\sigma_3} - I)(m^{\text{mod}})^{-1}$; thus, since $g_+ - g_-$ is constant on $\gamma_{(\tilde{\beta}, \beta)}$ and $g_+(\beta) - g_-(\beta) = O(|\alpha - \beta|^2)$ (cf. (6.12)),

$$ \|\hat{w}\|_{(L^1 \cap L^2 \cap L^\infty)(\gamma_{(\tilde{\beta}, \beta)} \setminus \bar{\mathcal{D}})} \le Ct|\alpha - \beta|^2, \quad (x, t) \in \mathbb{S}. \qquad (7.2f) $$

Equations (7.2) show that

$$ \|\hat{w}\|_{L^1(\Sigma^{\text{dif}})} \le C E'_1(x, t), \quad (x, t) \in \mathbb{S}, \qquad (7.3a) $$

$$ \|\hat{w}\|_{L^\infty(\Sigma^{\text{dif}})} \le C E_\infty(x, t), \quad (x, t) \in \mathbb{S}, \qquad (7.3b) $$

where

$$ E'_1(x, t) := E_1(x, t) + t^{\lfloor \operatorname{Im} \tilde{v} \rfloor - \frac{1}{2}} + t^{\lfloor \operatorname{Im} \tilde{v} \rfloor + 1} |\alpha - \beta|^2. \qquad (7.4) $$

Since $|\operatorname{Im} \tilde{v}| < 1/2$, $E_\infty(x, t)$ goes uniformly to zero as $t \to +\infty$. Hence

$$ \|\hat{C}_{\tilde{w}}\|_{B(L^2(\Sigma^{\text{dif}}))} \le C \|\hat{w}\|_{L^\infty(\Sigma^{\text{dif}})} \to 0, \quad t \to +\infty, \qquad (7.5) $$

uniformly with respect to $(x, t) \in \mathbb{S}$, where $\hat{C}_{\tilde{w}}$ is defined by $\hat{C}_{\tilde{w}}f = \hat{C}_{-}(f\hat{w})$ as in [14, Section 7.2] ($\hat{C}_{-h}$ is the boundary value of $\hat{C}h$ from the right side of $\Sigma^{\text{dif}}$ with $\hat{C}$ the Cauchy operator associated with $\Sigma^{\text{dif}}$). Hence, increasing $T$ in (2.10) if necessary, we have

$$ \|\hat{C}_{\tilde{w}}\|_{B(L^2(\Sigma^{\text{dif}}))} \le C < 1, \quad (x, t) \in \mathbb{S}. $$

Then, in particular, $I - \hat{C}_{\tilde{w}}(x, t, \cdot) \in B(L^2(\Sigma^{\text{dif}}))$ is invertible for all $(x, t) \in \mathbb{S}$, so we can define $\hat{\mu}(x, t, k) \in I + L^2(\Sigma^{\text{dif}})$ by

$$ \hat{\mu} = I + (I - \hat{C}_{\tilde{w}})^{-1} \hat{C}_{\tilde{w}} I, \quad (x, t) \in \mathbb{S}. \qquad (7.6) $$

Standard estimates using the Neumann series show that

$$ \|\hat{\mu} - I\|_{L^2(\Sigma^{\text{dif}})} \le C \frac{\|\hat{w}\|_{L^2(\Sigma^{\text{dif}})}}{1 - \|\hat{C}_{\tilde{w}}\|_{B(L^2(\Sigma^{\text{dif}}))}}. $$

Thus

$$ \|\hat{\mu}(x, t, \cdot) - I\|_{L^2(\Sigma^{\text{dif}})} \le C \|\hat{w}\|_{L^2(\Sigma^{\text{dif}})} \le C \sqrt{E'_1 E_\infty}, \quad (x, t) \in \mathbb{S}. \qquad (7.7) $$

It follows that there exists a unique solution $m^{\text{dif}} \in I + \dot{E}^2(\hat{C} \setminus \Sigma^{\text{dif}})$ of the RH problem (7.1) for all sufficiently large $t$. This solution is given by
$$m^{\text{dif}}(x, t, k) = I + \hat{C}(\hat{\mu}\hat{w}) = I + \frac{1}{2\pi i} \int_{\Sigma^{\text{dif}}} \hat{\mu}(x, t, s) \hat{w}(x, t, s) \frac{ds}{s-k}, \quad k \in \mathbb{C} \setminus \Sigma^{\text{dif}}, (x, t) \in \mathbb{S}. \tag{7.8}$$

### 7.3. Asymptotics of $m^{\text{dif}}$

For each $(x, t) \in \mathbb{S}$, we have

$$\lim_{k \to \infty} k (m^{\text{dif}}(x, t, k) - I) = -\frac{1}{2\pi i} \int_{\Sigma^{\text{dif}}} \hat{\mu}(x, t, k) \hat{w}(x, t, k) dk. \tag{7.9}$$

Let us consider the contributions to the right-hand side of (7.9) from the different parts of $\Sigma^{\text{dif}}$. All error terms in what follows will be uniform with respect to $(x, t) \in \mathbb{S}$.
By (7.2a) and (7.7),

$$\int_{\Sigma'} \hat{\mu}(x, t, k) \hat{w}(x, t, k) dk = \int_{\Sigma'} \hat{w}(x, t, k) dk + \int_{\Sigma'} (\hat{\mu}(x, t, k) - I) \hat{w}(x, t, k) dk \\ = O(\|\hat{w}\|_{L^1(\Sigma')}) + O(\|\hat{\mu} - I\|_{L^2(\Sigma')} \|\hat{w}\|_{L^2(\Sigma')}) = O(e^{-ct}), \quad t \to +\infty.$$

Hence the contribution to the integral in (7.9) from $\Sigma'$ is $O(e^{-ct})$. The contribution from $\partial D_{\epsilon}(\mu)$ to the right-hand side of (7.9) is

$$\begin{aligned} & -\frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\mu)} \hat{w}(x, t, k) dk - \frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\mu)} (\hat{\mu}(x, t, k) - I) \hat{w}(x, t, k) dk \\ &= -\frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\mu)} \left( m^{\text{mod}}(m^{\mu})^{-1} - I \right) dk + O\left( \|\hat{\mu} - I\|_{L^2(\partial D_{\epsilon}(\mu))} \|\hat{w}\|_{L^2(\partial D_{\epsilon}(\mu))} \right) \\ &= \frac{T_{\mu}(x, t)}{\sqrt{t}} + O(t^{-1}) + O\left( \sqrt{E'_1 E_{\infty}} t^{-\frac{1}{2}} \right), \quad t \to +\infty, \end{aligned}$$

where

$$T_{\mu}(x, t) := - \frac{Y_{\mu}(x, t, \mu) m_{1}^{X}(\xi) Y_{\mu}(x, t, \mu)^{-1}}{\psi_{\mu}(\xi, \mu)}. \tag{7.10}$$

The function $\psi_{\mu}(k) \equiv \psi_{\mu}(\xi, k)$ and the constant $m_1^X \equiv m_1^X(\xi)$ are defined in [14, (6.17) & (6.31)], while $Y_{\mu}$ is defined as in [14, (6.26)] but with a different $m^{\text{mod}}$, which comes from Section 5:

$$Y_{\mu}(x, t, k) := m^{\text{mod}}(x, t, k) e^{-i h(k) \sigma_3} B(k) \delta(k)^{-\sigma_3} \delta_0(t)^{\sigma_3}.$$

The contribution from $\mathcal{X}$ to the right-hand side of (7.9) is
$$O(\|\hat{w}\|_{L^1(\mathcal{X})}) + O(\|\hat{\mu} - I\|_{L^2(\Sigma^{\text{dif}})} \|\hat{w}\|_{L^2(\mathcal{X})}) = O\left(t^{-\frac{1}{4}} \ln t + \sqrt{E_1' E_\infty} t^{-\frac{3}{4}} \ln t\right), \quad t \to +\infty.$$

By (6.46), (7.2b), and (7.7), the contribution from $\partial D_\epsilon(\beta)$ to the right-hand side of (7.9) is

$$\begin{aligned}
& -\frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\beta)} \hat{w}(x, t, k) dk - \frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\beta)} (\hat{\mu}(x, t, k) - I) \hat{w}(x, t, k) dk \\
&= -\frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\beta)} (m^{\text{mod}}(m^{\beta})^{-1} - I) dk + O(\|\hat{\mu} - I\|_{L^2(\partial D_{\epsilon}(\beta))} \|\hat{w}\|_{L^2(\partial D_{\epsilon}(\beta))}) \\
&= \frac{T_{\beta}(x, t)}{\sqrt{t}} + O\left(\sqrt{E_1' E_\infty} (t^{\text{Im}\tilde{v}}|^{-\frac{1}{2}} + t^{\text{Im}\tilde{v}+1}|\alpha - \beta|^2)\right), \quad t \to +\infty,
\end{aligned}$$

where

$$T_{\beta}(x, t) := - \frac{Y_{\beta}(x, t, \beta) t^{\frac{i\tilde{v}}{2}} \sigma_3 m_1^{\gamma}(y, \tilde{v}) t^{-\frac{i\tilde{v}}{2}} \sigma_3 Y_{\beta}(x, t, \beta)^{-1}}{\psi_{\beta}(\xi)}. \quad (7.11)$$

The function $Y_\beta(x, t, k)$ is defined in (6.41), $m_1^\gamma(y, \tilde{v})$ is defined in (A.6) with $y \equiv y(x, t)$ given by (6.8), and $\psi_\beta(\xi)$ is defined in (6.9). The symmetries $\hat{\mu}(x, t, k) = \sigma_3 \overline{\sigma_1 \hat{\mu}(x, t, k)} \sigma_1 \sigma_3$ and $\hat{w}(x, t, k) = \sigma_3 \sigma_1 \hat{w}(x, t, k) \sigma_1 \sigma_3$ imply that the contribution from $D_\epsilon(\bar{\beta})$ to the right-hand side of (7.9) is

$$\begin{aligned}
-\frac{1}{2\pi i} \int_{\partial D_{\epsilon}(\bar{\beta})} (\hat{\mu}\hat{w})(x, t, k) dk &= \frac{1}{2\pi i} \overline{\sigma_3 \sigma_1 \int_{\partial D_{\epsilon}(\beta)} (\hat{\mu}\hat{w})(x, t, k) dk \sigma_1 \sigma_3} \\
&= \frac{\sigma_3 \sigma_1 \overline{T_{\beta}(x, t)} \sigma_1 \sigma_3}{\sqrt{t}} + O\left(\sqrt{E_1' E_\infty} (t^{\text{Im}\tilde{v}}|^{-\frac{1}{2}} + t^{\text{Im}\tilde{v}+1}|\alpha - \beta|^2)\right), \quad t \to +\infty.
\end{aligned}$$

The contribution from $\mathcal{Y} = \Sigma^{(6)} \cap D_\epsilon(\beta)$ to the right-hand side of (7.9) is

$$O(\|\hat{w}\|_{L^1(\mathcal{Y})} + \|\hat{\mu} - I\|_{L^2(\mathcal{Y})} \|\hat{w}\|_{L^2(\mathcal{Y})}) = O\left(E_1 + \sqrt{E_1' E_\infty} \sqrt{E_1 E_\infty}\right), \quad t \to +\infty.$$

The contribution from $\Sigma^{(6)} \cap D_\epsilon(\bar{\beta})$ is of the same order. By (7.2f) and (7.7), the contribution from $\Sigma'' := \gamma_{(\bar{\beta}, \beta)} \setminus \bar{D}$ is

$$O(\|\hat{w}\|_{L^1(\Sigma'')}) + O(\|\hat{\mu} - I\|_{L^2(\Sigma'')}) \|\hat{w}\|_{L^2(\Sigma'')}) = O\left(t|\alpha - \beta|^2 + \sqrt{E_1' E_\infty} t|\alpha - \beta|^2\right), \quad t \to +\infty.$$

Collecting the above contributions, we find from (7.9) that
$$
\begin{equation}
\begin{aligned}
\lim_{k \to \infty} k(m^{\text{dif}}(x, t, k) - I) = & \frac{T_{\mu}(x, t) + T_{\beta}(x, t) + \sigma_3 \sigma_1 \overline{T_{\beta}(x, t)} \sigma_1 \sigma_3}{\sqrt{t}} + O\left(\sqrt{E'_1 E_{\infty}} t^{-\frac{1}{2}}\right) \\
& + O\left(t^{-1} \ln t + \sqrt{E'_1 E_{\infty}} t^{-\frac{3}{4}} \ln t\right) + O\left(\sqrt{E'_1 E_{\infty}} \left(t^{\lvert \text{Im } \tilde{v} \rvert - \frac{1}{2}} + t^{\lvert \text{Im } \tilde{v} \rvert + 1} \rvert \alpha - \beta \rvert^2\right)\right) \\
& + O\left(E_1 + \sqrt{E'_1 E_{\infty}} \sqrt{E_1 E_{\infty}}\right) + O\left(t \rvert \alpha - \beta \rvert^2 + \sqrt{E'_1 E_{\infty}} t \rvert \alpha - \beta \rvert^2\right), \quad t \to +\infty.
\end{aligned}
\tag{7.12}
\end{equation}
$$

Using that

$$
E_{\infty} \leq F, \quad E_{1} \leq t^{-\frac{1}{2}} F, \quad E'_{1} \leq F,
$$

where $F$ is given by (2.14), we can replace the error term with the simpler (and only slightly less
sharp) expression

$$
O(F^2 + t|\alpha - \beta|^2).
$$

In particular, if $\xi = \xi_0$, then the error is $O(t^{2|\text{Im } \tilde{v}_0| - 1}(\ln t)^{2|\text{Im } \tilde{v}_0| + 4})$.

7.4. Asymptotics of q

Recalling the various transformations of Section 4, we have

$$
\hat{m} = e^{i \tilde{g}^{(0)} \sigma_3} e^{i \tilde{h}(\xi, \infty) \sigma_3} m^{\text{dif}} m^{\text{mod}} e^{-i \tilde{h} \sigma_3} \tilde{\delta} \sigma_3 \delta \sigma_3 e^{-i \tilde{t} (g(k) - \theta(k)) \sigma_3}
$$

for all large $k$ in $i\mathbb{R}_+$. It follows that

$$
\begin{align*}
& \lim_{k \to \infty} k (\hat{m}(x, t, k) - I)_{12} \\
&= e^{i \tilde{g}^{(0)} \sigma_3} e^{i \tilde{h}(\xi, \infty) \sigma_3} \lim_{k \to \infty} k \left( m^{\text{mod}} - I + (m^{\text{dif}} - I) m^{\text{mod}} \right)_{12} e^{-i \tilde{h}(\xi, \infty) \sigma_3} e^{-i \tilde{g}^{(0)} \sigma_3} \\
&= e^{i (\tilde{g}^{(0)} + \tilde{h}(\xi, \infty)) \hat{\sigma}_3} \left( \lim_{k \to \infty} k \left( m^{\text{mod}} - I \right)_{12} + \lim_{k \to \infty} k (m^{\text{dif}} - I)_{12} \right).
\end{align*}
$$

Hence

$$
\begin{align*}
q(x,t) &= 2i \lim_{k \to \infty} k (\hat{m}(x,t,k))_{12} \\
&= 2ie^{2i(\tilde{g}^{(0)}(\xi) + \tilde{h}(\xi,\infty))} \left( \lim_{k \to \infty} k (m^{\text{mod}}(x,t,k))_{12} + \lim_{k \to \infty} k (m^{\text{dif}}(x,t,k))_{12} \right).
\end{align*}
$$

In view of (5.6) and (7.12), this yields (2.13) and completes the proof of Theorem 2.1.
<figure><img src="image_12.png" /><figcaption>Fig. A.1. The contour $Y = \cup_{j=1}^{9} Y_j$ and the domains $\{S_j\}_1^6$ in the complex $\lambda$-plane.</figcaption></figure>

## Acknowledgments

J. Lenells acknowledges support from the Göran Gustafsson Foundation, the Ruth and Nils-Erik Stenbäck Foundation, the Swedish Research Council, Grant No. 2021-03877, and The European Research Council, Grant Agreement No. 682537.

## Appendix A. An exactly solvable RH problem

We define the contour $Y \subset \mathbb{C}$ by $Y = \cup_{j=1}^{9} Y_j$, where $Y_1, \dots, Y_4$ denote the four rays

$$ Y_j = \{re^{\frac{(j-1)\pi}{2}} \mid 1 \le r < \infty\}, \quad j = 1, \dots, 4, \qquad (A.1) $$

and $Y_5, \dots, Y_9$ denote the following arcs whose union is the unit circle:

$$ \begin{aligned} Y_5 &= \{e^{i\varphi} \mid -\frac{\pi}{2} \le \varphi \le 0\}, & Y_6 &= \{e^{i\varphi} \mid 0 \le \varphi \le \frac{\pi}{2}\}, & Y_7 &= \{e^{i\varphi} \mid \frac{\pi}{2} \le \varphi \le \pi\}, \\ Y_8 &= \{e^{i\varphi} \mid \pi \le \varphi \le \frac{5\pi}{4}\}, & Y_9 &= \{e^{i\varphi} \mid -\frac{3\pi}{4} \le \varphi \le -\frac{\pi}{2}\}. \end{aligned} $$

We orient $Y$ as in Fig. A.1.

We let $\mathcal{C}$ denote a branch cut going from $0$ to $\infty$ in the third quadrant (see Fig. A.1). We let $\ln_C \lambda$ denote the function $\ln \lambda$ with cut along $\mathcal{C}$ and branch fixed by the condition that $\ln_C \lambda > 0$ for $\lambda > 0$.
Given $y \in \mathbb{R}$ and $\tilde{v} \in \mathbb{C} \setminus \mathbb{Z}$, we define the jump matrix $v^Y(y, \tilde{v}, \lambda)$ for $\lambda \in Y$ by

$$ v^Y(y, \tilde{v}, \lambda) = \begin{cases} \begin{pmatrix} 1 & 0 \\ \frac{\rho\rho^*}{1+\rho\rho^*} \lambda 2i\tilde{v} e^{-\lambda^2-2y\lambda} & 1 \end{pmatrix} & \lambda \in Y_1, \\ \begin{pmatrix} 1 & -\lambda-2i\tilde{v}e^{\lambda^2+2y\lambda} \\ 0 & 1 \end{pmatrix} & \lambda \in Y_2, \\ \begin{pmatrix} 1 & 0 \\ \rho\rho^*\lambda 2i\tilde{v}e^{-\lambda^2-2y\lambda} & 1 \end{pmatrix} & \lambda \in Y_3, \\ \begin{pmatrix} 1 & -(1+\rho\rho^*)\lambda-2i\tilde{v}e^{\lambda^2+2y\lambda} \\ 0 & 1 \end{pmatrix} & \lambda \in Y_4, \\ \lambda^{-i\tilde{v}\sigma_3}, & \lambda \in Y_5, \quad (A.2) \\ \lambda^{-i\tilde{v}\sigma_3} \begin{pmatrix} 1 & 0 \\ -\frac{\rho\rho^*}{1+\rho\rho^*}e^{-\lambda^2-2y\lambda} & 1 \end{pmatrix} & \lambda \in Y_6, \\ \lambda^{-i\tilde{v}\sigma_3} \begin{pmatrix} \frac{1}{1+\rho\rho^*} & e^{\lambda^2+2y\lambda} \\ -\frac{\rho\rho^*}{1+\rho\rho^*}e^{-\lambda^2-2y\lambda} & 1 \end{pmatrix} & \lambda \in Y_7, \\ \lambda^{-i\tilde{v}\sigma_3} \begin{pmatrix} \frac{1}{1+\rho\rho^*} & e^{\lambda^2+2y\lambda} \\ 0 & 1+\rho\rho^* \end{pmatrix} & \lambda \in Y_8, \\ \lambda^{-i\tilde{v}\sigma_3} \begin{pmatrix} 1 & (1+\rho\rho^*)e^{\lambda^2+2y\lambda} \\ 0 & 1 \end{pmatrix} & \lambda \in Y_9, \end{cases} $$

where $\rho\rho^* := e^{2\pi\tilde{v}} - 1$ and the branch cut runs along $C$, so that $\lambda^a = e^{a \ln C} \lambda$ for $a \in \mathbb{C}$.

We consider the following family of RH problems parametrized by $y \in \mathbb{R}$ and $\tilde{v} \in \mathbb{C} \setminus \mathbb{Z}$:

$$ \begin{cases} m^Y(y, \tilde{v}, \cdot) \in I + \dot{E}^2(\mathbb{C} \setminus Y), \\ m_+^Y(y, \tilde{v}, \lambda) = m_-^Y(y, \tilde{v}, \lambda)v^Y(y, \tilde{v}, \lambda) \quad \text{for a.e. } \lambda \in Y. \end{cases} \quad (A.3) $$

Define the function $\beta^Y(\tilde{v})$ by

$$ \beta^Y(\tilde{v}) = \frac{(2i)^{i\tilde{v}+1} \sqrt{\pi} e^{\frac{5\pi i \tilde{v}}{2}}}{(e^{2\pi i \tilde{v}} - 1) \Gamma(i\tilde{v})}, \quad \tilde{v} \in \mathbb{C} \setminus \mathbb{Z}. \quad (A.4) $$

**Theorem A.1.** For any choice of $y \in \mathbb{R}$ and $\tilde{v} \in \mathbb{C} \setminus \mathbb{Z}$, the RH problem (A.3) has a unique solution $m^Y(y, \tilde{v}, \lambda)$. This solution satisfies

$$ m^Y(y, \tilde{v}, \lambda) = I + \frac{m_1^Y(y, \tilde{v})}{\lambda} + O(\lambda^{-2}), \quad \lambda \to \infty, \quad (A.5) $$

where $m_1^Y$ is defined by

$$ m_1^Y(y, \tilde{v}) := \begin{pmatrix} -i\tilde{v}y & -\frac{\beta^Y(\tilde{v})}{2}e^{-y^2} \\ -\frac{i\tilde{v}}{\beta^Y(\tilde{v})}e^{y^2} & i\tilde{v}y \end{pmatrix}, \quad y \in \mathbb{R}, \tilde{v} \in \mathbb{C} \setminus \mathbb{Z}, \quad (A.6) $$
and the error term is uniform with respect to $\arg \lambda \in [0, 2\pi]$ and $\tilde{v}$ and $y$ in compact subsets of $\mathbb{C} \setminus i\mathbb{Z}$ and $\mathbb{R}$, respectively. Moreover, for any compact subsets $K_1 \subset \mathbb{R}$ and $K_2 \subset \mathbb{C} \setminus i\mathbb{Z}$, we have

$$ \sup_{y \in K_1} \sup_{\tilde{v} \in K_2} \sup_{\lambda \in \mathbb{C} \setminus \mathcal{Y}} |m^Y(y, \tilde{v}, \lambda)| < \infty. \quad (A.7) $$

**Proof.** Uniqueness of the solution $m^Y$ follows because $\det v^Y = 1$. Fix $y \in \mathbb{R}$ and $\tilde{v} \in \mathbb{C} \setminus i\mathbb{Z}$. Let the branch cut run along $\mathcal{C}$. Define the $2 \times 2$-matrix valued function $\psi(y, \tilde{v}, \lambda)$ by

$$ \psi = e^{-\frac{\lambda^2}{2}\sigma_3} \begin{pmatrix} e^{-\frac{\pi \tilde{v}}{2}} D_{-i\tilde{v}}(i\sqrt{2}(y+\lambda)) & -\frac{\beta^Y(\tilde{v})}{\sqrt{2}} D_{i\tilde{v}-1}(\sqrt{2}(y+\lambda)) \\ \frac{\sqrt{2}}{\beta^Y(\tilde{v})} e^{-\frac{\pi \tilde{v}}{2}} \tilde{v} D_{-i\tilde{v}-1}(i\sqrt{2}(y+\lambda)) & D_{i\tilde{v}}(\sqrt{2}(y+\lambda)) \end{pmatrix} 2^{\frac{\tilde{v}}{2}\sigma_3}, \quad (A.8) $$

where $D_a(z)$ denotes the parabolic cylinder function. Define the matrices $R_j \equiv R_j(\tilde{v})$, $j = 1, \dots, 4$, by

$$ \begin{aligned} R_1 &= \begin{pmatrix} 1 & 0 \\ \frac{\rho\rho^*}{1+\rho\rho^*} & 1 \end{pmatrix}, & R_2 &= \begin{pmatrix} 1 & -1 \\ 0 & 1 \end{pmatrix}, \\ R_3 &= \begin{pmatrix} 1 & 0 \\ \rho\rho^* & 1 \end{pmatrix}, & R_4 &= \begin{pmatrix} 1 & -(1+\rho\rho^*) \\ 0 & 1 \end{pmatrix}. \end{aligned} $$

We claim that the solution of (A.3) is given explicitly in terms of parabolic cylinder functions by

$$ m^Y(y, \tilde{v}, \lambda) = \begin{cases} \Psi(y, \tilde{v}, \lambda)e^{-(\frac{\lambda^2}{2} + y\lambda)\sigma_3} \lambda^{i\tilde{v}\sigma_3}, & \lambda \in S_1 \cup \dots \cup S_5, \\ \Psi(y, \tilde{v}, \lambda)e^{-(\frac{\lambda^2}{2} + y\lambda)\sigma_3}, & \lambda \in S_6, \end{cases} \quad (A.9) $$

where the sectionally analytic function $\Psi$ is defined by

$$ \Psi(y, \tilde{v}, \lambda) = \begin{cases} \psi, & \lambda \in S_1, \\ \psi R_1, & \lambda \in S_2, \\ \psi R_1 R_2, & \lambda \in S_3, \\ \psi R_1 R_2 R_3^{-1}, & \lambda \in S_4, \\ \psi R_4, & \lambda \in S_5, \\ \psi, & \lambda \in S_6. \end{cases} \quad (A.10) $$

Recall that $D_a(z)$ is an entire function of both $a$ and $z$. In particular, $m^Y(y, \tilde{v}, \lambda)$ is an analytic function of $\lambda \in \mathbb{C} \setminus Y$. It is easily seen from the definition (A.10) of $\Psi$ that $m^Y$ satisfies the jump condition in (A.3).

For each $\delta > 0$, the parabolic cylinder function satisfies the asymptotic formula [38]

$$ D_a(z) = z^a e^{-\frac{z^2}{4}} \left( 1 - \frac{a(a-1)}{2z^2} + O(z^{-4}) \right) $$
$$ -\frac{\sqrt{2\pi} e^{\frac{z^2}{4}} z^{-a-1}}{\Gamma(-a)} \left(1 + \frac{(a+1)(a+2)}{2z^2} + O(z^{-4})\right) \\ \times \begin{cases} 0, & \arg z \in \left[-\frac{3\pi}{4} + \delta, \frac{3\pi}{4} - \delta\right], \\ e^{i\pi a}, & \arg z \in \left[\frac{\pi}{4} + \delta, \frac{5\pi}{4} - \delta\right], \\ e^{-i\pi a}, & \arg z \in \left[-\frac{5\pi}{4} + \delta, -\frac{\pi}{4} - \delta\right], \end{cases} \quad z \to \infty, a \in \mathbb{C}, \quad (A.11) $$

where the error terms are uniform with respect to $a$ in compact subsets and $\arg z$ in the given ranges. Tedious computations using this formula in the explicit formula (A.9) show that $m^Y$ satisfies (A.5), together with (A.6), uniformly for $\tilde{v}$ and $y$ in compact subsets and for $\arg \lambda \in [0, 2\pi]$. It follows that $m^Y$ satisfies (A.7) and that $m^Y(y, \tilde{v}, \cdot) \in I + \hat{E}^2(\mathbb{C} \setminus Y)$. This completes the proof of the theorem. $\square$

**Remark A.2.** The explicit formula for $m^Y$ given in (A.9) can be derived as follows. Suppose $m^Y$ is a solution of (A.3). Let

$$ \Psi := m^Y e^{\left(\frac{\lambda^2}{2} + y\lambda\right)\sigma_3} \lambda^{-i\tilde{v}\sigma_3}. $$

Then $\Psi$ satisfies the jump condition

$$ \Psi_+ = \Psi_- v^\Psi, \quad \lambda \in Y \cup \{\arg \lambda = -\frac{3\pi}{4}\}, $$

where the jump matrix $v^\Psi$ is given on $Y$ by (for $j = 1, \dots, 9$, $v_j^\Psi$ denotes the restriction of $v^\Psi$ to $Y_j$)

$$ \begin{gathered} v_j^\Psi = R_j, \quad j = 1, \dots, 4, \qquad v_5^\Psi = I, \qquad v_6^\Psi = \begin{pmatrix} 1 & 0 \\ -\frac{\rho\rho^*}{1+\rho\rho^*} & 1 \end{pmatrix}, \\ v_7^\Psi = \begin{pmatrix} \frac{1}{1+\rho\rho^*} & 1 \\ -\frac{\rho\rho^*}{1+\rho\rho^*} & 1 \end{pmatrix}, \qquad v_8^\Psi = \begin{pmatrix} \frac{1}{1+\rho\rho^*} & 0 \\ 0 & 1+\rho\rho^* \end{pmatrix}, \qquad v_9^\Psi = \begin{pmatrix} 1 & 1+\rho\rho^* \\ 0 & 1 \end{pmatrix}, \end{gathered} $$

and on the branch cut (oriented towards the origin) by

$$ v^\Psi = \begin{pmatrix} 1 + \rho\rho^* & 0 \\ 0 & \frac{1}{1+\rho\rho^*} \end{pmatrix}, \quad \arg \lambda = -\frac{3\pi}{4}. $$

Since $v^\Psi$ is independent of $\lambda$ and $y$, we conclude that

$$ A := \Psi_{\lambda} \Psi^{-1} = m_{\lambda}^{Y} (m^{Y})^{-1} + (\lambda + y) m^{Y} \sigma_{3} (m^{Y})^{-1} - i \tilde{v} \lambda^{-1} m^{Y} \sigma_{3} (m^{Y})^{-1} $$

and

$$ U := \Psi_y \Psi^{-1} = m_y^Y (m^Y)^{-1} + \lambda m^Y \sigma_3 (m^Y)^{-1} $$

are entire functions. Assuming that
$$m^Y(y, \tilde{v}, \lambda) = I + \frac{m_1^Y(y, \tilde{v})}{\lambda} + O(\lambda^{-2}), \quad \lambda \to \infty,$$

we deduce that

$$A(y, \tilde{v}, \lambda) = \lambda A_1(y, \tilde{v}) + A_0(y, \tilde{v}),$$

$$U(y, \tilde{v}, \lambda) = \lambda U_1(y, \tilde{v}) + U_0(y, \tilde{v}).$$

The terms of order $O(\lambda)$ show that $A_1 = U_1 = \sigma_3$ while the terms of order $O(1)$ show that

$$A_0 = y\sigma_3 + [m_1^Y, \sigma_3], \quad U_0 = [m_1^Y, \sigma_3].$$

Defining $w(y, \tilde{v})$ and $z(y, \tilde{v})$ by $w := -2(m_1^Y)_{12}$ and $z := i\tilde{v} + w(m_1^Y)_{21}$, we find that $\Psi$ satisfies the Lax pair equations

$$\Psi_{\lambda} = A\Psi, \quad \Psi_{y} = U\Psi, \qquad (A.12)$$

where

$$A = \begin{pmatrix} \lambda + y & w(y) \\ 2 \frac{z(y)-i\tilde{v}}{w(y)} & -\lambda - y \end{pmatrix}, \quad U = \begin{pmatrix} \lambda & w(y) \\ 2 \frac{z(y)-i\tilde{v}}{w(y)} & -\lambda \end{pmatrix}.$$

The (12) element of the compatibility condition $A_y - U_\lambda + [A, U] = 0$ of this Lax pair implies that $w_y + 2yw = 0$. This shows that $w$ has the form

$$w(y, \tilde{v}) = \beta^Y(\tilde{v})e^{-y^2},$$

where $\beta^Y(\tilde{v})$ is a function which is independent of $y$. The (21) element of the compatibility condition then yields $z_y = 0$; hence $z = z(\tilde{v})$. Substituting these expressions for $w$ and $z$ into the first column of the Lax pair equation $\Psi_\lambda = A\Psi$ yields

$$\Psi_{11\lambda} - (\lambda + y)\Psi_{11} - \beta^Y e^{-y^2} \Psi_{21} = 0, \quad \Psi_{21\lambda} + (\lambda + y)\Psi_{21} - \frac{2e^{y^2}(z - i\tilde{v})}{\beta^Y} \Psi_{11} = 0.$$

This linear system has the general solution

$$\Psi_{11}(y, \tilde{v}, \lambda) = C_1(y, \tilde{v}) D_{i\tilde{v}-1-z} (\sqrt{2}(\lambda + y)) + C_2(y, \tilde{v}) D_{z-i\tilde{v}} (i\sqrt{2}(y + \lambda)),$$

$$\Psi_{21}(y, \tilde{v}, \lambda) = C_3(y, \tilde{v}) D_{i\tilde{v}-z} (\sqrt{2}(\lambda + y)) + C_4(y, \tilde{v}) D_{z-i\tilde{v}-1} (i\sqrt{2}(y + \lambda)),$$

where $C_j(y, \tilde{v})$, $j = 1, \dots, 4$, are locally independent of $\lambda$ (but, in general, the values of the $C_j$ change as $\lambda$ crosses one of the contours $Y_j$). We determine the $y$-dependence of the $C_j$ by substituting these expressions into the first column of the equation $\Psi_y = U\Psi$. The first column of $\Psi_y = U\Psi$ is satisfied provided that

$$C_j(y, \tilde{v}) = e^{-\frac{y^2}{2}} B_j(\tilde{v}), \quad j = 1, 2; \quad C_j(y, \tilde{v}) = e^{\frac{y^2}{2}} B_j(\tilde{v}), \quad j = 3, 4,$$
where the functions $B_j(\tilde{v})$ are independent of $y$.

Let us first suppose that $\lambda \in S_1$. Utilizing the asymptotic formula (A.11) for the range $\arg z \in [-\frac{3\pi}{4} + \delta, \frac{3\pi}{4} - \delta]$ we see that the condition

$$ \Psi_{11}e^{-\frac{\lambda^2}{2}-y\lambda}\lambda^{i\tilde{v}} = 1 + O(\lambda^{-1}), \quad \lambda \to \infty, \qquad (A.13) $$

implies that $z(\tilde{v}) = 0$ and

$$ \begin{cases} C_1 = 0, \\ C_2 = e^{-\frac{y^2}{2} 2^{\frac{i\tilde{v}}{2}}} e^{-\frac{\pi \tilde{v}}{2}}, \end{cases} \quad \text{in } S_1. $$

Similarly, the condition that

$$ \Psi_{21}e^{-\frac{\lambda^2}{2}-y\lambda}\lambda^{i\tilde{v}} = \frac{(m_1^Y)_{21}}{\lambda} + O(\lambda^{-2}), \quad \lambda \to \infty, \qquad (A.14) $$

implies that

$$ \begin{cases} C_3 = 0, \\ C_4 = e^{\frac{y^2}{2} \frac{1+i\tilde{v}}{2}} e^{-\frac{\pi \tilde{v}}{2}} \frac{\beta^Y(\tilde{v})}{\beta^Y}, \end{cases} \quad \text{in } S_1. $$

This yields the first column of the definition (A.8) of $\psi$. The second column is derived in a similar way using the second columns of the Lax pair equations (A.12).

Let us now assume that $\lambda \in S_2$. Then we use the asymptotic formula (A.11) for the range $\arg z \in [-\frac{3\pi}{4} + \delta, \frac{3\pi}{4} - \delta]$ to compute the asymptotics of $D_{i\tilde{v}-z}(\sqrt{2}(\lambda+y))$ and $D_{i\tilde{v}-1-z}(\sqrt{2}(\lambda+y))$, whereas we use (A.11) for the range $\arg z \in [\frac{\pi}{4} + \delta, \frac{5\pi}{4} - \delta]$ to compute the asymptotics of $D_{z-i\tilde{v}}(i\sqrt{2}(\lambda+y))$ and $D_{z-i\tilde{v}-1}(i\sqrt{2}(\lambda+y))$. The conditions (A.13) and (A.14) then imply

$$ C_1 = e^{-\frac{y^2}{2} \frac{\sqrt{\pi} 2^{\frac{1}{2}+\frac{i\tilde{v}}{2}} \tilde{v}}{\Gamma(i\tilde{v}+1)}}, \quad C_2 = e^{-\frac{y^2}{2} 2^{\frac{i\tilde{v}}{2}}} e^{-\frac{\pi \tilde{v}}{2}}, \quad C_3 = e^{\frac{y^2}{2} \frac{i \sqrt{\pi} 2^{1+\frac{i\tilde{v}}{2}}}{\beta^Y \Gamma(i\tilde{v})}}, \quad C_4 = e^{\frac{y^2}{2} \frac{2^{\frac{1}{2}+\frac{i\tilde{v}}{2}} e^{-\frac{\pi \tilde{v}}{2}} \tilde{v}}{\beta^Y}} $$

in $S_2$. Similar computations apply to the second column. This gives an expression for $\Psi$ in $S_2$. We then use, for example, the (21) element of the jump relation $(\Psi_-)^{-1}\Psi_+ = R_1$ valid for $\lambda \in Y_1$ to determine $\beta^Y$ in terms of $\tilde{v}$. This yields the expression (A.4) for $\beta^Y(\tilde{v})$. Finally, we can use the jump matrix $v^\Psi$ to obtain an expression for $\Psi$ also for $\lambda \in S_j, j = 3, \dots, 6$. This leads to the expression (A.9) for $m^Y$.

**Remark A.3.** The RH problem (A.3) is closely related to the RH problem associated with Painlevé IV. More precisely, consider the Painlevé IV equation

$$ u_{yy} = \frac{u_y^2}{2u} + \frac{3}{2}u^3 + 4yu^2 + (2+2y^2-4\Theta_\infty)u - \frac{8\Theta^2}{u}, \qquad (A.15) $$

where $\Theta_\infty$ and $\Theta$ are constant parameters. The jump matrix $v^{P_{IV}}(y, \tilde{v}, \lambda)$ of the RH problem associated with (A.15) has the form (see [27, p. 184]):
$$
v^{\text{PIV}} =
\begin{cases}
\begin{pmatrix} 1 & 0 \\ s_1 \lambda^{2\Theta_{\infty}} e^{-\lambda^2 - 2y\lambda} & 1 \end{pmatrix}, & \lambda \in Y_1, \\[1.5ex]
\begin{pmatrix} 1 & s_2 \lambda^{-2\Theta_{\infty}} e^{\lambda^2 + 2y\lambda} \\ 0 & 1 \end{pmatrix}, & \lambda \in Y_2, \\[1.5ex]
\begin{pmatrix} 1 & 0 \\ -s_3 \lambda^{2\Theta_{\infty}} e^{-\lambda^2 - 2y\lambda} & 1 \end{pmatrix}, & \lambda \in Y_3, \\[1.5ex]
\begin{pmatrix} 1 & -s_4(\lambda^{-2\Theta_{\infty}})_+ e^{\lambda^2 + 2y\lambda} \\ 0 & 1 \end{pmatrix}, & \lambda \in Y_4, \\[1.5ex]
\lambda^{-\Theta_{\infty}\sigma_3} E^{-1} \lambda^{-\Theta\sigma_3}, & \lambda \in Y_5, \\[1.5ex]
\lambda^{-\Theta_{\infty}\sigma_3} \begin{pmatrix} 1 & 0 \\ -s_1 e^{-\lambda^2 - 2y\lambda} & 1 \end{pmatrix} E^{-1} \lambda^{-\Theta\sigma_3}, & \lambda \in Y_6, \\[1.5ex]
\lambda^{-\Theta_{\infty}\sigma_3} \begin{pmatrix} 1 + s_1 s_2 & -s_2 e^{\lambda^2 + 2y\lambda} \\ -s_1 e^{-\lambda^2 - 2y\lambda} & 1 \end{pmatrix} E^{-1} \lambda^{-\Theta\sigma_3}, & \lambda \in Y_7, \\[1.5ex]
\lambda^{-\Theta_{\infty}\sigma_3} \begin{pmatrix} 1 + s_1 s_2 & -s_2 e^{\lambda^2 + 2y\lambda} \\ (-s_1 - (1 + s_1 s_2)s_3) e^{-\lambda^2 - 2y\lambda} & 1 + s_2 s_3 \end{pmatrix} E^{-1} \lambda^{-\Theta\sigma_3}, & \lambda \in Y_8 \cup Y_9,
\end{cases}
\quad (A.16)
$$

where the branch cut in (A.16) runs along iR− (so that λa = e a(ln|λ|+iargλ) with arg λ ∈ (−π/ 2 , 3 π/ 2] for a ∈ C ) and ( λ −2Θ ∞ ) + denotes the boundary value of λ −2Θ ∞ from the left. The complex constants { s$_{j}$ } 4 1 in (A.16) parametrize the solutions of (A.15) and must obey the relation

$$
(1 + s_2 s_3) e^{2i\pi \Theta_{\infty}} + (s_1 s_4 + (1 + s_3 s_4)(1 + s_1 s_2)) e^{-2i\pi \Theta_{\infty}} = 2 \cos(2\pi \Theta),
$$

while $E$ is a certain unimodular eigenmatrix. Letting $\Theta_{\infty} = i\tilde{\nu}$, $\Theta = 0$, and

$$
s_1 = \frac{\rho \rho^*}{1 + \rho \rho^*}, \quad s_2 = -1, \quad s_3 = -\rho \rho^*, \quad s_4 = \frac{1}{1 + \rho \rho^*}, \quad (A.17)
$$

we can take $E$ to be the identity matrix. Then, after shifting the branch cut from $i\mathbb{R}_-$ to $C$, the jump matrix $v^{\text{PIV}}$ defined in (A.16) reduces exactly to the jump matrix $v^Y$ in (A.2). Thus the RH problem (A.3) can be viewed as a special case of the RH problem associated with Painlevé IV. However, the connection with Painlevé IV only applies for $\Theta \neq n/2$, $n \in \mathbb{Z}$. In our case, this connection breaks down because $\Theta = 0$.

Data availability

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References

[1] M.J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, 1981.

[2] K. Andreiev, I. Egorova, T.L. Lange, G. Teschl, Rarefaction waves of the Korteweg–de Vries equation via nonlinear steepest descent, J. Differ. Equ. 261 (10) (2016) 5371–5410.

[3] M. Bertola, P. Giavedoni, A degeneration of two-phase solutions of the focusing nonlinear Schrödinger equation via Riemann–Hilbert problems, J. Math. Phys. 56 (2015) 061507.
[4] M. Bertola, A. Tovbis, First colonization of a spectral outpost in random matrix theory, Constr. Approx. 30 (2009) 225–263.

[5] M. Bertola, A. Tovbis, Universality in the profile of the semiclassical limit solutions to the focusing nonlinear Schrödinger equation at the first breaking curve, Int. Math. Res. Not. 2010 (2010) 2119–2167.

[6] M. Bertola, A. Tovbis, Maximal amplitudes of finite-gap solutions for the focusing nonlinear Schrödinger equation, Commun. Math. Phys. 354 (2017) 525–547.

[7] D. Bilman, R. Buckingham, D.-S. Wang, Far-field asymptotics for multiple-pole solitons in the large-order limit, J. Differ. Equ. 297 (2021) 320–369.

[8] G. Biondini, G. Kovačič, Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions, J. Math. Phys. 55 (3) (2014) 031506.

[9] G. Biondini, D. Mantzavinos, Universal nature of the nonlinear stage of modulational instability, Phys. Rev. Lett. 116 (4) (2016) 043902.

[10] G. Biondini, D. Mantzavinos, Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability, Commun. Pure Appl. Math. 70 (12) (2017) 2300–2365.

[11] G. Biondini, S. Li, D. Mantzavinos, Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions in the presence of a discrete spectrum, Commun. Math. Phys. 382 (3) (2021) 1495–1577.

[12] M. Borghese, R. Jenkins, K.D.T.-R. McLaughlin, Long time asymptotic behavior of the focusing nonlinear Schrödinger equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 35 (4) (2018) 887–920.

[13] A. Boutet de Monvel, J. Lenells, D. Shepelsky, The focusing NLS equation with step-like oscillating background: scenarios of long-time asymptotics, Commun. Math. Phys. 383 (2021) 893–952.

[14] A. Boutet de Monvel, J. Lenells, D. Shepelsky, The focusing NLS equation with step-like oscillating background: the genus 3 sector, Commun. Math. Phys. 390 (2022) 1081–1148.

[15] A. Boutet de Monvel, D. Shepelsky, Long-time asymptotics of the Camassa-Holm equation on the line, in: Integrable Systems and Random Matrices, 2008, pp. 99–116.

[16] R.J. Buckingham, P.D. Miller, Large-degree asymptotics of rational Painlevé-II functions: critical behaviour, Nonlinearity 28 (2015) 1539–1596.

[17] R. Buckingham, S. Venakides, Long-time asymptotics of the nonlinear Schrödinger equation shock problem, Commun. Pure Appl. Math. 60 (9) (2007) 1349–1414.

[18] Q. Cheng, E. Fan, Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time, J. Differ. Equ. 309 (2022) 883–948.

[19] P.-J. Cheng, S. Venakides, X. Zhou, Long-time asymptotics for the pure radiation solution of the sine-Gordon equation, Commun. Partial Differ. Equ. 24 (7–8) (1999) 1195–1262.

[20] T. Claeys, T. Grava, Solitonic asymptotics for the Korteweg–de Vries equation in the small dispersion limit, SIAM J. Math. Anal. 42 (2010) 2132–2154.

[21] P. Deift, S. Venakides, X. Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Commun. Pure Appl. Math. 47 (1994) 199–206.

[22] P. Deift, S. Venakides, X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Not. 6 (1997) 286–299.

[23] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math. (2) 137 (2) (1993) 295–368.

[24] F. Demontis, B. Prinari, C. van der Mee, F. Vitale, The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions, J. Math. Phys. 55 (2014) 101505.

[25] I. Egorova, J. Michor, A. Pryimak, G. Teschl, Long-time asymptotics for Toda shock waves in the modulation region, J. Math. Phys. Anal. Geom. 19 (2) (2023) 396–442.

[26] I. Egorova, M. Piorkowski, G. Teschl, Asymptotics of KdV shock waves via the Riemann-Hilbert approach, Indiana Univ. Math. J. 73 (2) (2024) 645–690.

[27] A.S. Fokas, A.R. Its, A.A. Kapaev, V.Yu. Novokshenov, Painlevé Transcendents, Mathematical Surveys and Monographs, vol. 128, American Mathematical Society, Providence, RI, 2006. The Riemann-Hilbert approach.

[28] S. Fromm, J. Lenells, R. Quirchmayr, The defocusing nonlinear Schrödinger equation with steplike oscillatory initial data, Adv. Differ. Equ. 30 (2025) 455–525.

[29] A.A. Himonas, D. Mantzavinos, The nonlinear Schrödinger equation on the half-line with a Robin boundary condition, Anal. Math. Phys. 11 (4) (2021) 157.

[30] L. Huang, J. Xu, E. Fan, Long-time asymptotic for the Hirota equation via nonlinear steepest descent method, Nonlinear Anal., Real World Appl. 26 (2015) 229–262.
[31] R. Jenkins, Regularization of a sharp shock by the defocusing nonlinear Schrödinger equation, Nonlinearity 28 (7) (2015) 2131–2180.

[32] S. Kamvissis, Long time behavior for the focusing nonlinear Schroedinger equation with real spectral singularities, Commun. Math. Phys. 180 (2) (1996) 325–341.

[33] S. Kamvissis, K.D.T-R. McLaughlin, P.D. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrödinger Equation, Annals of Mathematics Studies, vol. 169, Princeton University Press, Cambridge, 2003.

[34] J. Lenells, The nonlinear steepest descent method for Riemann–Hilbert problems of low regularity, Indiana Math. J. 66 (4) (2017) 1287–1332.

[35] J. Lenells, Matrix Riemann–Hilbert problems with jumps across Carleson contours, Monatshefte Math. 186 (1) (2018) 111–152.

[36] N. Liu, B. Guo, Long-time asymptotics for the Sasa-Satsuma equation via nonlinear steepest descent method, J. Math. Phys. 60 (1) (2019) 011504.

[37] B.-Y. Lu, P. Miller, Universality near the gradient catastrophe point in the semiclassical sine-Gordon equation, Commun. Pure Appl. Math. 75 (7) (2022) 1517–1641.

[38] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, 2010.

[39] A. Tovbis, S. Venakides, Nonlinear steepest descent asymptotics for semiclassical limit of integrable systems: continuation in the parameter space, Commun. Math. Phys. 295 (1) (2010) 139–160.

[40] Y.-F. Wang, N. Liu, B.-L. Guo, Long-time asymptotic behavior for a fourth-order dispersive nonlinear Schrödinger equation, J. Math. Anal. Appl. 506 (1) (2022) 125560.

[41] K. Xu, Y. Yang, E. Fan, The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: long-time and Painlevé asymptotics, J. Differ. Equ. 380 (2024) 24–91.