Boussinesq's equation for water waves: the soliton resolution conjecture for Sector IV

TL;DR

This paper derives an asymptotic formula for the Boussinesq equation in Sector IV with solitons, providing insights into soliton-radiation interactions and supporting the soliton resolution conjecture for water waves.

Contribution

It extends previous work by deriving the leading asymptotic behavior in Sector IV with solitons, confirming the soliton resolution conjecture for this case.

Findings

Exact expression for soliton-radiation interaction

Verification of the soliton resolution conjecture in Sector IV

Asymptotic behavior characterized by specific x/t ratio

Abstract

We consider the Boussinesq equation on the line for a broad class of Schwartz initial data relevant for water waves. In a recent work, we identified ten main sectors describing the asymptotic behavior of the solution, and for each of these sectors we gave an exact expression for the leading asymptotic term in the case when no solitons are present. In this paper, we derive an asymptotic formula in Sector IV, characterized by $\frac{x}{t}\in (\frac{1}{\sqrt{3}},1)$, in the case when solitons are present. In particular, our results provide an exact expression for the soliton-radiation interaction to leading order and a verification of the soliton resolution conjecture for the Boussinesq equation in Sector IV.

Equations

$$u_{tt}=u_{xx}+(u^2{)}_{xx}+u_{xxxx},$$

$$u(x,0)=u_0(x),u_t(x,0)=u_1(x),$$

$$l_j(k)=i\frac{{\omega }^{j}k+({\omega }^{j}k{)}^{-1}}{2\sqrt{3}},z_j(k)=i\frac{({\omega }^{j}k{)}^2+({\omega }^{j}k{)}^{-2}}{4\sqrt{3}},k\in \mathbb{C}\setminus \{0\}.$$

$$P(k)=\left(\begin{array}{ccc}1& 1& 1\\ l_1(k)& l_2(k)& l_3(k)\\ l_1(k{)}^2& l_2(k{)}^2& l_3(k{)}^2\end{array}\right),\mathsf{U}(x,k)=P(k{)}^{-1}\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ -\frac{u_{0x}}{4}-\frac{i{v}_0}{4\sqrt{3}}& -\frac{u_0}{2}& 0\end{array}\right)P(k),$$

$$X(x,k)=I-{\int }_x^{\infty }{e}^{(x-x^{\prime })\widehat{\mathcal{L}(k)}}(\mathsf{U}X)(x^{\prime },k)d{x}^{\prime },$$

$$X^A(x,k)=I+{\int }_x^{\infty }{e}^{-(x-x^{\prime })\widehat{\mathcal{L}(k)}}({\mathsf{U}}^T{X}^A)(x^{\prime },k)d{x}^{\prime },$$

$$Y(x,k)=I+{\int }_{-\infty }^x{e}^{(x-x^{\prime })\widehat{\mathcal{L}(k)}}(\mathsf{U}Y)(x^{\prime },k)d{x}^{\prime },$$

$$Y^A(x,k)=I-{\int }_{-\infty }^x{e}^{-(x-x^{\prime })\widehat{\mathcal{L}(k)}}({\mathsf{U}}^T{Y}^A)(x^{\prime },k)d{x}^{\prime },$$

$$s(k)=I-{\int }_{\mathbb{R}}{e}^{-x\widehat{\mathcal{L}(k)}}(\mathsf{U}X)(x,k)dx,$$

$$\{\begin{array}{ll}{r}_1(k)=\frac{(s(k){)}_{12}}{(s(k){)}_{11}},& k\in {\hat{\Gamma }}_1\setminus \hat{\mathcal{Q}},\\ r_2(k)=\frac{(s^A(k){)}_{12}}{(s^A(k){)}_{11}},& k\in {\hat{\Gamma }}_4\setminus \hat{\mathcal{Q}}.\end{array}$$

$$\displaystyle D_{\mathrm{reg}}:=D_{2}\cap\big{(}\{k\,|\,|k|>1,\text{Im\,}k>0\}\cup\{k\,|\,|k|<1,\text{Im\,}k<0\}\big{)},$$

$$\displaystyle D_{\mathrm{sing}}:=D_{2}\cap\big{(}\{k\,|\,|k|>1,\text{Im\,}k<0\}\cup\{k\,|\,|k|<1,\text{Im\,}k>0\}\big{)},$$

$$c_{k_0}:=\{\begin{array}{ll}-\frac{s_{13}(k_0)}{{\dot{s}}_{11}(k_0)},& k_0\in \mathsf{Z}\setminus \mathbb{R},\\ -\frac{s_{12}(k_0)}{{\dot{s}}_{11}(k_0)},& k_0\in \mathsf{Z}\cap \mathbb{R}.\end{array}$$

$$w=\left(\begin{array}{c}{Y}_{21}^A{X}_{32}^A-Y_{31}^A{X}_{22}^A\\ Y_{31}^A{X}_{12}^A-Y_{11}^A{X}_{32}^A\\ Y_{11}^A{X}_{22}^A-Y_{21}^A{X}_{12}^A\end{array}\right).$$

$$\frac{w(x,k_0)}{{\dot{s}}_{11}(k_0)}=c_{k_0}{e}^{(l_1(k_0)-l_3(k_0))x}[X(x,k_0){]}_1\text{for all x∈R}.$$

$$\frac{[Y(x,k_0){]}_2}{{\dot{s}}_{22}^A(k_0)}=c_{k_0}{e}^{(l_1(k_0)-l_2(k_0))x}[X(x,k_0){]}_1\text{for all x∈R}.$$

$$\underset{k\to k_{\star }}{\lim }(k-k_{\star })s(k{)}_{11}\ne 0,$$

$$\underset{k\to k_{\star }}{\lim }(k-k_{\star })s^A(k{)}_{11}\ne 0,$$

$${\Phi }_{ij}(\zeta ,k)=(l_i(k)-l_j(k))\zeta +(z_i(k)-z_j(k)),$$

$$\displaystyle k_{2}=\frac{1}{4}\bigg{(}\zeta-\sqrt{8+\zeta^{2}}-i\sqrt{2}\sqrt{4-\zeta^{2}+\zeta\sqrt{8+\zeta^{2}}}\bigg{)},$$

$$\displaystyle k_{4}=\frac{1}{4}\bigg{(}\zeta+\sqrt{8+\zeta^{2}}-i\sqrt{2}\sqrt{4-\zeta^{2}-\zeta\sqrt{8+\zeta^{2}}}\bigg{)},$$

$$\displaystyle\delta_{1}(\zeta,k)=\exp\bigg{\{}\frac{-1}{2\pi i}\int_{i}^{\omega k_{4}}\frac{\ln(1+r_{1}(s)r_{2}(s))}{s-k}ds\bigg{\}},\;\delta_{4}(\zeta,k)=\exp\bigg{\{}\frac{1}{2\pi i}\int_{\omega^{2}k_{2}}^{\omega}\frac{\ln f(s)}{s-k}ds\bigg{\}},$$

$$\displaystyle\delta_{2}(\zeta,k)=\exp\bigg{\{}\frac{1}{2\pi i}\int_{\omega k_{4}}^{\omega^{2}k_{2}}\frac{\ln(1+r_{1}(s)r_{2}(s))}{s-k}ds\bigg{\}},\;\delta_{5}(\zeta,k)=\exp\bigg{\{}\frac{1}{2\pi i}\int_{\omega^{2}k_{2}}^{\omega}\frac{\ln f(\omega^{2}s)}{s-k}ds\bigg{\}},$$

$$\displaystyle\delta_{3}(\zeta,k)=\exp\bigg{\{}\frac{1}{2\pi i}\int_{\omega k_{4}}^{\omega^{2}k_{2}}\frac{\ln f(s)}{s-k}ds\bigg{\}},$$

$$f(k):=1+r_1(k)r_2(k)+r_1(\frac{1}{{\omega }^{2}k})r_2(\frac{1}{{\omega }^{2}k}),k\in \partial \mathbb{D}.$$

$$\mathcal{P}(k)=\prod _{k_0\in \mathsf{Z}\cap D_{\mathrm{reg}}^R}\frac{(k-k_0)(k-k_0^{-1})(k-{\omega }^2{\bar{k}}_0)(k-\omega {\bar{k}}_0^{-1})}{(k-{\bar{k}}_0)(k-{\bar{k}}_0^{-1})(k-\omega k_0)(k-{\omega }^2{k}_0^{-1})}\cdot \prod _{k_0\in \mathsf{Z}\cap (1,\infty )}\frac{(k-{\omega }^2{k}_0)(k-\omega k_0^{-1})}{(k-\omega k_0)(k-{\omega }^2{k}_0^{-1})},$$

$${\mathcal{D}}_1(k)=\frac{{\delta }_1(\omega k){\delta }_1(\frac{1}{{\omega }^{2}k}{)}^2}{{\delta }_1({\omega }^{2}k{)}^2{\delta }_1(\frac{1}{\omega k}){\delta }_1(\frac{1}{k})}\frac{{\delta }_2({\omega }^{2}k){\delta }_2(\frac{1}{k}{)}^2}{{\delta }_2(\omega k{)}^2{\delta }_2(\frac{1}{{\omega }^{2}k}){\delta }_2(\frac{1}{\omega k})}\frac{{\delta }_3(\omega k){\delta }_3({\omega }^{2}k){\delta }_3(\frac{1}{\omega k}{)}^2}{{\delta }_3(\frac{1}{k}){\delta }_3(\frac{1}{{\omega }^{2}k})}$$

$$\times \frac{{\delta }_4({\omega }^{2}k{)}^2{\delta }_4(\frac{1}{k}){\delta }_4(\frac{1}{\omega k})}{{\delta }_4(k){\delta }_4(\omega k){\delta }_4(\frac{1}{{\omega }^{2}k}{)}^2}\frac{{\delta }_5(\omega k{)}^2{\delta }_5(\frac{1}{\omega k}){\delta }_5(\frac{1}{{\omega }^{2}k})}{{\delta }_5(k){\delta }_5(\frac{1}{k}{)}^2{\delta }_5({\omega }^{2}k)},$$

$${\mathcal{D}}_2(k)=\frac{{\delta }_1(\omega k{)}^2{\delta }_1(\frac{1}{{\omega }^{2}k}){\delta }_1(\frac{1}{k})}{{\delta }_1({\omega }^{2}k){\delta }_1(\frac{1}{\omega k}{)}^2{\delta }_1(k)}\frac{{\delta }_2(\frac{1}{k}){\delta }_2(\frac{1}{\omega k})}{{\delta }_2(\omega k){\delta }_2(\frac{1}{{\omega }^{2}k}{)}^2{\delta }_2({\omega }^{2}k)}\frac{{\delta }_3({\omega }^{2}k{)}^2{\delta }_3(\frac{1}{\omega k}){\delta }_3(\frac{1}{{\omega }^{2}k})}{{\delta }_3(\frac{1}{k}{)}^2{\delta }_3(\omega k)}$$

$$\times \frac{{\delta }_4({\omega }^{2}k){\delta }_4(\frac{1}{\omega k}{)}^2}{{\delta }_4(\omega k{)}^2{\delta }_4(\frac{1}{{\omega }^{2}k}){\delta }_4(\frac{1}{k})}\frac{{\delta }_5(\omega k){\delta }_5(\frac{1}{{\omega }^{2}k}{)}^2{\delta }_5({\omega }^{2}k)}{{\delta }_5(\frac{1}{k}){\delta }_5(\frac{1}{\omega k})}.$$

$${\chi }_1(\zeta ,k)=$$

$${\chi }_2(\zeta ,k)=$$

$${\chi }_3(\zeta ,k)=$$

$${\tilde{\chi }}_2(\zeta ,k)=$$

$${\tilde{\chi }}_3(\zeta ,k)=$$

$${\tilde{\chi }}_4(\zeta ,k)=$$

$$:=$$

$${\tilde{\chi }}_5(\zeta ,k)=$$

$$:=$$

$${\nu }_1(k)=-\frac{1}{2\pi }\ln (1+r_1(\omega k)r_2(\omega k)),{\nu }_2(k)=-\frac{1}{2\pi }\ln (1+r_1({\omega }^{2}k)r_2({\omega }^{2}k)),$$

$${\nu }_3(k)=-\frac{1}{2\pi }\ln (f(\omega k)),{\nu }_4(k)=-\frac{1}{2\pi }\ln (f({\omega }^{2}k)),$$

Full text

AMS Subject Classification (2020): 35C08, 35G25, 35Q15, 37K40, 76B15.

Keywords: Boussinesq equation, long-time asymptotics, Riemann-Hilbert problem.

1. Introduction

The Boussinesq equation [3]

[TABLE]

where $u(x,t)$ is a real-valued function and subscripts denote partial derivatives, models small-amplitude dispersive waves in shallow water propagating in both the right and left directions, see e.g. [12]. Equation (1.1) supports solitons [10, 1, 2] and admits a Lax pair [16]. Until recently [4], the determination of the long-time asymptotics for the solution of (1.1) was still an open problem, see e.g. Deift’s list of open problems in [8].

In [4], we studied the direct and inverse scattering problems for (1.1). Among other things, we established that for a wide class of initial data relevant for water waves, the solution is unique and exists globally. We identified ten main asymptotic sectors and for each of these sectors, we computed the leading asymptotics of the solution. The proofs of some of these asymptotic formulas were omitted in [4] for conciseness. Moreover, only solitonless solutions were considered.

The purpose of this paper is to derive an asymptotic formula for the solution in the sector $\frac{x}{t}\in(\frac{1}{\sqrt{3}},1)$ in the case when solitons are present. This sector was referred to as Sector IV in [4]. In the special case when solitons are absent, our formula reduces to the formula announced in [4], thus providing a proof of this formula. In the case when solitons are present, our formula quantifies the asymptotic effect of solitons on the solution in Sector IV.

The soliton resolution conjecture for an integrable equation expresses the expectation that a solution of the equation with generic initial data eventually evolves into a set of solitons superimposed on a dispersive radiative background. Since $x/t<1$ in Sector IV and all one-soliton solutions of (1.1) travel with speeds greater than $1$ [10], the soliton resolution conjecture for (1.1) suggests that there should be no order $1$ contributions to the asymptotics in Sector IV stemming from solitons. This is indeed what we find: our formula shows that the solution of (1.1) in Sector IV is $O(t^{-1/2})$ as $t\to\infty$. Our results therefore verify the soliton resolution conjecture for the Boussinesq equation in this sector. We stress that although the solitons themselves are not directly observable in Sector IV for large times, they do have a nontrivial effect on the radiation in Sector IV; we compute the leading order contribution of this effect exactly.

It is important to note that we restrict ourselves to initial data which contain no high-frequency modes (in a sense made precise in Assumption $(\ref{nounstablemodesassumption})$ below). This assumption is consistent with the derivation of the Boussinesq equation as a model for water waves, which assumes that high-frequency Fourier modes are absent, or at least highly suppressed. In fact, if high-frequency modes are present, the solution will contain components that grow exponentially in time and the soliton resolution conjecture will automatically fail.

Our main result is presented in Section 2. Its proof relies on a Deift–Zhou [9] steepest descent analysis of a row-vector Riemann–Hilbert (RH) problem, and this RH problem is stated in Section 3. Section 4 contains a brief overview of the proof. In Sections 5 and 6, we carry out two transformations of the RH problem. After introducing a global parametrix in Section 7, we perform two further transformations in Sections 8 and 9. Using two local parametrices constructed in Sections 10 and 11, respectively, we arrive at a small-norm RH problem whose large $t$ behavior is analyzed in Section 12. Finally, the long-time asymptotics of the solution $u(x,t)$ of (1.1) is obtained in Section 13.

1.1. Notation

We use the following notation throughout the paper.

1. $-$

$C>0$ and $c>0$ denote generic constants that may change within a computation. 2. $-$

$[A]_{1}$, $[A]_{2}$, and $[A]_{3}$ denote the first, second, and third columns of a $3\times 3$ matrix $A$. 3. $-$

If $A$ is an $n\times m$ matrix, we define $|A|\geq 0$ by $|A|^{2}=\Sigma_{i,j}|A_{ij}|^{2}$. For a piecewise smooth contour $\gamma\subset{\mathbb{C}}$ and $1\leq p\leq\infty$, we define $\|A\|_{L^{p}(\gamma)}:=\||A|\|_{L^{p}(\gamma)}$. 4. $-$

${\mathbb{D}}=\{k\in{\mathbb{C}}\,|\,|k|<1\}$ denotes the open unit disk and $\partial{\mathbb{D}}=\{k\in{\mathbb{C}}\,|\,|k|=1\}$ denotes the unit circle. 5. $-$

$\mathcal{S}({\mathbb{R}})$ denotes the Schwartz space of rapidly decreasing functions on ${\mathbb{R}}$. 6. $-$

$\mathcal{Q}:=\{\kappa_{j}\}_{j=1}^{6}$ and $\hat{\mathcal{Q}}:=\mathcal{Q}\cup\{0\}$, where $\{\kappa_{j}=e^{\frac{\pi i(j-1)}{3}}\}_{1}^{6}$ are the sixth roots of unity. 7. $-$

$\{D_{n}\}_{n=1}^{6}$ denote the open subsets of the complex plane shown in Figure 2. 8. $-$

$\Gamma=\cup_{j=1}^{9}\Gamma_{j}$ denotes the contour shown and oriented as in Figure 2. 9. $-$

$\hat{\Gamma}_{j}=\Gamma_{j}\cup\partial{\mathbb{D}}$ denotes the union of $\Gamma_{j}$ and the unit circle.

2. Main results

We consider the initial value problem for (1.1) with initial data

[TABLE]

where $u_{0},u_{1}\in\mathcal{S}(\mathbb{R})$ are real-valued functions in the Schwartz class. We assume that $u_{1}$ obeys the condition $\int_{\mathbb{R}}u_{1}(x)dx=0$; this assumption is natural because it ensures that the total mass $\int_{\mathbb{R}}u(x,t)dx$ is conserved in time. A result of [4] states that the solution to the initial value problem (1.1)–(2.1) can be expressed in terms of the solution $n$ of a row-vector RH problem. The jump matrix of this RH problem is expressed in terms of two scalar reflection coefficients, $r_{1}(k)$ and $r_{2}(k)$, which are defined as follows.

2.1. Definition of $r_{1}$ and $r_{2}$.

Let $\omega:=e^{\frac{2\pi i}{3}}$ and define $\{l_{j}(k),z_{j}(k)\}_{j=1}^{3}$ by

[TABLE]

Let $P(k)$ and $\mathsf{U}(x,k)$ be given by

[TABLE]

where $v_{0}(x)=\int_{-\infty}^{x}u_{1}(x^{\prime})dx^{\prime}$. Let $X,X^{A},Y,Y^{A}$ be the unique solutions of the Volterra integral equations

[TABLE]

where $\mathcal{L}=\text{diag\,}(l_{1},l_{2},l_{3})$, $(\cdot)^{T}$ denotes the transpose operation, and $e^{\hat{\mathcal{L}}}$ acts on a $3\times 3$ matrix $M$ by $e^{\hat{\mathcal{L}}}M=e^{\mathcal{L}}Me^{-\mathcal{L}}$. Define $s(k)$ and $s^{A}(k)$ by

[TABLE]

The two spectral functions $\{r_{j}(k)\}_{1}^{2}$ are defined by

[TABLE]

2.2. Solitons

In [6], we extended the approach of [4] to the case when solitons are present. In this case, in addition to $r_{1}(k)$ and $r_{2}(k)$, the formulation of the RH problem also involves a set of poles $\mathsf{Z}$ and a set of corresponding residue constants $\{c_{k_{0}}\}_{k_{0}\in\mathsf{Z}}\subset{\mathbb{C}}$. In what follows, we describe how $\mathsf{Z}$ and $\{c_{k_{0}}\}_{k_{0}\in\mathsf{Z}}\subset{\mathbb{C}}$ are defined.

Solitons are generated by the possible zeros of the functions $s_{11}(k)$ and $s^{A}_{11}(k)$. As shown in [4], $s_{11}$ and $s^{A}_{11}$ have analytic extensions to the interiors of $\bar{D}_{1}\cup\bar{D}_{2}$ and $\bar{D}_{4}\cup\bar{D}_{5}$, respectively. We will restrict ourselves to the generic case when $s_{11}$ and $s^{A}_{11}(k)$ have no zeros on the contour $\Gamma$. The symmetries $s_{11}(k)=s_{11}(\omega/k)$ and $s_{11}^{A}(k)=\overline{s_{11}(\bar{k}^{-1})}$ (also established in [4]) then imply that it is enough to consider the zeros of $s_{11}$ in $D_{2}$. We decompose the open set $D_{2}$ into three parts: $D_{2}=D_{\mathrm{reg}}\sqcup D_{\mathrm{sing}}\sqcup(D_{2}\cap{\mathbb{R}})$, where

[TABLE]

and $D_{2}\cap{\mathbb{R}}=(-1,0)\cup(1,\infty)$. We denote the right and left parts of $D_{\mathrm{reg}}$ by $D_{\mathrm{reg}}^{R}$ and $D_{\mathrm{reg}}^{L}$, respectively, and similarly for $D_{\mathrm{sing}}$, see Figure 1.

We showed in [6] that zeros in $D_{\mathrm{sing}}$ give rise to solitons with singularities. We will therefore assume that the zero-set $\mathsf{Z}$ of $s_{11}(k)$ is contained in $D_{\mathrm{reg}}\cup(-1,0)\cup(1,\infty)$. We will consider the generic case of a finite number of simple zeros. In the case when the initial data $u_{0}$ and $u_{1}$ have compact support, the associated residue constants are then defined by

[TABLE]

If $u_{0}$ and $v_{0}$ do not have compact support, then the expressions in (2.4) are in general not well-defined and the following more involved definition is required: Define the vector-valued function $w(x,k)$ by

[TABLE]

If $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$, then $c_{k_{0}}$ is defined as the unique complex constant such that

[TABLE]

It turns out that simple zeros of $s_{11}(k)$ in $D_{\mathrm{reg}}^{R}$ and $D_{\mathrm{reg}}^{L}$ generate right- and left-moving breather solitons, respectively. Similarly, simple zeros of $s_{11}(k)$ in $(1,\infty)$ and $(-1,0)$ generate right- and left-moving solitons, respectively, and it was shown in [6] that a soliton generated by a zero in $(-1,0)\cup(1,\infty)$ is non-singular if and only if the associated residue constant $c_{k_{0}}$ satisfies $i(\omega^{2}k_{0}^{2}-\omega)c_{k_{0}}\notin(-\infty,0)$. This motivates the following assumptions on $\mathsf{Z}$ and $\{c_{k_{0}}\}_{k_{0}\in\mathsf{Z}}$.

2.3. Assumptions

As in [6, Theorem 2.11], our results will be valid under the following assumptions:

1. $(i)$

Finite number of non-singular solitons: suppose that $s_{11}(k)$ has a simple zero at each point in $\mathsf{Z}$, where $\mathsf{Z}\subset D_{\mathrm{reg}}\cup(-1,0)\cup(1,\infty)$ is a set of finite cardinality, and that $s_{11}(k)$ has no other zeros in $k\in(\bar{D}_{2}\cup\partial{\mathbb{D}})\setminus\hat{\mathcal{Q}}$. If $k_{0}\in\mathsf{Z}\setminus{\mathbb{R}}$, then we also assume that $s_{22}^{A}(k_{0})\neq 0$. If $k_{0}\in\mathsf{Z}\cap{\mathbb{R}}$, then we also assume that $i(\omega^{2}k_{0}^{2}-\omega)c_{k_{0}}\notin(-\infty,0)$. 2. $(ii)$

The spectral functions $s$ and $s^{A}$ have generic behavior near $k=1$ and $k=-1$: we suppose for $k_{\star}=1$ and $k_{\star}=-1$ that

[TABLE] 3. $(iii)$

No high-frequency modes: we suppose that $r_{1}(k)=0$ for all $k\in[0,i]$, where $[0,i]$ is the vertical segment from [math] to $i$.

We emphasize that Assumption $(\ref{originassumption})$ is generic. Assumption $(\ref{nounstablemodesassumption})$ ensures that the solution belongs to the physically relevant class of solutions which do not contain exponentially growing high-frequency modes. If Assumptions $(\ref{solitonassumption})$–$(\ref{nounstablemodesassumption})$ hold true, then the solution $u(x,t)$ of the initial value problem (1.1)–(2.1) exists globally [6].

2.4. Statement of main result

The formulation of our main theorem involves a number of quantities that we now introduce. Let $\zeta:=x/t$ and assume that $\zeta\in(\frac{1}{\sqrt{3}},1)$. For $k\in\mathbb{C}\setminus\{0\}$ and $1\leq j<i\leq 3$, define $\Phi_{ij}(\zeta,k)$ by

[TABLE]

where $l_{j}(k)$ and $z_{j}(k)$ are as in (2.2). The function $k\mapsto\Phi_{21}(\zeta,k)$ has four saddle points $\{k_{j}=k_{j}(\zeta)\}_{j=1}^{4}$ given by

[TABLE]

$k_{1}=\bar{k}_{2}$, and $k_{3}=\bar{k}_{4}$. Note that $|k_{2}|=|k_{4}|=1$, $\arg k_{2}\in(-\frac{3\pi}{4},-\frac{2\pi}{3})$, and $\arg k_{4}\in(-\frac{\pi}{6},0)$. Define also $\delta_{j}(\zeta,k)$, $j=1,\dots,5$, by

[TABLE]

where the paths follow the unit circle in the counterclockwise direction, the principal branch is used for the logarithms, and

[TABLE]

By [4, Theorem 2.3 and Lemma 2.13], $r_{1},r_{2},f$ are well-defined on $\partial{\mathbb{D}}\setminus\mathcal{Q}$ and the arguments of all logarithms appearing in the above definitions of $\{\delta_{j}(\zeta,k)\}_{j=1}^{5}$ are $>0$. Define

[TABLE]

Define $\chi_{j}(\zeta,k)$, $j=1,2,3$, and $\tilde{\chi}_{j}(\zeta,k)$, $j=2,3,4,5$, by

[TABLE]

where the paths follow $\partial\mathbb{D}$ in the counterclockwise direction. For $s\in\{e^{i\theta}\,|\,\theta\in[\frac{\pi}{2},\frac{2\pi}{3}]\}$, $k\mapsto\ln_{s}(k-s)=\ln|k-s|+i\arg_{s}(k-s)$ has a cut along $\{e^{i\theta}\,|\,\theta\in[\frac{\pi}{2},\arg s]\}\cup(i,i\infty)$ and the branch is such that $\arg_{s}(1)=2\pi$. Also, for $s\in\{e^{i\theta}\,|\,\theta\in[\frac{\pi}{2},\frac{2\pi}{3}]\}$, $k\mapsto\tilde{\ln}_{s}(k-s):=\ln|k-s|+i\tilde{\arg}_{s}(k-s)$ has a cut along $\{e^{i\theta}\,|\,\theta\in[\arg s,\pi]\}\cup(-\infty,0)$, and satisfies $\tilde{\arg}_{s}(1)=0$. Regularized integrals are needed in the definitions of $\tilde{\chi}_{4},\tilde{\chi}_{5}$ because $f(1)=f(\omega)=0$ (see [4, Lemma 2.13 $(i)$]). Define $\{\nu_{j}\}_{j=1}^{4}$, $\hat{\nu}_{1}$, and $\hat{\nu}_{2}$ by

[TABLE]

The following functions $d_{1,0}$ and $d_{2,0}$ appear in our final result:

[TABLE]

where $z_{1,\star}$, $z_{2,\star}$ are given by

[TABLE]

and the branches for the complex powers $z_{j,\star}^{\alpha}=e^{\alpha\ln z_{j,\star}}$ are fixed by

[TABLE]

with $\arg(\omega k_{4})\in(\tfrac{\pi}{2},\tfrac{2\pi}{3})$ and $\arg(\omega^{2}k_{2})\in(\tfrac{\pi}{2},\tfrac{2\pi}{3})$. Define also

[TABLE]

where

[TABLE]

We now state our main result. The Gamma function $\Gamma(k)$ appears in the statement, as well as the square roots of $\hat{\nu}_{1}(k_{4})$ and $\hat{\nu}_{2}(k_{2})$. These square roots are well-defined and $\geq 0$ thanks to the inequalities $\hat{\nu}_{1}(k_{4})\geq 0$ and $\hat{\nu}_{2}(k_{2})\geq 0$, which follow from [4, Lemma 2.13] and the fact that $\arg k_{2}\in(-\frac{3\pi}{4},-\frac{2\pi}{3})$ and $\arg k_{4}\in(-\frac{\pi}{6},0)$.

Theorem 2.1** (Asymptotics in Sector IV).**

Let $u_{0},u_{1}\in\mathcal{S}({\mathbb{R}})$ be real-valued and such that $\int_{\mathbb{R}}u_{1}dx=0$. Let $v_{0}(x)=\int_{-\infty}^{x}u_{1}(x^{\prime})dx^{\prime}$ and suppose Assumptions $(\ref{solitonassumption})$–$(\ref{nounstablemodesassumption})$ are fulfilled. Let $\mathcal{I}$ be a fixed compact subset of $(\smash{\frac{1}{\sqrt{3}}},1)$. Then the global solution $u(x,t)$ of the initial value problem for (1.1) with initial data $u_{0},u_{1}$ enjoys the following asymptotics as $t\to\infty$:

[TABLE]

uniformly for $\zeta:=\frac{x}{t}\in\mathcal{I}$, where

[TABLE]

The effect of the solitons on the asymptotic behavior (2.16) of $u(x,t)$ is encoded in the phase shifts $\arg\tfrac{\mathcal{P}(\omega k_{4})}{\mathcal{P}(\omega^{2}k_{4})}$ and $\arg\tfrac{\mathcal{P}(\omega^{2}k_{2})}{\mathcal{P}(\omega k_{2})}$ that appear in the definitions of $\alpha_{1}$ and $\alpha_{2}$, respectively. If there are no solitons (i.e., if the set $\mathsf{Z}$ is empty), then these phase shifts are absent, and the formula (2.16) reduces to the formula for Sector IV given in [4]. Indeed, the only place where the set $\mathsf{Z}$ appears in the above definitions is in the definition (2.11) of $\mathcal{P}(k)$, and if $\mathsf{Z}$ is empty, then $\mathcal{P}(k)$ is identically equal to $1$. Thus, these phase shifts describe the leading order soliton-radiation interaction in Sector IV. We observe that the products in the definition of $\mathcal{P}$ only run over the zeros in $\mathsf{Z}$ with positive real part; this corresponds to the fact that only right-moving solitons generate a phase shift in Sector IV.

3. The RH problem for $n$

We will prove Theorem 2.1 by analyzing a row-vector RH problem, whose solution is denoted by $n$. In what follows, we recall this RH problem from [6]. Let $\theta_{ij}(x,t,k)=t\,\Phi_{ij}(\zeta,k)$. For $j=1,\dots,6$, we write $\Gamma_{j}=\Gamma_{j^{\prime}}\cup\Gamma_{j^{\prime\prime}}$, where $\Gamma_{j^{\prime}}=\Gamma_{j}\setminus{\mathbb{D}}$ and $\Gamma_{j^{\prime\prime}}:=\Gamma_{j}\setminus\Gamma_{j^{\prime}}$ with $\Gamma_{j}$ as in Figure 2. The jump matrix $v(x,t,k)$ is defined for $k\in\Gamma$ by

[TABLE]

where $v_{j},v_{j^{\prime}},v_{j^{\prime\prime}}$ are the restrictions of $v$ to $\Gamma_{j}$, $\Gamma_{j^{\prime}}$, and $\Gamma_{j^{\prime\prime}}$, respectively. Let $\Gamma_{\star}=\{i\kappa_{j}\}_{j=1}^{6}\cup\{0\}$ be the set of intersection points of $\Gamma$. Let $\mathsf{Z}$ be as in Assumption $(\ref{solitonassumption})$ and let

[TABLE]

with $\mathsf{Z}^{-1}=\{k_{0}^{-1}\,|\,k_{0}\in\mathsf{Z}\}$, $\mathsf{Z}^{*}=\{\bar{k}_{0}\,|\,k_{0}\in\mathsf{Z}\}$, and $\mathsf{Z}^{-*}=\{\bar{k}_{0}^{-1}\,|\,k_{0}\in\mathsf{Z}\}$.

RH problem 3.1** (RH problem for $n$).**

Find a $1\times 3$-row-vector valued function $n(x,t,k)$ with the following properties:

1. $(a)$

$n(x,t,\cdot):{\mathbb{C}}\setminus(\Gamma\cup\hat{\mathsf{Z}})\to\mathbb{C}^{1\times 3}$* is analytic.* 2. $(b)$

The limits of $n(x,t,k)$ as $k$ approaches $\Gamma\setminus\Gamma_{\star}$ from the left and right exist, are continuous on $\Gamma\setminus\Gamma_{\star}$, and are denoted by $n_{+}$ and $n_{-}$, respectively. Furthermore, they are related by

[TABLE] 3. $(c)$

$n(x,t,k)=O(1)$* as $k\to k_{\star}\in\Gamma_{\star}$.* 4. $(d)$

For $k\in{\mathbb{C}}\setminus(\Gamma\cup\hat{\mathsf{Z}})$, $n$ obeys the symmetries

[TABLE]

where $\mathcal{A}$ and $\mathcal{B}$ are the matrices defined by

[TABLE] 5. $(e)$

$n(x,t,k)=(1,1,1)+O(k^{-1})$* as $k\to\infty$.* 6. $(f)$

At each point of $\hat{\mathsf{Z}}$, one entry of $n$ has (at most) a simple pole while two entries are analytic. The following residue conditions hold at the points in $\hat{\mathsf{Z}}$: for each $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$,

[TABLE]

and, for each $k_{0}\in\mathsf{Z}\cap\mathbb{R}$,

[TABLE]

where the $(x,t)$-dependence of $n$ and of $\theta_{ij}$ has been omitted for conciseness and the complex constants $d_{k_{0}}$ are defined by

[TABLE]

If $k_{0}\in\mathsf{Z}$ is such that $c_{k_{0}}=0$, then the pole of $n$ at $k_{0}$ is removable so $k_{0}$ can be removed from $\mathsf{Z}$ without affecting RH problem 3.1. In the rest of this paper, we will therefore assume that $c_{k_{0}}\neq 0$ for all $k_{0}\in\mathsf{Z}$.

4. Brief overview of the proof

Under the assumptions of Theorem 2.1, RH problem 3.1 has a unique solution $n(x,t,k)$ for each $(x,t)\in{\mathbb{R}}\times[0,\infty)$, the function

[TABLE]

is well-defined and smooth for $(x,t)\in{\mathbb{R}}\times[0,\infty)$, and

[TABLE]

is a Schwartz class solution of (1.1) on ${\mathbb{R}}\times[0,\infty)$ with initial data $u_{0},u_{1}$ [6]. By using Deift–Zhou steepest descent arguments, we will derive an asymptotic formula for $n_{3}^{(1)}$. Substituting this formula into (4.1), we will obtain the formula of Theorem 2.1.

The steepest descent analysis of RH problem 3.1 involves the saddle points of the three phase functions $\Phi_{21}$, $\Phi_{31}$, and $\Phi_{32}$ whose signature tables are displayed in Figure 3. The saddle points $\{k_{j}\}_{j=1}^{4}$ of $\Phi_{21}$ are given in (2.8). Using the relations

[TABLE]

we see that $\{\omega k_{j}\}_{j=1}^{4}$ are the saddle points of $\Phi_{31}$ and that $\{\omega^{2}k_{j}\}_{j=1}^{4}$ are the saddle points of $\Phi_{32}$. It turns out that the main contributions to the long-time asymptotics of $u(x,t)$ in Sector IV are generated by a global parametrix $\Delta^{-1}$ and from twelve local parametrices near the saddle points $\{k_{j},\omega k_{j},\omega^{2}k_{j}\}_{j=1}^{4}$.

Our proof uses a series of transformations $n\to n^{(1)}\to n^{(2)}\to n^{(3)}\to n^{(4)}\to\hat{n}$ to bring RH problem 3.1 to a small-norm RH problem. These transformations are invertible, implying that the RH problems satisfied by $\{n^{(j)}\}_{1}^{4}$ and $\hat{n}$ are equivalent to RH problem 3.1. The jump contours for the RH problems for $\{n^{(j)}\}_{1}^{4}$ and $\hat{n}$ will be denoted by $\{\Gamma^{(j)}\}_{1}^{4}$ and $\hat{\Gamma}$ and the associated jump matrices by $\{v^{(j)}\}_{1}^{4}$ and $\hat{v}$, respectively.

The jump matrix $v$ in (3.1) obeys the following symmetries in accordance with (3.3):

[TABLE]

We will preserve the symmetries (3.3) and (4.3) at each step, so that, for $j=1,2,3,4$,

[TABLE]

and similarly for $\hat{n}$ and $\hat{v}$. These symmetries imply that we only need to construct the local parametrices near $\omega k_{4}$ and $\omega^{2}k_{2}$, and that we only need to define the transformations $n\to n^{(1)}$, $n^{(j)}\to n^{(j+1)}$, and $n^{(4)}\to\hat{n}$ in the sector $\mathsf{S}:=\{k\in\mathbb{C}|\arg k\in[\frac{\pi}{3},\frac{2\pi}{3}]\}$. In $\mathbb{C}\setminus\mathsf{S}$, we will define the local parametrices and the transformations using the $\mathcal{A}$- and $\mathcal{B}$-symmetries.

We know from [4, Theorem 2.3] that the functions $r_{1}$ and $r_{2}$ satisfy the following properties: $r_{1}\in C^{\infty}(\hat{\Gamma}_{1})$, $r_{2}\in C^{\infty}(\hat{\Gamma}_{4}\setminus\{\omega^{2},-\omega^{2}\})$, $r_{1}(\kappa_{j})\neq 0$ for $j=1,\ldots,6$, $r_{2}(k)$ has simple poles at $k=\pm\omega^{2}$, and simple zeros at $k=\pm\omega$, and $r_{1},r_{2}$ are rapidly decreasing as $|k|\to\infty$. Moreover,

[TABLE]

These properties will be used repeatedly throughout the proof.

The residue conditions (3.5) involve the exponentials $e^{-\theta_{31}(x,t,k_{0})}$ and $e^{\theta_{32}(x,t,\bar{k}_{0})}$. We see from Figure 3 that these exponentials are small as $t\to\infty$ for $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$ if $\text{Re\,}k_{0}<0$ (recall that $\mathsf{Z}\setminus\mathbb{R}\subset D_{\mathrm{reg}}$). Similarly, the residue conditions (3.6) involve the exponential $e^{-\theta_{21}(x,t,k_{0})}$, and we see from Figure 3 that this exponential is small as $t\to\infty$ for $k_{0}\in\mathsf{Z}\cap\mathbb{R}$ if $\text{Re\,}k_{0}<0$ (recall that $\mathsf{Z}\cap\mathbb{R}\subset(-1,0)\cup(1,\infty)$). This suggests that points $k_{0}$ in $\mathsf{Z}$ with $\text{Re\,}k_{0}<0$ (i.e., points corresponding to left-moving solitons) will not contribute to the asymptotics in Sector IV at any finite order, and this is indeed what we will find. On the other hand, for $k_{0}\in\mathsf{Z}$ with $\text{Re\,}k_{0}>0$, the above exponentials are large as $t\to\infty$. By including an appropriate factor in the global parametrix, we can flip the signs of the exponents in these exponentials, thereby making them small for large $t$. The factor included in the global parametrix gives rise to the phase shifts $\arg\tfrac{\mathcal{P}(\omega k_{4})}{\mathcal{P}(\omega^{2}k_{4})}$ and $\arg\tfrac{\mathcal{P}(\omega^{2}k_{2})}{\mathcal{P}(\omega k_{2})}$ generated by right-moving solitons seen in Theorem 2.1.

5. The $n\to n^{(1)}$ transformation

We will open lenses around $\partial\mathbb{D}\setminus\mathcal{Q}$. Define

[TABLE]

and let

[TABLE]

The domains of definition of these spectral functions are (in general) only subsets of $\Gamma$. To open lenses around $\partial\mathbb{D}$, it is therefore necessary to decompose them into an analytic part and a small remainder. Let $M>1$ and define open sets $\{U_{1}^{\ell}=U_{1}^{\ell}(\zeta,M)\}_{\ell=1}^{5}$ by (see Figure 4)

[TABLE]

It follows from [4, Lemma 2.13] that $f(k)=0$ if and only if $k\in\{\pm 1,\pm\omega\}$, and that $1+r_{1}(k)r_{2}(k)>0$ for all $k\in\partial\mathbb{D}$ with $\arg k\in(\pi/3,\pi)\cup(-2\pi/3,0)$. Therefore, for each $\ell\in\{1,2,3\}$, $r_{1,\ell}$ is well-defined for $k\in\partial U_{1}^{\ell}\cap\partial\mathbb{D}$; $r_{1,4}$ is well-defined for $k\in(\partial U_{1}^{4}\cap\partial\mathbb{D})\setminus\{1\}$; and $r_{1,5}$ is well-defined for $k\in(\partial U_{1}^{5}\cap\partial\mathbb{D})\setminus\{\omega\}$. Let $n_{1}\geq 0$ denote the smallest integer such that $(k-1)^{n_{1}}r_{1,4}(k)=O(1)$ as $k\to 1$, $k\in\partial\mathbb{D}$, and let $n_{\omega}\geq 0$ denote the smallest integer such that $(k-\omega)^{n_{\omega}}r_{1,5}(k)=O(1)$ as $k\to\omega$, $k\in\partial\mathbb{D}$.

We now find decompositions of $\{r_{1,\ell}\}_{\ell=1}^{5}$; the decompositions of $\{r_{2,\ell}\}_{\ell=1}^{5}$ will then be obtained using the symmetry (4.7). Let $N\geq 1$ be an integer.

Lemma 5.1** (Decomposition lemma).**

There exist $M>1$ and decompositions

[TABLE]

such that $\{r_{1,\ell,a},r_{1,\ell,r}\}_{\ell=1}^{5}$ satisfy the following properties:

1. $(a)$

For each $\zeta\in\mathcal{I}$, $t\geq 1$, and $\ell\in\{1,\dots,5\}$, $r_{1,\ell,a}(x,t,k)$ is defined and continuous for $k\in\bar{U}_{1}^{\ell}$ and analytic for $k\in U_{1}^{\ell}$. 2. $(b)$

For each $\zeta\in\mathcal{I}$, $t\geq 1$, and $\ell\in\{1,\dots,5\}$, $r_{1,\ell,a}$ obeys

[TABLE]

where $\mathcal{R}_{1}=\{\omega k_{3},e^{\pm 5\pi i/6},-1,-i,-\omega,e^{\pm\frac{\pi i}{6}},k_{1}\}$, $\mathcal{R}_{2}=\{i,\omega k_{4},\omega^{2}k_{2}\}$, $\mathcal{R}_{3}=\{k_{1},\omega k_{3}\}$, $\mathcal{R}_{4}=\{\omega k_{2},\omega^{2}k_{1},k_{3}\}$, $\mathcal{R}_{5}=\{\omega^{2}k_{2},k_{1}\}$, and the constant $C$ is independent of $\zeta,t,k$. Moreover, for $\zeta\in\mathcal{I}$ and $t\geq 1$,

[TABLE]

and these estimates hold for $k\in\bar{U}_{1}^{4}$ and $k\in\bar{U}_{1}^{5}$, respectively. 3. $(c)$

For each $1\leq p\leq\infty$ and $\ell\in\{1,\dots,5\}$, the $L^{p}$-norm of $r_{1,\ell,r}(x,t,\cdot)$ on $\partial\bar{U}_{1}^{\ell}\cap\partial\mathbb{D}$ is $O(t^{-N})$ uniformly for $\zeta\in\mathcal{I}$ as $t\to\infty$.

Proof.

The function $\theta\mapsto-i\Phi_{21}(\zeta,e^{i\theta})=(\zeta-\cos\theta)\sin\theta$ is real-valued and bijective on each connected component of $\partial U_{1}^{\ell}\cap\partial\mathbb{D}$, $\ell=1,\ldots,5$. Thus the statement can be proved using similar arguments as in [9]. Since these arguments are rather standard by now, we omit details. ∎

In what follows, shrinking $M>1$ if necessary, we assume that $|k_{0}|>M$ for all k_{0}\in\mathsf{Z}\cap\big{(}D_{\mathrm{reg}}^{R}\cup(1,\infty)\big{)} and that $|k_{0}|<M^{-1}$ for all k_{0}\in\mathsf{Z}\cap\big{(}D_{\mathrm{reg}}^{L}\cup(-1,0)\big{)}.

It is easy to check (using the second equation in (4.7)) that $\tilde{r}(k)\in\mathbb{R}$ for $k\in\partial\mathbb{D}\setminus\{-\omega^{2},\omega^{2}\}$ and that $\tilde{r}(k)=\tilde{r}(\frac{1}{\omega k})\tilde{r}(\frac{1}{\omega^{2}k})$. Using also the first equation in (4.7), we infer that $r_{2,\ell}(k)=\tilde{r}(k)\overline{r_{1,\ell}(\bar{k}^{-1})}$, $k\in\partial\mathbb{D}\cap U_{1}^{\ell}$, $\ell=1,\ldots,5$. Hence, for each $\ell\in\{1,\ldots,5\}$ we let $U_{2}^{\ell}:=\{k|\bar{k}^{-1}\in U_{1}^{\ell}\}$ and define a decomposition $r_{2,\ell}=r_{2,\ell,a}+r_{2,\ell,r}$ by

[TABLE]

We now define the first transformation $n\to n^{(1)}$, which consists in opening lenses around $\partial\mathbb{D}\setminus\mathcal{Q}$. As mentioned in Section 4, we only need to define it explicitly in the sector $\mathsf{S}$. We introduce a new contour $\Gamma^{(0)}$, which coincides as a set with $\Gamma\cap\mathsf{S}$, but is oriented and labeled differently, see Figure 5. We let $\Gamma_{j}^{(0)}$ denote the subcontour of $\Gamma^{(0)}$ labeled by $j$ in Figure 5. For example,

[TABLE]

We will open lenses differently on the four parts of $\Gamma^{(0)}$. On $\Gamma_{9_{r}}^{(0)}$, we will use the factorization

[TABLE]

where

[TABLE]

and

[TABLE]

On $\Gamma_{2}^{(0)}$, we will use the factorization

[TABLE]

On $\Gamma_{5}^{(0)}$, we will use the factorization

[TABLE]

where

[TABLE]

and $a_{ij,a},a_{ij,r}$ are the analytic continuation and the remainder of $a_{ij}$ from Lemma 5.1, e.g., $a_{13,a}=-r_{1,4,a}(\frac{1}{\omega^{2}k})$ and $a_{23,a}=r_{2,3,a}(\frac{1}{\omega k})$.

Finally, on $\Gamma_{8}^{(0)}$, we will use the factorization

[TABLE]

where

[TABLE]

and $b_{ij,a},b_{ij,r}$ are the analytic continuation and the remainder of $b_{ij}$ from Lemma 5.1, e.g., $b_{12,a}=-r_{1,5,a}(k)$ and $b_{23,a}=r_{2,5,a}(\frac{1}{\omega k})$. We do not write down the long expressions for $v_{2,r}^{(1)},v_{5,r}^{(1)}$ and $v_{8,r}^{(1)}$ which are similar to the expression for $v_{2,r}^{(1)}$ given in [4, Section 8]; the only property of these matrices that is important for us is that

[TABLE]

as a consequence of Lemma 5.1.

Let $\Gamma^{(1)}$ be the contour shown in Figure 6, and define $G^{(1)}$ for $k\in\mathsf{S}$ by

[TABLE]

We now appeal to the $\mathcal{A}$- and $\mathcal{B}$-symmetries to extend the definition of $G^{(1)}$ to the whole complex plane:

[TABLE]

Define the sectionally meromorphic function $n^{(1)}$ by

[TABLE]

The functions $G^{(1)}$ and $n^{(1)}$ are analytic on ${\mathbb{C}}\setminus\Gamma^{(1)}$. Indeed, let us look for example at the region on the $-$ side of $\Gamma_{8}^{(1)}$ that is inside the lens; in this region $G^{(1)}=v_{7}^{(1)}$ is given in terms of $r_{1,5,a}(k)$, $r_{2,5,a}(\frac{1}{\omega k})$, and $r_{2,4,a}(\omega^{2}k)$, and it follows from Lemma 5.1 and the definitions of $U_{1}^{5}$, $U_{2}^{5}$, and $U_{2}^{4}$ that these functions are all analytic in this region.

The following lemma is a direct consequence of Figure 3. In this lemma, the disks $D_{\epsilon}(\omega^{j})$ have been excluded; this is because $v_{7}^{(1)},v_{8}^{(1)},v_{9}^{(1)}$ (and hence also $n^{(1)}(x,t,k)$) are singular at $k=\omega$, due to the fact that $f(\omega)=f(1)=0$.

Lemma 5.2**.**

For any $\epsilon>0$, $G^{(1)}(x,t,k)$ and $G^{(1)}(x,t,k)^{-1}$ are uniformly bounded for $k\in\mathbb{C}\setminus(\Gamma^{(1)}\cup\cup_{j=0}^{2}D_{\epsilon}(\omega^{j}))$, $t\geq 1$, and $\zeta\in\mathcal{I}$. Furthermore, $G^{(1)}(x,t,k)=I$ for all large enough $|k|$.

Let $\Gamma^{(1)}_{\star}$ be the set of self-intersection points of $\Gamma^{(1)}$. The jump matrix $v^{(1)}$ for $n^{(1)}$ is given on $\Gamma^{(1)}\setminus\Gamma^{(1)}_{\star}$ as follows: For $j=9_{L},9_{r},9_{R},1,\dots,9$, the matrix $v_{j}^{(1)}$ is given by (5.2)–(5.5); for $j=1^{\prime\prime},1_{r}^{\prime\prime},4_{r}^{\prime},4^{\prime}$ it is given by

[TABLE]

for $j=1_{s}$ it is given by

[TABLE]

and similar expressions are valid for $j=2_{s},3_{s},4_{s}$. Using the symmetries in (4.4) we extend $v^{(1)}$ to all of $\Gamma^{(1)}\setminus\Gamma^{(1)}_{\star}$.

Since $n^{(1)}=n$ near all the points of $\hat{\mathsf{Z}}$, the residue conditions (3.5)–(3.6) are not affected by the transformation $n\to n^{(1)}$. The function $n^{(1)}$ therefore satisfies the following RH problem for $j=1$.

RH problem 5.3** (RH problem for $n^{(j)}$).**

Find $n^{(j)}(x,t,k)$ with the following properties:

1. $(a)$

$n^{(j)}(x,t,\cdot):{\mathbb{C}}\setminus(\Gamma^{(j)}\cup\hat{\mathsf{Z}})\to\mathbb{C}^{1\times 3}$* is analytic.* 2. $(b)$

On $\Gamma^{(j)}\setminus\Gamma^{(j)}_{\star}$, the boundary values of $n^{(j)}$ exist, are continuous, and satisfy $n^{(j)}_{+}=n^{(j)}_{-}v^{(j)}$. 3. $(c)$

$n^{(j)}(x,t,k)=O(1)$* as $k\to k_{\star}\in\Gamma^{(j)}_{\star}\setminus\{1,\omega,\omega^{2}\}$.* 4. $(d)$

$n^{(j)}$* obeys the symmetries $n^{(j)}(x,t,k)=n^{(j)}(x,t,\omega k)\mathcal{A}^{-1}=n^{(j)}(x,t,k^{-1})\mathcal{B}$ for $k\in{\mathbb{C}}\setminus\Gamma^{(j)}$.* 5. $(e)$

$n^{(j)}(x,t,k)=(1,1,1)+O(k^{-1})$* as $k\to\infty$.* 6. $(f)$

At each point of $\hat{\mathsf{Z}}$, one entry of $n^{(j)}$ has (at most) a simple pole while two entries are analytic. Moreover, $n^{(j)}$ satisfies the residue conditions (3.5)–(3.6) with $n$ replaced by $n^{(j)}$.

Note that $n^{(1)}$ is singular at the points $1,\omega,\omega^{2}$ in general. The behavior of $n^{(1)}$ near these singularities is as follows: As $k\to\omega$ from the right side of $\Gamma_{8}^{(1)}$ (i.e. $k\in D_{4}$ and $|k|>1$),

[TABLE]

and as $k\to\omega$ from the left side of $\Gamma_{8}^{(1)}$ (i.e. $k\in D_{1}$ and $|k|<1$),

[TABLE]

We know from [4, Lemma 2.13] that $f(\pm 1)=f(\pm\omega)=0$. Since $f\geq 0$ on $\partial{\mathbb{D}}$, these zeros must be at least double zeros. One could try to make the solution of this singular RH problem unique by specifying the pole structure at the points $1,\omega,\omega^{2}$ in further detail, but since the singularities at $1,\omega,\omega^{2}$ will be removed by the global parametrix below, we do not need to do this.

6. The $n^{(1)}\to n^{(2)}$ transformation

The jump matrix $v^{(1)}$ admits the following factorizations on $\Gamma^{(1)}\cap\mathsf{S}$ (the subscripts $u$ and $d$ indicate that the corresponding matrix will be deformed up or down):

[TABLE]

where

[TABLE]

Let $\Gamma^{(2)}$ be the contour shown in Figure 7, and let $\Gamma_{j}^{(2)}$ be the subcontour of $\Gamma^{(2)}$ labeled by $j$ in Figure 7. We emphasize that $\Gamma_{10}^{(2)}:=\{e^{i\theta}\,|\,\theta\in(\arg(\omega^{2}k_{2}),\tfrac{2\pi}{3})\}$ ends at $\omega$. Define the sectionally meromorphic function $n^{(2)}$ by

[TABLE]

where $G^{(2)}$ is defined for $k\in\mathsf{S}$ by

[TABLE]

and $G^{(2)}$ is extended to all of ${\mathbb{C}}\setminus\Gamma^{(2)}$ using the $\mathcal{A}$- and $\mathcal{B}$-symmetries (as in (5.7)). Using Lemma 5.1 and Figure 3, we infer that the following holds.

Lemma 6.1**.**

For any $\epsilon>0$, $G^{(2)}(x,t,k)$ and $G^{(2)}(x,t,k)^{-1}$ are uniformly bounded for $k\in\mathbb{C}\setminus(\Gamma^{(2)}\cup\cup_{j=0}^{2}D_{\epsilon}(\omega^{j}))$, $t>0$, and $\zeta\in\mathcal{I}$. Furthermore, $G^{(2)}(x,t,k)=I$ whenever $|k|$ is large enough.

The jumps $v_{j}^{(2)}$ of $n^{(2)}$ are given for $j=1,\dots,11$ by (6.1) and

[TABLE]

All other jumps on $\Gamma^{(2)}\cap\mathsf{S}$ are small as $t\to\infty$. Indeed, this follows from the signature tables in Figure 3 together with the following observations. On $\Gamma_{9_{r}}^{(2)}$, $v^{(2)}-I$ is small because it involves small remainders. On $\Gamma_{4_{r}^{\prime}}^{(2)}$ and $\Gamma_{1_{r}^{\prime\prime}}^{(2)}$, $v^{(2)}-I$ is small as a consequence of the following lemma which follows in the same way as [4, Lemma 8.5].

Lemma 6.2**.**

The $L^{\infty}$-norm of $v^{(2)}-I$ on $\Gamma_{4_{r}^{\prime}}^{(2)}\cup\Gamma_{1_{r}^{\prime\prime}}^{(2)}$ is $O(t^{-N-1})$ as $t\to\infty$ uniformly for $\zeta\in\mathcal{I}$.

We finally consider the jumps $v_{j}^{(2)}=v_{j}^{(1)}$, $j=1_{s},\dots,4_{s}$ and

[TABLE]

If the initial data $u_{0},v_{0}$ have compact support so that all the spectral functions have analytic continuations and we can choose $b_{ij,a}=b_{ij}$, then a straightforward calculation shows that these jumps are absent (i.e., $v_{j}^{(2)}=I$, $j=1_{s},\dots,8_{s}$) as a consequence of the relation (4.6) and the definition (2.10) of $f(k)$. In general, it seems difficult to construct analytic approximations which preserve the nonlinear relation (4.6). Therefore, the jump matrices $v_{j}^{(2)}$, $j=1_{s},\dots,8_{s}$, will generally be nontrivial. However, as the next lemma demonstrates, it is not difficult to choose the analytic approximations so that these jumps are uniformly small for large $t$.

Lemma 6.3**.**

It is possible to choose the analytic approximations so that the $L^{\infty}$-norm of $v_{j}^{(2)}-I$ on $\Gamma_{j}^{(2)}$, $j=1_{s},\dots,8_{s}$, is $O(t^{-N})$ as $t\to\infty$ uniformly for $\zeta\in\mathcal{I}$.

Proof.

Let us consider $v_{1_{s}}^{(2)}$; the other jumps are handled similarly. We see from (5.10) that $v_{1_{s}}^{(2)}=v_{1_{s}}^{(1)}$ has ones along the diagonal and that each of its off-diagonal entries is suppressed by an exponential of the form $e^{-t|\text{Re\,}\Phi_{ij}|}$ where $|\text{Re\,}\Phi_{ij}(\zeta,k)|\geq c|k-\omega|$ on $\Gamma_{1_{s}}^{(2)}$. Thus, by Lemma 5.1, $v_{1_{s}}^{(2)}-I$ is small as $t\to\infty$ everywhere on $\Gamma_{1_{s}}^{(2)}$ except possibly near $\omega$. On the other hand, we know that the matrix $v_{1_{s}}^{(2)}-I$ vanishes identically on $\Gamma_{1_{s}}^{(2)}$ if all the spectral functions have analytic continuations thanks to (4.6). This means that if we choose all the analytic approximations such that they agree with the functions they are approximating to sufficiently high order at each of the sixth roots of unity $\kappa_{j}$ (in the sense of (6.5) below), then we can ensure that

[TABLE]

uniformly for $\zeta\in\mathcal{I}$, and the lemma follows.

To construct analytic approximations of the above type, we proceed as follows. For definiteness, let us consider $b_{13}$. The analytic approximation $b_{13,a}$ of $b_{13}$ is needed in a region $\mathcal{U}:=\{k\,|\,\arg k\in(\arg(\omega^{2}k_{2}),\tfrac{2\pi}{3}),\;1<|k|<M\}$ to the right of $\Gamma_{8}^{(1)}=\Gamma_{10}^{(2)}$. Since $f(1)=0$ and $r_{2}(1)=-1$, $b_{13}$ has a singularity at $\omega$. Thus, we first choose $m\geq 1$ such that $B(k):=(k-\omega)^{m}b_{13}(k)$ is regular at $\omega$. Applying the method of [9], we construct a decomposition $B=B_{a}+B_{r}$ such that

[TABLE]

This implies in particular that $B_{r}$ has a zero of order at least $N_{1}$ at $k=\omega$. Then we define $b_{13,a}:=(k-\omega)^{-m}B_{a}$ and $b_{13,r}:=(k-\omega)^{-m}B_{r}$. By choosing $N_{1}$ large enough, we can ensure that the $L^{\infty}$-norm of $b_{13,r}$ is uniformly small for large $t$ and that

[TABLE]

where $\sum_{j=-m}^{N}b_{j}(k-\omega)^{j}$ denotes the expansion of $b_{13}$ to order $N$ at $\omega$, i.e., the expansion such that $b_{13}(k)-\sum_{j=-m}^{N}b_{j}(k-\omega)^{j}=O((k-\omega)^{N+1})$ as $k\to\omega$. This means that $b_{13,a}$ approximates $b_{13}$ to order $N$ at $\omega$. In a similar way, we construct analytic approximations which agree with the functions they are approximating to order $N$ at each $\kappa_{j}$ that lies in their domain of definition. ∎

The jumps on $\Gamma^{(2)}\setminus\mathsf{S}$ can be obtained using the symmetries (4.4).

The function $n^{(2)}$ satisfies RH problem 5.3 with $j=2$. The next transformation $n^{(2)}\to n^{(3)}$ will make $v^{(3)}-I$ small everywhere on a new contour $\Gamma^{(3)}$, except on twelve small crosses centered at the saddle points $\{k_{j},\omega k_{j},\omega^{2}k_{j}\}_{j=1}^{4}$. It will also change the residue conditions (3.5)–(3.6). To define this transformation, we first need to construct a global parametrix.

7. Global parametrix

For each $\zeta\in\mathcal{I}$, define the analytic functions

[TABLE]

by (2.9). The functions $\{\delta_{j}\}_{1}^{5}$ obey the jump relations

[TABLE]

where the contours $\Gamma_{2}^{(2)}$, $\Gamma_{5}^{(2)}$, and $\Gamma_{10}^{(2)}$ are oriented as in Figure 7, and

[TABLE]

Further properties of the functions $\delta_{j}$ are collected in the following lemma.

Lemma 7.1**.**

The functions $\delta_{j}(\zeta,k)$, $j=1,\ldots,5$, have the following properties:

1. $(a)$

The functions $\{\delta_{j}(\zeta,k)\}_{j=1}^{5}$ can be written as

[TABLE]

where for $s\in\{e^{i\theta}:\theta\in[\frac{\pi}{2},\frac{2\pi}{3}]\}$, $k\mapsto\ln_{s}(k-s):=\ln|k-s|+i\arg_{s}(k-s)$ has a cut along $\{e^{i\theta}:\theta\in[\frac{\pi}{2},\arg s]\}\cup(i,i\infty)$, and satisfies $\arg_{s}(1)=2\pi$. The $\nu_{j}$ are defined by

[TABLE]

and the functions $\chi_{j}$ are defined by

[TABLE]

and the integration paths starting at $\omega^{2}k_{2}$ are subsets of $\partial\mathbb{D}$ oriented in the counterclockwise direction. 2. $(b)$

The functions $\{\delta_{j}(\zeta,k)\}_{j=1}^{5}$ can be written as

[TABLE]

where for $s\in\{e^{i\theta}:\theta\in[\frac{\pi}{2},\frac{2\pi}{3}]\}$, $k\mapsto\tilde{\ln}_{s}(k-s):=\ln|k-s|+i\tilde{\arg}_{s}(k-s)$ has a cut along $\{e^{i\theta}:\theta\in[\arg s,\pi]\}\cup(-\infty,0)$, satisfies $\tilde{\arg}_{s}(1)=0$, and $\tilde{\chi}_{j}(\zeta,k)$ is defined in the same way as $\chi_{j}(\zeta,k)$ except that $\ln_{s}$ is replaced by $\tilde{\ln}_{s}$ and $\ln_{\omega}$ is replaced by $\tilde{\ln}_{\omega}$. 3. $(c)$

For each $\zeta\in\mathcal{I}$ and $j\in\{1,\ldots,5\}$, $\delta_{j}(\zeta,k)$ and $\delta_{j}(\zeta,k)^{-1}$ are analytic functions of $k$ in their respective domains of definition. Moreover, for any $\epsilon>0$,

[TABLE] 4. $(d)$

As $k\to\omega k_{4}$ along a path which is nontangential to $\partial\mathbb{D}$, we have

[TABLE]

and as $k\to\omega^{2}k_{2}$ along a path which is nontangential to $\partial\mathbb{D}$, we have

[TABLE]

where $C$ is independent of $\zeta\in\mathcal{I}$.

Proof.

All assertions follow from the definitions (2.9) of the functions $\delta_{j}$. ∎

We will also need the following result.

Lemma 7.2**.**

For all $\zeta\in(\frac{1}{\sqrt{3}},1)$, the following inequalities hold:

[TABLE]

Proof.

For $\zeta\in(\frac{1}{\sqrt{3}},1)$, we have $-\frac{\pi}{6}<\arg k_{4}<0$ and $-\frac{3\pi}{4}<\arg k_{2}<-\frac{2\pi}{3}$. Thus the claim follows from [4, Lemma 2.13]. ∎

In what follows, we often write $\delta_{j}(k)$ for $\delta_{j}(\zeta,k)$ for conciseness. For $k\in\mathbb{C}\setminus(\partial\mathbb{D}\cup\hat{\mathsf{Z}})$, we define

[TABLE]

where $\mathcal{P}$ is given by (2.11). The function $\Delta_{33}$ satisfies $\Delta_{33}(\zeta,\frac{1}{k})=\Delta_{33}(\zeta,k)$, and

[TABLE]

Define $\Delta_{11}(\zeta,k)=\Delta_{33}(\zeta,\omega k)$ and $\Delta_{22}(\zeta,k)=\Delta_{33}(\zeta,\omega^{2}k)$, and define

[TABLE]

Then $\Delta$ obeys the symmetries

[TABLE]

and satisfies the jump relations

[TABLE]

and

[TABLE]

The following lemma is a direct consequence of (7.6) and of Lemma 7.1.

Lemma 7.3**.**

For any $\epsilon>0$, $\Delta(\zeta,k)$ and $\Delta(\zeta,k)^{-1}$ are uniformly bounded for $k\in\mathbb{C}\setminus(\cup_{j=0}^{2}D_{\epsilon}(\omega^{j})\cup\partial\mathbb{D}\cup\cup_{k_{0}\in\hat{\mathsf{Z}}}D_{\epsilon}(k_{0}))$ and $\zeta\in\mathcal{I}$. Furthermore, $\Delta(\zeta,k)=I+O(k^{-1})$ as $k\to\infty$.

8. The $n^{(2)}\to n^{(3)}$ transformation

Define the sectionally meromorphic function $n^{(3)}$ by

[TABLE]

where $\Gamma^{(3)}=\Gamma^{(2)}$. The jumps for $n^{(3)}$ on $\Gamma^{(3)}$ are given by $v^{(3)}=\Delta_{-}^{-1}v^{(2)}\Delta_{+}$.

Lemma 8.1**.**

The jump matrix $v^{(3)}$ converges to the identity matrix $I$ as $t\to\infty$ uniformly for $\zeta\in\mathcal{I}$ and $k\in\Gamma^{(3)}$ except near the twelve saddle points $\{k_{j},\omega k_{j},\omega^{2}k_{j}\}_{j=1}^{4}$, i.e., for $\zeta\in\mathcal{I}$,

[TABLE]

Proof.

From the expressions for $v_{j}^{(1)}$, $j=2,5,8$, in (5.3)–(5.5) together with (6.4), (7.5), (7.7), (7.8), we deduce that

[TABLE]

Using that the matrices $\Delta_{-}^{-1}v_{j,r}^{(1)}\Delta_{+}$, $j=2,5,8$, are small as $t\to\infty$, and recalling the symmetries (4.4) and Lemmas 6.2 and 6.3, we infer that $v^{(3)}-I$ tends to [math] as $t\to\infty$, uniformly for $k\in\Gamma^{\prime(3)}$. ∎

We next consider the residue conditions at the points in $\hat{\mathsf{Z}}$.

Lemma 8.2** (Residue conditions for $n^{(3)}$).**

The function $n^{(3)}$ obeys the following residue conditions at the points in $\hat{\mathsf{Z}}$:

1. $(a)$

for each $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$ with $\text{Re\,}k_{0}>0$,

[TABLE] 2. $(b)$

for each $k_{0}\in\mathsf{Z}\cap\mathbb{R}$ with $\text{Re\,}k_{0}>0$,

[TABLE] 3. $(c)$

for each $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$ with $\text{Re\,}k_{0}<0$,

[TABLE] 4. $(d)$

for each $k_{0}\in\mathsf{Z}\cap\mathbb{R}$ with $\text{Re\,}k_{0}<0$,

[TABLE]

where the $(x,t)$-dependence has been suppressed for clarity. Moreover, at each point of $\hat{\mathsf{Z}}$, two entries of $n^{(3)}$ are analytic while one entry has (at most) a simple pole.

Proof.

Since $\Delta(k)=\Delta(\zeta,k)$ is analytic at each point of $k_{0}\in\mathsf{Z}\cap\{\text{Re\,}k_{0}<0\}$, the claims $(c)$ and $(d)$ regarding points $k_{0}\in\mathsf{Z}$ with $\text{Re\,}k_{0}<0$ follow easily from RH problem 3.1 and the fact that $n^{(3)}=n\Delta$ near every point of $\mathsf{Z}$. It remains to consider the points $k_{0}\in\mathsf{Z}$ with $\text{Re\,}k_{0}>0$. We prove the first residue condition in (8.4); the other conditions are proved in a similar way. Let $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$ be such that $\text{Re\,}k_{0}>0$. Recall that $n^{(2)}$ obeys the same residue conditions as $n$; in particular, by (3.5),

[TABLE]

Since $\Delta_{33}$ has a simple zero at $k_{0}$ and $\Delta_{11}(k)=\Delta_{33}(\omega k)$ has a simple pole at $k_{0}$, $n_{3}^{(3)}$ has a removable singularity at $k_{0}$ and $n_{1}^{(3)}$ has a simple pole at $k_{0}$. Using (8.8), we obtain

[TABLE]

and hence

[TABLE]

This shows that only the first entry of $n^{(3)}$ is singular at $k_{0}$ and that the corresponding residue is as stated in (8.4). ∎

Using Lemma 8.2, we infer that $n^{(3)}$ satisfies RH problem 5.3 for $j=3$ except that item $(\ref{RHnjitemf})$ describing the behavior near points in $\hat{\mathsf{Z}}$ must be replaced by the following:

1. $(f^{\prime})$

At each point of $\hat{\mathsf{Z}}$, two entries of $n^{(3)}$ are analytic while one entry has (at most) a simple pole. Moreover, for $k_{0}\in\mathsf{Z}$ such that $\text{Re\,}k_{0}>0$, $n^{(3)}$ satisfies the residue conditions (8.4)–(8.5), while for $k_{0}\in\mathsf{Z}$ such that $\text{Re\,}k_{0}<0$, $n^{(3)}$ satisfies the residue conditions (8.6)–(8.7).

The residue conditions in (8.4)–(8.7) involve the following exponentials: $(a)$ $e^{t\Phi_{31}(x,t,k_{0})}$ and $e^{-t\Phi_{32}(x,t,\bar{k}_{0})}$ for $k_{0}\in D_{\mathrm{reg}}^{R}$, $(b)$ $e^{t\Phi_{21}(x,t,k_{0})}$ for $k_{0}\in(1,\infty)$, $(c)$ $e^{-t\Phi_{31}(x,t,k_{0})}$ and $e^{t\Phi_{32}(x,t,\bar{k}_{0})}$ for $k_{0}\in D_{\mathrm{reg}}^{L}$, and $(d)$ $e^{-t\Phi_{21}(x,t,k_{0})}$ for $k_{0}\in(-1,0)$. It follows from the signature tables of $\Phi_{21}$, $\Phi_{31}$, and $\Phi_{32}$ (see Figure 3) that the real parts of all the exponents in these exponentials are negative. Thus, all the residues of $n^{(3)}$ at points in $\hat{\mathsf{Z}}$ are suppressed by exponentially small factor as $t\to\infty$. This suggests that the poles in $\hat{\mathsf{Z}}$ have no effect on the leading large $t$ asymptotics of $n^{(3)}$. Our next transformation will help us make this claim precise.

9. The $n^{(3)}\to n^{(4)}$ transformation

The fourth transformation $n^{(3)}\to n^{(4)}$ replaces the residue conditions (8.4)–(8.7) by jumps on small circles. For each $k_{0}\in\mathsf{Z}$, we let $D_{\epsilon}(k_{0})$ be a small open disk centered at $k_{0}$ of radius $\epsilon>0$. We let $D_{\epsilon}(k_{0})^{-1}$ and $D_{\epsilon}(k_{0})^{*}$ be the images of $D_{\epsilon}(k_{0})$ under the maps $k\mapsto k^{-1}$ and $k\mapsto\bar{k}$, respectively. If $k_{0}\in\mathbb{R}$, then $D_{\epsilon}(k_{0})^{*}=D_{\epsilon}(k_{0})$. Let $\partial D_{\epsilon}(k_{0})$ and $\partial D_{\epsilon}(k_{0})^{*}$ be oriented counterclockwise, and let $\partial D_{\epsilon}(k_{0})^{-1}$ be oriented clockwise. Let

[TABLE]

so that $\partial\mathcal{D}_{\text{sol}}$ is the union of $6|\mathsf{Z}\cap\mathbb{R}|+12|\mathsf{Z}\setminus\mathbb{R}|$ small circles. By shrinking $\epsilon>0$ if necessary, we may assume that these circles do not intersect each other, and that they do not intersect $\Gamma^{(3)}$. We let $\Gamma^{(4)}:=\Gamma^{(3)}\cup\partial\mathcal{D}_{\text{sol}}$ be the union of $\Gamma^{(3)}$ and the small circles $\partial\mathcal{D}_{\text{sol}}$, see Figure 8. Let $\Gamma_{\star}^{(4)}$ be the set of intersection points of $\Gamma^{(4)}$. Clearly, $\Gamma_{\star}^{(4)}=\Gamma_{\star}^{(3)}$.

The function $n^{(4)}$ is defined to equal $n^{(3)}$ except in $\mathcal{D}_{\text{sol}}$ where we define $n^{(4)}(x,t,k)$ as follows. If $k_{0}\in\mathsf{Z}\setminus\mathbb{R}$, we define $n^{(4)}$ for $k\in D_{\epsilon}(k_{0})\cup D_{\epsilon}(k_{0})^{*}$ by

[TABLE]

where

[TABLE]

and if $k_{0}\in\mathsf{Z}\cap\mathbb{R}$, we define $n^{(4)}$ for $k\in D_{\epsilon}(k_{0})$ by

[TABLE]

where

[TABLE]

We then extend $n^{(4)}$ to all of $\mathcal{D}_{\text{sol}}$ by means of the $\mathcal{A}$- and $\mathcal{B}$-symmetries (4.5).

It follows from the above definition of $n^{(4)}$ and Lemma 8.2 that $n^{(4)}$ has no poles at the points in $\hat{\mathsf{Z}}$. Moreover, on $\Gamma^{(4)}\setminus\Gamma^{(4)}_{\star}$, the boundary values of $n^{(4)}$ exist, are continuous, and satisfy $n^{(4)}_{+}=n^{(4)}_{-}v^{(4)}$, where $v^{(4)}$ equals $v^{(3)}$ on $\Gamma$ and $v^{(4)}$ is defined on $\partial\mathcal{D}_{\text{sol}}$ by setting

[TABLE]

and then extending it to all of $\partial\mathcal{D}_{\text{sol}}$ by means of the $\mathcal{A}$- and $\mathcal{B}$-symmetries (4.4).

Lemma 9.1**.**

The jump matrix $v^{(4)}$ converges to the identity matrix $I$ as $t\to\infty$ uniformly for $k\in\partial\mathcal{D}_{\text{sol}}$ and $\zeta\in\mathcal{I}$. More precisely, for $\zeta\in\mathcal{I}$,

[TABLE]

Proof.

The statement is a direct consequence of the uniform exponential decay of the exponentials appearing in the definitions of $Q_{1}^{R},Q_{7}^{R},Q_{1}^{L},Q_{7}^{L},P_{1}^{R},P_{1}^{L}$, together with the $\mathcal{A}$- and $\mathcal{B}$-symmetries. ∎

Due to Lemma 8.1 and the symmetries (4.5), it is sufficient to construct local parametrices approximating $n^{(4)}$ near $\omega k_{4}$ and $\omega^{2}k_{2}$.

10. Local parametrix near $\omega k_{4}$

As $k\to\omega k_{4}$, we have

[TABLE]

We define $z_{1}=z_{1}(\zeta,t,k)$ by

[TABLE]

where the principal branch is chosen for $\hat{z}_{1}=\hat{z}_{1}(\zeta,k)$, and

[TABLE]

Note that $\hat{z}_{1}(\zeta,\omega k_{4})=1$, and $t(\Phi_{31}(\zeta,k)-\Phi_{31}(\zeta,\omega k_{4}))=\frac{iz_{1}^{2}}{2}$. Let $\epsilon>0$ be small and independent of $t$. The map $z_{1}$ is conformal from $D_{\epsilon}(\omega k_{4})$ to a neighborhood of [math] and its expansion as $k\to\omega k_{4}$ is given by

[TABLE]

For all $k\in D_{\epsilon}(\omega k_{4})$, we have

[TABLE]

where $\ln_{0}(k):=\ln|k|+i\arg_{0}k$, $\arg_{0}(k)\in(0,2\pi)$, and $\ln$ is the principal logarithm. Shrinking $\epsilon>0$ if necessary, $k\mapsto\ln\hat{z}_{1}$ is analytic in $D_{\epsilon}(\omega k_{4})$, $\ln\hat{z}_{1}=O(k-\omega k_{4})$ as $k\to\omega k_{4}$, and

[TABLE]

where $\arg(\omega k_{4})\in(\tfrac{\pi}{2},\tfrac{2\pi}{3})$. By Lemma 7.1, we have

[TABLE]

For simplicity, for $a\in{\mathbb{C}}$, we will write $z_{1,(0)}^{a}:=e^{a\ln_{0}z_{1}}$, $z_{1,\star}^{a}:=e^{a\ln z_{1,\star}}$, $\hat{z}_{1}^{a}:=e^{a\ln\hat{z}_{1}}$, and $z_{1}^{a}:=e^{a\ln z_{1}}$. Using the above expressions for $\{\delta_{j}(\zeta,k)\}_{j=1}^{3}$, we obtain

[TABLE]

where $d_{1,0}(\zeta,t)$ is defined in (2.13) and

[TABLE]

We summarize in Lemma 10.1 below some properties of $d_{1,0}(\zeta,t)$ and $d_{1,0}(\zeta,k)$. Define

[TABLE]

where $\sigma=\text{diag\,}(1,0,-1)$ and $\lambda_{1}\in\mathbb{C}$ is a parameter that will be chosen later. For $k\in D_{\epsilon}(\omega k_{4})$, define

[TABLE]

Let $\tilde{v}$ be the jump matrix of $\tilde{n}$, and let $\tilde{v}_{j}$ denote the restriction of $\tilde{v}$ to $\Gamma_{j}^{(4)}\cap D_{\epsilon}(\omega k_{4})$. In view of (6.1), (8.1) and (10.1), the matrices $\tilde{v}_{j}$ can be written as

[TABLE]

Let $\mathcal{X}_{1}^{\epsilon}:=\cup_{j=1}^{4}\mathcal{X}_{1,j}^{\epsilon}$, where

[TABLE]

Let $k_{\star}=\omega k_{4}$. The above expressions for $\tilde{v}_{j}$ suggest that we approximate $\tilde{n}$ by $(1,1,1)Y_{1}\tilde{m}^{\omega k_{4}}$, where $\tilde{m}^{\omega k_{4}}(x,t,k)$ is analytic for $k\in D_{\epsilon}(\omega k_{4})\setminus\mathcal{X}_{1}$, obeys the jump relation $\tilde{m}^{\omega k_{4}}_{+}=\tilde{m}^{\omega k_{4}}_{-}\tilde{v}_{\mathcal{X}_{1}^{\epsilon}}^{\omega k_{4}}$ on $\mathcal{X}_{1}^{\epsilon}$ where

[TABLE]

and satisfies $\tilde{m}^{\omega k_{4}}\to I$ as $t\to\infty$ uniformly for $k\in\partial D_{\epsilon}(\omega k_{4})$. We choose our free parameter $\lambda_{1}$ as follows:

[TABLE]

and define the local parametrix $m^{\omega k_{4}}$ by

[TABLE]

where $m^{X,(1)}(\cdot)=m^{X,(1)}(q_{1},q_{3},\cdot)$ is the solution of the model RH problem of Lemma A.1 with

[TABLE]

Lemma 10.1**.**

The function $Y_{1}(\zeta,t)$ is uniformly bounded:

[TABLE]

Moreover, the functions $d_{1,0}(\zeta,t)$ and $d_{1,1}(\zeta,k)$ satisfy

[TABLE]

Proof.

Standard estimates give (10.6) and (10.8). To prove (10.7), we first note that

[TABLE]

We deduce from Lemma 7.1 that if $k\in\partial{\mathbb{D}}\setminus\{\arg k\in(\pi/2,2\pi/3)\}$ then

[TABLE]

and

[TABLE]

where $\arg_{i}(k)\in(\pi/2,5\pi/2)$. Hence, for $k\in\partial{\mathbb{D}}\setminus\{\arg k\in(\pi/2,2\pi/3)\}$,

[TABLE]

Note that the right-hand sides in (10.10) are independent of $k$. Since the points $k_{4},\omega^{2}k_{4}$, $\frac{1}{k_{4}}$, $\frac{1}{\omega k_{4}}$, $\frac{1}{\omega^{2}k_{4}}$ all lie in $\partial{\mathbb{D}}\setminus\{\arg k\in(\pi/2,2\pi/3)\}$, we obtain

[TABLE]

Similar arguments show that

[TABLE]

Plugging the above expressions into (10.9) and using that

[TABLE]

straightforward calculations show that $|d_{1,0}(\zeta,t)|=e^{-\pi\nu_{1}}$. ∎

Lemma 10.2**.**

The function $m^{\omega k_{4}}(x,t,k)$ defined in (10.5) is an analytic function of $k\in D_{\epsilon}(\omega k_{4})\setminus\mathcal{X}_{1}^{\epsilon}$ which is uniformly bounded for $t\geq 2$, $\zeta\in\mathcal{I}$, and $k\in D_{\epsilon}(\omega k_{4})\setminus\mathcal{X}_{1}^{\epsilon}$. Across $\mathcal{X}_{1}^{\epsilon}$, $m^{\omega k_{4}}$ obeys the jump condition $m_{+}^{\omega k_{4}}=m_{-}^{\omega k_{4}}v^{\omega k_{4}}$, where $v^{\omega k_{4}}$ satisfies

[TABLE]

Furthermore, as $t\to\infty$,

[TABLE]

uniformly for $\zeta\in\mathcal{I}$, and $m_{1}^{X,(1)}=m_{1}^{X,(1)}(q_{1},q_{3})$ is given by (A.3).

Proof.

By (10.2) and (10.5), we have

[TABLE]

where we recall that $\tilde{v}$ is given by (10.3). Thus, recalling (10.6), the bounds (10.11) follow if we can show that

[TABLE]

for $j=1,\dots,4$. We give the proof of (10.14) for $j=1$; similar arguments apply when $j=2,3,4$.

For $k\in\mathcal{X}_{1,1}^{\epsilon}$, only the $(31)$ element of the matrix $\tilde{v}-v^{X,1}$ is nonzero. Since $-\bar{q}_{3}=\tfrac{\mathcal{P}(\omega^{2}k_{4})}{\mathcal{P}(\omega k_{4})}\lambda_{1}^{2}r_{2}(\frac{1}{k_{4}})$ and

[TABLE]

we can estimate $|(\tilde{v}-v^{X,(1)})_{31}|$ as follows:

[TABLE]

Using (10.8), this gives

[TABLE]

Hence

[TABLE]

The variable $z_{1}$ tends to infinity as $t\to\infty$ if $k\in\partial D_{\epsilon}(\omega k_{4})$, because

[TABLE]

Thus equation (A.3) yields

[TABLE]

uniformly with respect to $k\in\partial D_{\epsilon}(\omega k_{4})$ and $\zeta\in\mathcal{I}$. Recalling the definition (10.5) of $m^{\omega k_{4}}$, this gives

[TABLE]

uniformly for $k\in\partial D_{\epsilon}(\omega k_{4})$ and $\zeta\in\mathcal{I}$. This proves (10.12) and (10.13). ∎

11. Local parametrix near $\omega^{2}k_{2}$

As $k\to\omega^{2}k_{2}$, we have

[TABLE]

We define $z_{2}=z_{2}(\zeta,t,k)$ by

[TABLE]

where the principal branch is chosen for $\hat{z}_{2}=\hat{z}_{2}(\zeta,k)$, and

[TABLE]

Note that $\hat{z}_{2}(\zeta,\omega^{2}k_{2})=1$, and $t(\Phi_{32}(\zeta,k)-\Phi_{32}(\zeta,\omega^{2}k_{2}))=\frac{iz_{2}^{2}}{2}$. Provided $\epsilon>0$ is chosen small enough, the map $z_{2}$ is conformal from $D_{\epsilon}(\omega^{2}k_{2})$ to a neighborhood of [math] and its expansion as $k\to\omega^{2}k_{2}$ is given by

[TABLE]

For all $k\in D_{\epsilon}(\omega^{2}k_{2})$, we have

[TABLE]

where $\ln_{0}(k):=\ln|k|+i\arg_{0}k$, $\arg_{0}(k)\in(0,2\pi)$, and $\ln$ is the principal logarithm. Shrinking $\epsilon>0$ if necessary, $k\mapsto\ln\hat{z}_{2}$ is analytic in $D_{\epsilon}(\omega^{2}k_{2})$, $\ln\hat{z}_{2}=O(k-\omega^{2}k_{2})$ as $k\to\omega^{2}k_{2}$, and

[TABLE]

where $\arg(\omega^{2}k_{2})\in(\tfrac{\pi}{2},\tfrac{2\pi}{3})$. By Lemma 7.1, we have

[TABLE]

where we have used the notation $z_{2,(0)}^{a}:=e^{a\ln_{0}z_{2}}$, $\hat{z}_{2}^{a}:=e^{a\ln\hat{z}_{2}}$, $z_{2,\star}^{a}:=e^{a\ln z_{2,\star}}$, and $z_{2}^{a}:=e^{a\ln z_{2}}$ for $a\in{\mathbb{C}}$. We obtain

[TABLE]

where $d_{2,0}(\zeta,t)$ is defined in (2.14) and

[TABLE]

We state in Lemma 11.1 below some properties of $d_{2,0}(\zeta,t)$ and $d_{2,1}(\zeta,k)$. Define

[TABLE]

where $\tilde{\sigma}:=\text{diag\,}(0,1,-1)$ and $\lambda_{2}\in\mathbb{C}$ is a parameter that will be chosen below. For $k\in D_{\epsilon}(\omega^{2}k_{2})$, define

[TABLE]

Let $\tilde{v}$ be the jump matrix of $\tilde{n}$, and let $\tilde{v}_{j}$ be $\tilde{v}$ on $\Gamma_{j}^{(4)}\cap D_{\epsilon}(\omega^{2}k_{2})$. By (6.1), (8.1) and (11.1), the matrices $\tilde{v}_{j}$ can be written as

[TABLE]

Let $\mathcal{X}_{2}^{\epsilon}:=\cup_{j=1}^{4}\mathcal{X}_{2,j}^{\epsilon}$, where

[TABLE]

are oriented outwards from $\omega^{2}k_{2}$. Let $k_{\star}=\omega^{2}k_{2}$. The above expressions for $\tilde{v}_{j}$ suggest that we approximate $\tilde{n}$ by $(1,1,1)Y_{2}\tilde{m}^{\omega^{2}k_{2}}$, where $\tilde{m}^{\omega^{2}k_{2}}(x,t,k)$ is analytic for $k\in D_{\epsilon}(\omega^{2}k_{2})\setminus\mathcal{X}_{2}$, obeys the jump relation $\tilde{m}^{\omega^{2}k_{2}}_{+}=\tilde{m}^{\omega^{2}k_{2}}_{-}\tilde{v}_{\mathcal{X}_{2}^{\epsilon}}^{\omega^{2}k_{2}}$ on $\mathcal{X}_{2}^{\epsilon}$ where

[TABLE]

and satisfies $\tilde{m}^{\omega^{2}k_{2}}\to I$ as $k\to\infty$. We choose our free parameter $\lambda_{2}$ as follows:

[TABLE]

and define the local parametrix $m^{\omega^{2}k_{2}}$ by

[TABLE]

where $m^{X,(2)}(\cdot)=m^{X,(2)}(q_{2},q_{4},q_{5},q_{6},\cdot)$ is the solution of the model RH problem of Lemma A.2 with

[TABLE]

Using the relation (4.6) together with the fact that

[TABLE]

we infer that $q_{4}-\bar{q}_{5}-q_{2}\bar{q}_{6}=0$, so that (A.5) is fulfilled as it must. The proof of the following lemma is similar to the proof of Lemma 10.1.

Lemma 11.1**.**

The function $Y_{2}(\zeta,t)$ is uniformly bounded, i.e., $\sup_{\zeta\in\mathcal{I}}\sup_{t\geq 2}|Y_{2}(\zeta,t)^{\pm 1}|<C$. Moreover, the functions $d_{2,0}(\zeta,t)$ and $d_{2,1}(\zeta,k)$ satisfy

[TABLE]

The next lemma follows from the same kind of arguments used to prove Lemma 10.2.

Lemma 11.2**.**

The function $m^{\omega^{2}k_{2}}(x,t,k)$ defined in (11.4) is an analytic function of $k\in D_{\epsilon}(\omega^{2}k_{2})\setminus\mathcal{X}_{2}^{\epsilon}$ which is uniformly bounded for $t\geq 2$, $\zeta\in\mathcal{I}$, and $k\in D_{\epsilon}(\omega^{2}k_{2})\setminus\mathcal{X}_{2}^{\epsilon}$. Across $\mathcal{X}_{2}^{\epsilon}$, $m^{\omega^{2}k_{2}}$ obeys the jump condition $m_{+}^{\omega^{2}k_{2}}=m_{-}^{\omega^{2}k_{2}}v^{\omega^{2}k_{2}}$, where $v^{\omega^{2}k_{2}}$ satisfies

[TABLE]

Furthermore, as $t\to\infty$,

[TABLE]

uniformly for $\zeta\in\mathcal{I}$, where $\partial D_{\epsilon}(\omega^{2}k_{2})$ is oriented clockwise, and $m_{1}^{X,(2)}$ is given by (A.7).

12. The $n^{(4)}\to\hat{n}$ transformation

We use the symmetries

[TABLE]

to extend the domains of definition of $m^{\omega k_{4}}$ and $m^{\omega^{2}k_{2}}$ from $D_{\epsilon}(\omega k_{4})$ and $D_{\epsilon}(\omega^{2}k_{2})$ to $\mathcal{D}_{\omega k_{4}}$ and $\mathcal{D}_{\omega^{2}k_{2}}$, respectively, where

[TABLE]

We will show that the solution $\hat{n}(x,t,k)$ defined by

[TABLE]

is small for large $t$. Let $\hat{\Gamma}=\Gamma^{(4)}\cup\partial\mathcal{D}$ with $\mathcal{D}:=\mathcal{D}_{\omega k_{4}}\cup\mathcal{D}_{\omega^{2}k_{2}}$ be the contour displayed in Figure 9, where each circle that is part of $\partial\mathcal{D}$ is oriented negatively, and define the jump matrix $\hat{v}$ by

[TABLE]

The function $\hat{n}$ is analytic on ${\mathbb{C}}\setminus\hat{\Gamma}$ and satisfies $\hat{n}_{+}=\hat{n}_{-}\hat{v}$ for $k\in\hat{\Gamma}\setminus\hat{\Gamma}_{\star}$, where $\hat{\Gamma}_{\star}$ denotes the points of self-intersection of $\hat{\Gamma}$. As $k\to\infty$, $\hat{n}(x,t,k)=(1,1,1)+O(k^{-1})$.

Let \hat{\mathcal{X}}^{\epsilon}=\bigcup_{j=1}^{2}\big{(}\mathcal{X}_{j}^{\epsilon}\cup\omega\mathcal{X}_{j}^{\epsilon}\cup\omega^{2}\mathcal{X}_{j}^{\epsilon}\cup(\mathcal{X}_{j}^{\epsilon})^{-1}\cup(\omega\mathcal{X}_{j}^{\epsilon})^{-1}\cup(\omega^{2}\mathcal{X}_{j}^{\epsilon})^{-1}\big{)}.

Lemma 12.1**.**

Let $\hat{w}=\hat{v}-I$. The following estimates hold uniformly for $t\geq 2$ and $\zeta\in\mathcal{I}$:

[TABLE]

Proof.

On $\hat{\Gamma}\setminus\bar{\mathcal{D}}$, the estimate (12.3a) follows from (12.2) and Lemmas 8.1 and 9.1; on $(\hat{\Gamma}\cap\mathcal{D})\setminus\hat{\mathcal{X}}^{\epsilon}$, it follows from (12.2) and Lemmas 6.3, 7.3, 10.2, and 11.2. The estimate (12.3b) follows from (12.2), (10.12), and (11.8). Finally, (12.3c) and (12.3d) follow from (12.2), (10.11), and (11.7). ∎

The estimates in Lemma 12.1 show that

[TABLE]

and hence, employing the general identity $\|f\|_{L^{p}}\leq\|f\|_{L^{1}}^{1/p}\|f\|_{L^{\infty}}^{(p-1)/p}$,

[TABLE]

for each $1\leq p\leq\infty$. The estimates (12.5) imply that $\hat{w}\in(L^{2}\cap L^{\infty})(\hat{\Gamma})$ and that there is a $T$ such that $I-\hat{\mathcal{C}}_{\hat{w}(x,t,\cdot)}\in\mathcal{B}(L^{2}(\hat{\Gamma}))$ is invertible whenever $\zeta\in\mathcal{I}$ and $t\geq T$. So standard RH theory gives

[TABLE]

for $\zeta\in\mathcal{I}$ and $t\geq T$, where

[TABLE]

Lemma 12.2**.**

Let $1<p<\infty$. For all sufficiently large $t$, we have

[TABLE]

Proof.

Suppose $t$ is sufficiently large so that $\|\hat{w}\|_{L^{\infty}(\hat{\Gamma})}<K_{p}^{-1}$, where $K_{p}:=\|\hat{\mathcal{C}}_{-}\|_{\mathcal{B}(L^{p}(\hat{\Gamma}))}$ is finite because $p>1$. The estimate (12.5) implies that $\hat{w}\in L^{p}(\hat{\Gamma})$, and so $\hat{\mathcal{C}}_{\hat{w}}(1,1,1)\in L^{p}(\hat{\Gamma})$. Thus

[TABLE]

The claim is now a consequence of (12.4) and (12.5). ∎

We now turn to the asymptotics of $\hat{n}$. The following limit exists as $k\to\infty$:

[TABLE]

Lemma 12.3**.**

As $t\to\infty$,

[TABLE]

Proof.

Since

[TABLE]

the lemma follows from Lemmas 12.1 and 12.2 and straightforward estimates. ∎

We define the functions $\{F_{1}^{(l)},F_{2}^{(l)}\}_{l\in\mathbb{Z}}$ by

[TABLE]

We infer from (10.13) and (11.9) that $F_{1}^{(l)}$ and $F_{2}^{(l)}$ satisfy (recall that $\hat{z}_{1}(\zeta,\omega k_{4})=1=\hat{z}_{2}(\zeta,\omega^{2}k_{2})$ and that $\partial D_{\epsilon}(\omega k_{4})$ and $\partial D_{\epsilon}(\omega^{2}k_{2})$ are oriented negatively)

[TABLE]

where

[TABLE]

and

[TABLE]

and $\nu_{1}$, $\hat{\nu}_{1}$, $\nu_{2}$, $\hat{\nu}_{2}$, $\nu_{4}$ have been defined in Lemmas 7.1 and 7.2.

Lemma 12.4**.**

For $l\in\mathbb{Z}$ and $j=0,1,2$, we have

[TABLE]

Proof.

The symmetry properties of $\hat{v}$ imply that

[TABLE]

Therefore, for $j=0,1,2$ we have

[TABLE]

and

[TABLE]

This proves (12.13) and (12.14). The proofs of (12.15) and (12.16) are similar. ∎

Using Lemma 12.4, we find

[TABLE]

Therefore, (12.8), (12.9), and (12.10) imply that $\hat{n}^{(1)}=(1,1,1)\hat{m}^{(1)}$, where

[TABLE]

as $t\to\infty$ uniformly for $\zeta\in\mathcal{I}$.

13. Asymptotics of $u$

Recall from (4.1) that $u(x,t)=-i\sqrt{3}\frac{\partial}{\partial x}n_{3}^{(1)}(x,t)$, where $n_{3}(x,t,k)=1+n_{3}^{(1)}(x,t)k^{-1}+O(k^{-2})$ as $k\to\infty$. Recall also that $n^{(4)}$ is defined to equal $n^{(3)}$ except in $\mathcal{D}_{\text{sol}}$. Taking also the transformations (5.8), (6.2), (8.1), and (12.1) into account, we obtain

[TABLE]

for $k\in\mathbb{C}\setminus(\hat{\Gamma}\cup\bar{\mathcal{D}}\cup\bar{\mathcal{D}}_{\text{sol}})$, where $G^{(1)}$, $G^{(2)}$, $\Delta$ are defined in (5.6), (6.3), and (7.6), respectively. Thus

[TABLE]

as $t\to\infty$. By (12.17), as $t\to\infty$,

[TABLE]

Recall from Lemma 7.2 that $\hat{\nu}_{1}=\nu_{3}-\nu_{1}\geq 0$ and $\hat{\nu}_{2}=\nu_{2}+\nu_{5}-\nu_{4}\geq 0$. From (10.7), (11.5), (12.11), and (12.12), we see that $\overline{Z_{1,13}}=\lambda_{1}^{4}Z_{1,31}$ and $\overline{Z_{2,23}}=\lambda_{2}^{4}Z_{2,32}$. Moreover, by (10.4) and (11.3), we have \lambda_{1}^{4}=-\tilde{r}(k_{4}^{-1})^{-1}\big{(}\frac{\mathcal{P}(\omega k_{4})}{\mathcal{P}(\omega^{2}k_{4})}\big{)}^{2} and \lambda_{2}^{4}=-\tilde{r}(k_{2}^{-1})\big{(}\frac{\mathcal{P}(\omega^{2}k_{2})}{\mathcal{P}(\omega k_{2})}\big{)}^{2}. It follows that, as $t\to\infty$,

[TABLE]

The function $\mathcal{P}(k)$ obeys the symmetries $\mathcal{P}(k)=\mathcal{P}(k^{-1})=\overline{\mathcal{P}(\bar{k})}^{-1}$. It follows that $|\mathcal{P}(k)|=1$ for $k\in\partial{\mathbb{D}}$, and thus

[TABLE]

Using (12.12), (13.2), and the identities

[TABLE]

we obtain

[TABLE]

As in [7], it is possible to show that the above asymptotics can be differentiated with respect to $x$ without increasing the error term. Therefore, as $t\to\infty$,

[TABLE]

In view of the relations

[TABLE]

this completes the proof of Theorem 2.1.

Appendix A Model RH problems

Let $X\subset{\mathbb{C}}$ be the cross defined by $X=X_{1}\cup\cdots\cup X_{4}$, where the rays

[TABLE]

are oriented away from the origin. The proofs of the following two lemmas are similar to (but more complicated than) the proof for the parabolic cylinder model problem that appears in [9, 11]. A detailed proof of Lemma A.2 can be found in [5].

Lemma A.1** (Model RH problem needed near $k=\omega k_{4}$ for Sector IV).**

Let $q_{1},q_{3}\in\mathbb{C}$ be such that $1+|q_{1}|^{2}-|q_{3}|>0$, define

[TABLE]

and define the jump matrix $v^{X,(1)}(z)$ for $z\in X$ by

[TABLE]

where the branches for $z^{i\nu_{3}},z^{i\nu_{1}}$ lie on the negative real axis and the branch for $z_{(0)}^{i\nu_{1}}$ lies on the positive real axis, such that $z^{i\nu}=|z|^{i\nu}e^{-\nu\arg(z)}$, $\arg(z)\in(-\pi,\pi)$ and $z_{(0)}^{i\nu}=|z|^{i\nu}e^{-\nu\arg_{0}(z)}$, $\arg_{0}(z)\in(0,2\pi)$. Then the RH problem

1. $(a)$

$m^{X,(1)}(q_{1},q_{3},\cdot):{\mathbb{C}}\setminus X\to\mathbb{C}^{3\times 3}$* is analytic;* 2. $(b)$

on $X\setminus\{0\}$, the boundary values of $m^{X,(1)}$ exist, are continuous, and satisfy $m_{+}^{X,(1)}=m_{-}^{X,(1)}v^{X,(1)}$; 3. $(c)$

$m^{X,(1)}(q_{1},q_{3},z)=I+O(z^{-1})$* as $z\to\infty$, and $m^{X,(1)}(q_{1},q_{3},z)=O(1)$ as $z\to 0$;*

has a unique solution $m^{X,(1)}(q_{1},q_{3},z)$. This solution satisfies

[TABLE]

where the error term is uniform with respect to $\arg z\in[0,2\pi]$ and $q_{1},q_{3}$ in compact subsets of $\{q_{1},q_{3}\in\mathbb{C}\,|\,1+|q_{1}|^{2}-|q_{3}|>0\}$, and $\beta_{12}^{(1)},\beta_{21}^{(1)}$ are defined by

[TABLE]

where $\hat{\nu}_{1}:=\nu_{3}-\nu_{1}\geq 0$. (Note that $\beta_{12}^{(1)}\beta_{21}^{(1)}=\hat{\nu}_{1}$.)

Lemma A.2** (Model RH problem needed near $k=\omega^{2}k_{2}$).**

Let $q_{2},q_{4},q_{5},q_{6}\in\mathbb{C}$ be such that $1+|q_{2}|^{2}-|q_{4}|^{2}>0$, $1-|q_{5}|^{2}-|q_{6}|^{2}>0$, and

[TABLE]

Define $\nu_{2},\nu_{4},\nu_{5}\in{\mathbb{R}}$ by

[TABLE]

Define the jump matrix $v^{X,(2)}(z)$ for $z\in X$ by

[TABLE]

where $z^{i\nu}$ has a branch cut along $(-\infty,0]$ and $z_{(0)}^{i\nu}$ has a branch cut along $[0,+\infty)$ such that $z^{i\nu}=|z|^{i\nu}e^{-\nu\arg(z)}$, $\arg(z)\in(-\pi,\pi)$, and $z_{(0)}^{i\nu}=|z|^{i\nu}e^{-\nu\arg_{0}(z)}$, $\arg_{0}(z)\in(0,2\pi)$. Then the RH problem

1. $(a)$

$m^{X,(2)}(\cdot)=m^{X,(2)}(q_{2},q_{4},q_{5},q_{6},\cdot):{\mathbb{C}}\setminus X\to\mathbb{C}^{3\times 3}$* is analytic;* 2. $(b)$

on $X\setminus\{0\}$, the boundary values of $m^{X,(2)}$ exist, are continuous, and satisfy $m_{+}^{X,(2)}=m_{-}^{X,(2)}v^{X,(2)}$; 3. $(c)$

$m^{X,(2)}(z)=I+O(z^{-1})$* as $z\to\infty$, and $m^{X,(2)}(z)=O(1)$ as $z\to 0$;*

has a unique solution $m^{X,(2)}(z)$. This solution satisfies

[TABLE]

where the error term is uniform with respect to $\arg z\in[-\pi,\pi]$ and $q_{2},q_{4},q_{5},q_{6}$ in compact subsets of $\{q_{2},q_{4},q_{5},q_{6}\in\mathbb{C}\,|\,1+|q_{2}|^{2}-|q_{4}|^{2}>0,1-|q_{5}|^{2}-|q_{6}|^{2}>0,q_{4}-\bar{q}_{5}-q_{2}\bar{q}_{6}=0\}$, and

[TABLE]

where $\hat{\nu}_{2}:=\nu_{2}+\nu_{5}-\nu_{4}$. (Note that $\beta_{12}^{(2)}\beta_{21}^{(2)}=\hat{\nu}_{2}$.)

Acknowledgements

Support is acknowledged from the Novo Nordisk Fonden Project, Grant 0064428, the European Research Council, Grant Agreement No. 682537, the Swedish Research Council, Grant No. 2015-05430, Grant No. 2021-04626, and Grant No. 2021-03877, and the Ruth and Nils-Erik Stenbäck Foundation.