# Soliton resolution and asymptotic stability of $N$-loop-soliton solutions
for the Ostrovsky-Vakhnenko equation

Ruihong Ma  Engui Fan∗, School of Mathematical Sciences and Key Laboratory of
Mathematics for Nonlinear Science, Fudan University, Shanghai,200433, China
* Corresponding author and e-mail address<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the soliton resolution and asymptotic stability of
$N$-loop soliton solutions to the Cauchy problem for the Ostrovsky-Vakhenko
equation

$\displaystyle u_{txx}-3\kappa u_{x}+3u_{x}u_{xx}+uu_{xxx}=0,$ $\displaystyle
u(x,0)=u_{0}(x),$

where $u_{0}$ is assumed in the Schwartz space satisfying $-u_{0xx}+1>0$. It
is shown that the solution of the Cauchy problem can be characterized via a
Riemann-Hilbert (RH) problem in a new scale $(y,t)$. Using nonlinear steepest
descent method to deform the RH problem, we derive the leading order
approximation to the solution of OV equation for long times in the solitonic
region of space-time. Our results implies that $N$-loop soliton solutions of
the OV equation are symptotically stable.

###### keywords:

Ostrovsky-Vakhenko equation, Riemann-Hilbert problem,
$\bar{\partial}$-steepest descent method, Soliton resolution, Large time
asymptotics Mathematics Subject Classification: 35P25; 35Q51; 35Q15; 35A01;
35G25.

††journal:

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Soliton resolution
    2. 1.2 Asymptotic stability
    3. 1.3 Notations
    4. 1.4 Outline of this paper
  2. 2 Direct and inverse scattering transforms
    1. 2.1 Lax pair
    2. 2.2 Jost functions for large $\lambda$
    3. 2.3 Jost functions for small $\lambda$
    4. 2.4 Scattering data
    5. 2.5 Set up of a RH problem
    6. 2.6 Stationary saddle points and decay domains
  3. 3 Soliton resolution in region $x/t<\epsilon$
    1. 3.1 Conjugation
    2. 3.2 A mixed $\bar{\partial}$-RH problem
    3. 3.3 Analysis on the pure RH problem
      1. 3.3.1 A local model near the saddle point
      2. 3.3.2 A small-norm RH problem
      3. 3.3.3 Analysis on the $N$-loop soliton problem
    4. 3.4 Analysis on the remaining $\bar{\partial}$-problem
    5. 3.5 The proof of Theorem 1.1 in region $x/t<\epsilon$
  4. 4 Soliton resolution in region $x/t>\epsilon$
    1. 4.1 Conjugation
    2. 4.2 Analysis on the pure RH problem
    3. 4.3 Analysis on the $N$-loop soliton problem
    4. 4.4 The proof of Theorem 1.1 in region $x/t>\epsilon$

##  1 Introduction and main results

In this paper, we study soliton resolution and asymptotic stability of N-loop
soliton solutions for the Ostrovsky-Vakhenko (OV) equation on the line

$\displaystyle u_{txx}-3\kappa u_{x}+3u_{x}u_{xx}+uu_{xxx}=0,$ (1.1)
$\displaystyle u(x,0)=u_{0}(x),\quad(x,t)\in\mathbb{R}\times\mathbb{R}^{+},$
(1.2)

where $\kappa>0$ is a parameter and $u\equiv u(x,t)$ is a real valued
function. The equation (1.1) arises in the theory of propagation of surface
waves in deep water [1]. This equation can be named as the short wave model of
the Degasperis-Procesi equation [2]

$u_{t}-u_{xxt}+3\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}.$ (1.3)

Indeed under a scaling transformation

$x\to\epsilon x,\quad t\to t/\epsilon,\quad u\to\epsilon^{2}u,$

then (1.3) becomes (1.1) as $\epsilon\to 0$.

In recent years, much work has been done to study the various mathematical
properties of the OV equation. For example, as an integrable system, the OV
equation can be solved by the inverse scattering method [3]. The initial-
boundary value problem for the OV equation on the half-line was investigated
via the Fokas unified method [4]. A bi-Hamiltonian formulation for the OV
equation was established by using its higher order symmetry and a new
transformation to the Caudrey-Dodd-Gibbon-Sawada-Kotera equation [5]. The
shock solutions and singular soliton solution, such as peakon, cuspon and loop
solitons for OV equation were constructed by developing discontinuous Galerkin
method [6]. The well-posedness of the Cauchy problem for the OV equation and
its relatives (reduced Ostrovsky equation, generalized Ostrovsky equation,
etc.) in Sobolev spaces has been widely studies using analysis techniques [7,
8, 9, 10]. It was shown that the Cauchy problem for the OV equation has a
unique global solution $u\in C(\mathbb{R}_{+},H^{3}(\mathbb{R}))$ with initial
data $u_{0}\in H^{3}(\mathbb{R})$ and $-u_{0xx}(x)+1>0$, which plays an
important role in the integrability of OV equation [11].

For $\kappa=0$ and $\kappa=-1/3$, equation (1.1) respectively reduces to the
derivative Burgers equation

$(u_{t}+uu_{xx})_{xx}=0,$ (1.4)

and the differentiated Vakhnenko equation [12]

$(u_{t}+uu_{x})_{x}+u=0.$ (1.5)

The exact soliton solutions for the equation (1.5) were constructed by
applying Hirota bilinear method [13, 14, 15]. These solutions are multi-valued
functions having the form of a loop (1-soliton) or many loops (multi-soliton).
Without loss of generality, in our paper we assume that $\kappa=1$.

Recently it was shown that the OV equation (1.1) has smooth solitary wave
solutions and under the transformation [16, 17]

$y(x,t):=x-\int_{x}^{\infty}((-u_{xx}+1)^{1/3}-1)d\xi.$

In the original variable $(x,t)$, the soliton solution of OV equation (1.1) to
be a multivalued function having a loop shape, and under the change of
variable $y\to x$, which is not monotone, that makes the soliton in the
variable $(y,t)$ having a bell shape, which is typical for solitons of
integrable nonlinear evolution equations as shown in Fig.1.1.

(a) The soliton in the $(y,t)$ variables

(b) The soliton in the $(x,t)$ variables

Figure 1.1: Expression of soliton in different coordinates

If we assume there are no loop solitons, the pure radiation case is discussed
in [16]. It has shown that the OV equation (1.1) has different asymptotic
behaviors in different space-time regions. To deduce large-time asymptotics
and soliton resolution for (1.1) are established for initial data $u_{0}$ in
the following regions:

$x$$t$$x/t<-\epsilon$$x/t>\epsilon$modulatedoscillations regionsoliton region
Figure 1.2: Sectors in the $(x,t)$ half-plane.

  * $\blacktriangleright$

modulated oscillations region: In this region, we can observe loop solitons
traveling in the left direction as $t\to\infty$

  * $\blacktriangleright$

soliton region: In this region, we can observe loop solitons traveling in the
right direction as $t\to\infty$

From our brief discussion above, one should realize that general solution to
the OV equation will consist of loop solitons and radiation term which
differently in various space-time regions. In this generic setting, finitely
many loop solitons can appear and they interact with the radiation. One might
expect that as the consequence of integrability, these nonlinear modes
interact elastically during the dynamics but the ways they influence the
radiation are remarkably different.

###  1.1 Soliton resolution

Soliton resolution refers to the property that the solution decompose into the
sum of a finite number of separated solitons and a radiative part as
$|t|\to\infty$. The limiting soliton parameters are slightly modulated, due to
the soliton-soliton and soliton-radiation interactions. We fully describe the
dispersive part which contains two components, one coming from the continuous
spectrum and another one from the interaction of the discrete and continuous
spectrum. This decomposition ia s central feature in nonlinear wave dynamics
and has been the object of many theoretical and numerical studies. It has been
established in many perturbation contexts, that is when the initial condition
is close to a soliton or a multi-soliton. A direct consequence of this result
is that $N$-soliton solitons are asymptotically stable.

Our long-time asymptotic will also result in the verification of the soliton
resolution conjecture for the OV equation with generic data. This conjecture
asserts, roughly speaking, that any reasonable solution eventually resolves
into a superposition of a radiation component plus a finite number of
“nonlinear bound states” or “solitons”. Soliton resolution along a sequence of
times for the focusing energy critical wave equation was studied [18]. By
using $\bar{\partial}$-analysis and nonlinear steepest descent method, soliton
resolution for the derivative NLS equation was investigated in [19]. In [20],
soliton resolution for the focusing modified KdV equation is established.
Soliton resolution and large time behavior of solutions to the Cauchy problem
for the Novikov equation with a nonzero background was studied in [21]. Our
main result is the following detailed soliton resolution for the solution to
the OV equation.

######  Theorem 1.1.

Let $u$ be the solution of the OV equation (1.1), then $u$ can be written as
the loop solutions and the radiation as follows

$u(x,t)=\sum_{j=1}^{6N}u^{(lo)}(x,t)+R(x,t),$ (1.6)

where the loop soliton $u^{(lo)}(x,t)$ and radiation part $R(x,t)$ are given
by

  * $\blacktriangleright$

For the loop solitons part: We have the following asymptotics:

$\displaystyle u^{(lo)}(x,t)$
$\displaystyle=\sum_{j=1}^{N^{\prime}}\frac{\hat{e}}{\rho_{j}^{2}}\frac{(\cos(\phi_{j}+\frac{\pi}{3})-\hat{e}+\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2})}{(1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2})^{2}}$
$\displaystyle+\sum_{j=1}^{N^{\prime}}\frac{4}{\hat{e}}\frac{(j-1)\cos^{2}(\phi_{j}+\frac{\pi}{3})+\cos(\phi_{j}-\frac{\pi}{3})+\hat{e}}{(\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2}+\hat{e})^{2}},$

where $\rho,\phi$ is determined by constant $c_{j}$ of residue condition in RH
problem 2.1 and

$\displaystyle
x(y,t)=y+\frac{2\sqrt{3}}{\rho_{j}}\frac{-2\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}{1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}.$

We give the parameter representation for one loop soliton

$\displaystyle
u^{(lo)}(x,t)=\frac{12}{\rho^{2}}\frac{\hat{e}(\cos(\phi+\frac{\pi}{3})-\hat{e}+\cos(\phi+\frac{\pi}{3})\hat{e}^{2})}{(1-4\cos(\phi+\frac{\pi}{3})\hat{e}+\hat{e}^{2})^{2}},$
(1.7)

where

$\displaystyle\hat{e}(y,t)=\frac{C}{2\sqrt{3}\rho}e^{-\sqrt{3}\rho(y+\frac{t}{\rho^{2}}-\frac{1}{\sqrt{3}\rho}\log\frac{C}{2\sqrt{3}\rho})},$

and $z=\rho e^{i\theta}$ is spectrum parameter and $\phi$ and $C>0$ is
constant. The loop solitons have velocity $v=-\frac{1}{\rho^{2}}$, which means
loop solitons can travel in both directions.

  * $\blacktriangleright$

For the radiation part: We have the following asymptotics:

    1. $\bullet$

In the soliton region,

$|R(x,t)|\lesssim t^{-n},\quad n>1.$ (1.8)

    2. $\bullet$

In the modulated oscillations region,

$R(x,t)=u_{s}(x,t)+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$ (1.9)

where

$u_{s}(x,t)=\frac{k_{1}}{\sqrt{t}}\cos(\frac{2\sqrt{3}}{\varkappa}t+\nu\log
t+k_{2})-\frac{\delta(\omega^{2}z)-\delta(\omega z)}{2z\delta(\omega
z)\varkappa^{2}ct^{3/2}}+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$

and

$\displaystyle
k_{1}=-2^{3/2}3^{1/4}\sqrt{\frac{\nu}{\varkappa}}\sin\Big{(}\frac{\arg
r(\varkappa)-\arg r(-\varkappa)}{2}-\frac{2\pi}{3}\Big{)},$ $\displaystyle
k_{2}=\nu\log\frac{8\sqrt{3}}{\varkappa}+\frac{\arg r(\varkappa)+\arg
r(-\varkappa)}{2}+\arg\Gamma(-i\nu)+\frac{\pi}{4}$
$\displaystyle\qquad+\frac{3\varkappa}{\pi}\int_{\varkappa}^{\infty}\frac{\log(1-|r(s)|^{2})}{s^{2}}ds+\frac{1}{\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\log|\varkappa-s|d\log(1-|r(s)|^{2})$
$\displaystyle\qquad+\frac{1}{2\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\frac{\log(1-|r(s)|^{2})(2s+\varkappa)}{s^{2}+s\varkappa+\varkappa^{2}}$

where $\Gamma$ is the Euler Gamma function and $\nu$ is define in (3.3).

###  1.2 Asymptotic stability

The full asymptotic stability of multi-loop soliton of the OV equation (1.1)
which the by-products of our solitons resolution from the Theorem 1.1.

######  Theorem 1.2.

Given the loop solitons $u^{(lo)}(x,t,z_{0})$ with the discrete scattering
data $(z_{0},c_{0})$ such that $z=\rho e^{i\theta}$, $\rho,\theta>0$ with
velocity $v=-\frac{1}{\rho^{2}}<0$. Suppose
$\|R(0)\|_{L^{\infty}(\mathbb{R})}<\epsilon$ for $0\leq\epsilon\ll$ small
enough, consider the solution $u$ to the OV equation (1.1) with the initial
data

$u_{0}=u^{(lo)}(x,0,z_{0},c_{0})+R(0)$ (1.10)

Then there exist $z_{1}=\rho_{1}e^{i\theta_{1}}$ and the norming constant
$c_{1}$ such that

$|z_{1}-z_{0}|+|c_{1}-c_{0}|\lesssim\epsilon.$ (1.11)

Then, we obtain that

$u=u^{(lo)}(x,t)+R(x,t),$ (1.12)

where the loop solitons $u^{(lo)}(x,t)$ is given by the scattering data
$\\{r(z),\\{z_{1},c_{1}\\}\\}$ and the radiation term $R(x,t)$ has the
following asymptotics:

  1. 1.

In the soliton region,

$|R(x,t)|\lesssim t^{-n},\quad n>1.$ (1.13)

  2. 2.

In the modulated oscillations region,

$R(x,t)=u_{s}(x,t)+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$

where

$u_{s}(x,t)=\frac{k_{1}}{\sqrt{t}}\cos(\frac{2\sqrt{3}}{\varkappa}t+\nu\log
t+k_{2})-\frac{\delta(\omega^{2}z)-\delta(\omega z)}{2z\delta(\omega
z)\varkappa^{2}ct^{3/2}}+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$

and

$\displaystyle
k_{1}=-2^{3/2}3^{1/4}\sqrt{\frac{\nu}{\varkappa}}\sin\Big{(}\frac{\arg
r(\varkappa)-\arg r(-\varkappa)}{2}-\frac{2\pi}{3}\Big{)},$ $\displaystyle
k_{2}=\nu\log\frac{8\sqrt{3}}{\varkappa}+\frac{\arg r(\varkappa)+\arg
r(-\varkappa)}{2}+\arg\Gamma(-i\nu)+\frac{\pi}{4}$
$\displaystyle\qquad+\frac{3\varkappa}{\pi}\int_{\varkappa}^{\infty}\frac{\log(1-|r(s)|^{2})}{s^{2}}ds+\frac{1}{\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\log|\varkappa-s|d\log(1-|r(s)|^{2})$
$\displaystyle\qquad+\frac{1}{2\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\frac{\log(1-|r(s)|^{2})(2s+\varkappa)}{s^{2}+s\varkappa+\varkappa^{2}}$

where $\Gamma$ is the Euler Gamma function and $\nu$ is define in (3.3).

###  1.3 Notations

With regard to complex variables, given a variable $z$ or a function $f(z)$,
we denote by $\bar{z}$ and $\bar{f}(z)$ their respective complex conjugates.

The symbol $\bar{\partial}$ denoted the derivative with respect to $\bar{z}$,
i.e. if $z=x+iy$, then

$\bar{\partial}f=\frac{1}{2}(f_{x}+if_{y}).$

Let $C^{\pm}$ are the Cauchy projections:

$(C^{\pm}f)(z)=\underset{z\to\Sigma_{\pm}}{\lim}\frac{1}{2\pi
i}\int_{\Sigma}\frac{f(s)}{s-z}ds.$

Here $+(-)$ denotes taking the limit from the positive (negative) side of the
oriented contour.

As usual, $A:=B$ is definition of $A$ by means of the expression $B$. For
positive quantities $a$ and $b$, we write $a\lesssim b$ for $a\leq Cb$ where
$C$ is some prescribed constant. Throughout, we use
$u_{t}:=\frac{\partial}{\partial_{t}}u$,
$u_{x}:=\frac{\partial}{\partial_{x}}u$.

###  1.4 Outline of this paper

The paper is organized as follows. In Section 2, we report the results on the
analyticity, sysmmetries and asymptotics for the Jost functions and scattering
data via the inverse scattering transform for the spectral problem (2.1). Then
we set up the original RH problem associated with the Cauchy problem
(1.1)-(1.2) based on scattering data. In Section 3, we analyze long time
asymptotic behavior in region $x/t<\epsilon$ via the $\bar{\partial}$
analysis. We construct a global model solution which captures the leading
order asymptotic behavior of the solution, then removing this component of the
solution results in a small-norm $\bar{\partial}$ problem. In section 4 we
analyze long time asymptotic behavior in region $x/t>\epsilon$ via the
steepest descent method.

##  2 Direct and inverse scattering transforms

###  2.1 Lax pair

The equation (1.1) is integrable and admits Lax pair [22]

$\displaystyle\psi_{xxx}=\lambda(-u_{xx}+\kappa)\psi,$ (2.1a)
$\displaystyle\psi_{t}=\frac{1}{\lambda}\psi_{xx}-u\psi_{x}+u_{x}\psi,$ (2.1b)

where $\psi\equiv\psi(t,x;\lambda)$, let $\Phi\equiv\Phi(x,t,z)$ be the
matrix-valued function defined by
$\Phi=\begin{pmatrix}\psi,\psi_{x},\psi_{xx}\end{pmatrix}^{T}$, then the Lax
pair (2.1) can be written in the matrix form

$\displaystyle\Phi_{x}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\
z^{3}q^{3}&0&0\end{pmatrix}\Phi,$ (2.2a)
$\displaystyle\Phi_{t}=\begin{pmatrix}u_{x}&1&0\\\ 0&0&1\\\
z^{3}q^{3}&0&0\end{pmatrix}\Phi,$ (2.2b)

where $q=(-u_{xx}+1)^{1/3}>0$ and $\lambda=z^{3}$. Introduce

$\displaystyle D(x,t)=\begin{pmatrix}q^{-1}(x,t)&0&0\\\ 0&1&1\\\
0&0&q(x,t)\end{pmatrix},$ (2.3a) $\displaystyle P(z)=\begin{pmatrix}1&1&1\\\
\lambda_{1}&\lambda_{2}&\lambda_{3}\\\
\lambda_{1}^{2}&\lambda_{2}^{2}&\lambda_{3}^{2}\\\
\end{pmatrix}\equiv\begin{pmatrix}1&0&0\\\ 0&z&0\\\ 0&0&z^{2}\\\
\end{pmatrix}\begin{pmatrix}1&1&1\\\ \omega&\omega^{2}&1\\\
\omega^{2}&\omega&1\\\ \end{pmatrix},$ (2.3b)

where $\omega=e^{2\pi i/3}$, $\lambda_{j}(z)=z\omega^{j},j=1,2,3$.

###  2.2 Jost functions for large $\lambda$

Setting

$\tilde{\Phi}=P^{-1}D^{-1}\Phi,$ (2.4)

this leads (2.2) to another Lax pair

$\displaystyle\tilde{\Phi}_{x}-q\Lambda\tilde{\Phi}=U\tilde{\Phi},$ (2.5a)
$\displaystyle\tilde{\Phi}_{t}+(uq\Lambda(z)-\Lambda^{-1}(z))\tilde{\Phi}=V\tilde{\Phi},$
(2.5b)

where
$\tilde{\Phi}\equiv\begin{pmatrix}\tilde{\Phi}_{1}&\tilde{\Phi}_{2}&\tilde{\Phi}_{3}\end{pmatrix}^{T}$
with $\tilde{\Phi}_{j}=\tilde{\Phi}_{j}(x,t,z)$, $j=1,2,3$ and

$\displaystyle\Lambda=\begin{pmatrix}\lambda_{1}(z)&0&0\\\ 0&\lambda_{2}&0\\\
0&0&\lambda_{3}(z)\end{pmatrix}=z\begin{pmatrix}\omega&0&0\\\
0&\omega^{2}&0\\\ 0&0&1\end{pmatrix}=z\hat{\Lambda},$ $\displaystyle
U=\frac{q_{x}}{3q}\begin{pmatrix}0&1-\omega^{2}&1-\omega\\\
1-\omega&0&1-\omega^{2}\\\ 1-\omega^{2}&1-\omega&0\end{pmatrix},$
$\displaystyle
V=-uU+\frac{1}{3z}\Bigg{\\{}3\bigg{(}\frac{1}{q}-1\bigg{)}I+\bigg{(}q^{2}-\frac{1}{q}\bigg{)}\begin{pmatrix}1&1&1\\\
1&1&1\\\ 1&1&1\end{pmatrix}\Bigg{\\}}\begin{pmatrix}\omega^{2}&0&0\\\
0&\omega&0\\\ 0&0&1\end{pmatrix},$

where $\hat{\Lambda}=\mbox{diag}\\{\omega,\omega^{2},1\\}$ and $I$ is the
3$\times$3 identity matrix. The transformation (2.4) makes $U$ and $V$ are
bounded at $z=\infty$ which is appropriate for controlling the behavior of its
solutions for large $z$.

For the purpose of considering the eigenfunctions at $z=\infty$, introducing
the new variable

$y(x,t):=x-\int_{x}^{\infty}(q(\xi,t)-1)d\xi,$ (2.7)

by observing the formula (2.5), one defines $Q$ by

$Q(x,t,z)=y(x,t)\Lambda(z)+t\Lambda^{-1}(z),$ (2.8)

and introduce the 3$\times$3 matrix $M\equiv M(x,t,z)$ by

$M=\tilde{\Phi}e^{-Q}.$ (2.9)

Then (2.5) reduces to the system

$\displaystyle M_{x}-[Q_{x},M]=UM,$ (2.10a) $\displaystyle
M_{t}-[Q_{t},M]=VM,$ (2.10b)

whose solutions can be constructed as the Fredholm integral equation whose
exponential factors are bounded are determined by the signs of the difference
$\mbox{Re}\lambda_{j}(z)-\mbox{Re}\lambda_{l}(z)$, $1\leq j\neq l\leq 3$.
Namely, defining

$\infty_{jl}=\begin{cases}+\infty,\quad\mbox{Re}\lambda_{j}(z)\geq\mbox{Re}\lambda_{l}(z),\\\
-\infty,\quad\mbox{Re}\lambda_{j}(z)<\mbox{Re}\lambda_{l}(z),\end{cases}$
(2.11)

and introduce the set

$\Sigma=\\{z\,|\,\mbox{Re}\,\lambda_{j}(z)=\mbox{Re}\,\lambda_{l}(z)\,\,\mbox{for
some}\,\,j\neq l\\},$ (2.12)

that consists of six rays

$l_{v}=\mathbb{R}_{+}e^{\frac{\pi i}{3}(\nu-1)},\quad\nu=1,\dots,6,$ (2.13)

dividing the $z$-plane into six sectors,

$\Omega_{\mu}=\Big{\\{}z\,|\,\frac{\pi}{3}(\nu-1)<\mbox{arg}z<\frac{\pi}{3}\nu\Big{\\}},\quad\nu=1,\dots,6,$
(2.14)

and the matrix function $M(x,t;z)$ has to be understood as a collection of
scalar Fredholm integral equation

$M_{jl}(x,t;z)=I_{jl}+\int_{\infty_{jl}}^{x}e^{-\lambda_{j}(z)\int_{x}^{\xi}q(\zeta,t)d\zeta}[(UM)_{jl}(\xi,t;z)]e^{\lambda_{l}(z)\int_{x}^{\xi}q(\zeta,t)d\zeta}.$
(2.15)

which provide the boundedness as $z\to\infty$. It was shown that the
eigenfunction $M(z)$ defined by (2.9) has the following properties [16].

######  Proposition 2.1.

The equation (2.15) uniquely define 3$\times$3-matrix valued solution
$M(z)=M(x,t,z)$ of (2.10) with the following properties:

  * 1.

$\det M(z)\equiv 1$.

  * 2.

For spectral parameter $z$, the function $M(z)$ is piecewise meromorphic with
respect to $\Sigma$.

  * 3.

$M(x,t,z)$ satisfies the symmetry relations:

    * $\blacktriangleright$

(S1) $\Gamma_{1}\overline{M(x,t,\bar{z})}\Gamma_{1}=M(x,t,z)$ with
$\Gamma_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}$.

    * $\blacktriangleright$

(S2) $\Gamma_{2}\overline{M(x,t,\bar{z}\omega^{2})}\Gamma_{2}=M(x,t,z)$ with
$\Gamma_{2}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix}$.

    * $\blacktriangleright$

(S3) $\Gamma_{3}\overline{M(x,t,\bar{z}\omega)}\Gamma_{3}=M(x,t,z)$ with
$\Gamma_{3}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}=\Gamma_{2}\Gamma_{1}\Gamma_{2}$.

    * $\blacktriangleright$

(S4) $M(x,t,z\omega)=C^{-1}M(x,t,z)C$ with $C=\begin{pmatrix}0&0&1\\\ 1&0&0\\\
0&1&0\end{pmatrix}=\Gamma_{2}\Gamma_{1}.$

  * 4.

According to [23], $M(z)$ is at most a finite number of simple poles lying in
$\Omega_{\nu},\nu=1,\dots,6$. The poles may lie only on the lines $\rho
e^{\frac{i\pi}{6}+\frac{i\pi m}{3}},\,m=0,\dots,5$ for $\rho>0$ as shown in
Fig. 2.1 we denote them as

$\displaystyle\mathcal{Z}=\\{z_{j},\bar{z}_{j},\omega
z_{j},\omega\bar{z}_{j},\omega^{2}z_{j},\omega^{2}\bar{z}_{j}\\}_{j=1}^{N}.$

which mean for $1\leq j\leq N$, the simplest case involves 6 poles and it does
not depend on $(x,t)$.

  * 5.

$M(x,t,z)$ is bounded as a function of $x$, for all fixed $t$ and
$z\in\mathbb{C}$, $z\notin\Sigma\cup\mathcal{Z}$. Moreover, $M(x,t,z)\to I$ as
$x\to\infty$.

  * 6.

$M(x,t,z)\to I$ as $z\to\infty$.

$l_{1}$$l_{5}$$l_{3}$$l_{2}$$l_{4}$$l_{6}$$z_{j}$$\bullet$$\bullet$$\omega
z_{j}$$\bullet$$\omega^{2}\bar{z}_{j}$$\bullet$$\bar{z}_{j}$$\bullet$$\bullet$$\bullet$$\omega^{2}z_{j}$$\omega\bar{z}_{j}$$\Omega_{1}$$\Omega_{3}$$\Omega_{4}$$\Omega_{6}$$\Omega_{2}$$\Omega_{5}$
Figure 2.1: Distribution of the discrete spectrum $\mathcal{Z}$.

###  2.3 Jost functions for small $\lambda$

Setting

$\hat{\Phi}=P^{-1}\Phi,$ (2.16)

this leads (2.2) to another Lax pair

$\displaystyle\hat{\Phi}_{x}-\Lambda(z)\hat{\Phi}=\hat{U}\hat{\Phi},$ (2.17a)
$\displaystyle\hat{\Phi}_{t}-\Lambda^{-1}(z))\hat{\Phi}=\hat{V}\hat{\Phi},$
(2.17b)

where
$\hat{\Phi}\equiv\begin{pmatrix}\hat{\Phi}_{1}&\hat{\Phi}_{2}&\hat{\Phi}_{3}\end{pmatrix}^{T}$
with $\hat{\Phi}_{j}=\hat{\Phi}_{j}(x,t,z)$, $j=1,2,3$ and

$\displaystyle\hat{U}=-\frac{zu_{xx}}{3}\begin{pmatrix}w&0&0\\\ 0&w^{2}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&1&1\\\ 1&1&1\\\ 1&1&1\end{pmatrix},$
$\displaystyle\hat{V}=\frac{u_{x}}{3}\begin{pmatrix}0&1-w^{2}&1-w\\\
1-w&0&1-w^{2}\\\ 1-w^{2}&1-w&0\end{pmatrix}-zu\begin{pmatrix}w&0&0\\\
0&w^{2}&0\\\
0&0&1\end{pmatrix}\Bigg{\\{}I-\frac{u_{xx}}{3}\begin{pmatrix}1&1&1\\\ 1&1&1\\\
1&1&1\end{pmatrix}\Bigg{\\}}.$

The transformation (2.7) makes $\hat{U}$ and $\hat{V}$ are bounded at $z=0$
which is appropriate for controlling the behavior of its solutions as $z\to
0$. One defines $\hat{Q}$ by

$\hat{Q}=e^{x\Lambda+t\Lambda^{-1}},$ (2.19)

and introduce the 3$\times$3 matrix $\hat{M}\equiv\hat{M}(x,t;z)$ by

$\hat{M}=\hat{\Phi}e^{-\hat{Q}}.$ (2.20)

Then (2.5) reduces to the system

$\displaystyle\hat{M}_{x}-[\hat{Q}_{x},\hat{M}]=\hat{U}\hat{M},$ (2.21a)
$\displaystyle\hat{M}_{t}-[\hat{Q}_{t},\hat{M}]=\hat{V}\hat{M},$ (2.21b)

and the matrix function $\hat{M}(x,t;z)$ has to be understood as a collection
of scalar Fredholm integral equation

$\hat{M}{jl}(x,t;z)=I_{il}-\frac{z}{3}\int_{\infty_{jl}}^{x}e^{\lambda_{j}(z)(x-\xi)}[(u_{xx}\Omega\hat{M})_{jl}(\xi,t;z)]e^{-\lambda_{l}(z)(x-\xi)}d\xi.$
(2.22)

where $\infty_{jl}$ is defined in (2.11) and
$\Omega=\begin{pmatrix}\omega&\omega&\omega\\\
\omega^{2}&\omega^{2}&\omega^{2}\\\ 1&1&1\end{pmatrix}.$ Similarly to
Proposition 2.1, equation (2.22) determines a piecewise meromorphic,
3$\times$3 matrix-valued function $\hat{M}$. Moreover, the particular on $z$
in (2.22) implies a particular form of the first coefficients in the expansion
of $\hat{M}$ as $z\to 0$.

######  Proposition 2.2.

The equation (2.20) uniquely define 3$\times$3-matrix valued solution
$M(z)=M(x,t,z)$ of (2.21) with the following properties:

  * 1.

$\det M(z)\equiv 1$.

  * 2.

For spectral parameter $z$, the function $\hat{M}(z)$ is piecewise meromorphic
with respect to $\Sigma$.

  * 3.

$\hat{M}(x,t,z)$ satisfies the symmetry relations:

    * $\blacktriangleright$

(S1)$\Gamma_{1}\overline{\hat{M}(x,t,\bar{z})}\Gamma_{1}=\hat{M}(x,t,z)$ with
$\Gamma_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\ 0&0&1\end{pmatrix}$.

    * $\blacktriangleright$

(S2)$\Gamma_{2}\overline{\hat{M}(x,t,\bar{z}\omega^{2})}\Gamma_{2}=\hat{M}(x,t,z)$
with $\Gamma_{2}=\begin{pmatrix}0&0&1\\\ 0&1&0\\\ 1&0&0\end{pmatrix}$.

    * $\blacktriangleright$

(S3)$\Gamma_{3}\overline{\hat{M}(x,t,\bar{z}\omega)}\Gamma_{3}=\hat{M}(x,t,z)$
with $\Gamma_{3}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}=\Gamma_{2}\Gamma_{1}\Gamma_{2}$.

    * $\blacktriangleright$

(S4)$\hat{M}(x,t,z\omega)=C^{-1}\hat{M}(x,t,z)C$ with
$C=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix}=\Gamma_{2}\Gamma_{1}.$

  * 4.

$\hat{M}(x,t,z)$ is bounded as a function of $x$, for all fixed $t$ and
$z\in\mathbb{C}$, $z\notin\Sigma\cup\mathcal{Z}$, the set of poles of
$\hat{M}(z)$ is $\mathcal{Z}_{0}$. Moreover, $M(x,t,z)\to I$ as $x\to\infty$.

  * 5.

$M(x,t,z)\to I$ as $z\to 0$. Moreover,

$\hat{M}(x,t,z)=I+\hat{M}^{(1)}(x,t)z+M^{(2)}(x,t)z^{2}+\mathcal{O}(z^{3})\quad
z\to 0,$ (2.23)

where

$\hat{M}^{(1)}=-\frac{1}{3}u_{x}\Omega,\quad\hat{M}^{(2)}=-\frac{1}{3}u\tilde{\Omega},$

with $\tilde{\Omega}=\hat{\Lambda}\Omega-\Omega\hat{\Lambda}$ and
$\hat{\Lambda}=diag\\{\omega,\omega^{1},1\\}$.

Tracing back the linear system of PDEs (2.1), we notice that $M$ and $\hat{M}$
are related in the following way:

$M(x,t,z)=G(x,t)\hat{M}(x,t,z)e^{N(x,t)z\hat{\Lambda}},$ (2.24)

where

$F(x,t):=x-y(x,t)=\int_{0}^{+\infty}(q(\xi,t)-1)d\xi,$ (2.25)

and

$G(x,t):=P^{-1}(z)D^{-1}(x,t)P(z)=\begin{pmatrix}\alpha&\beta&\gamma\\\
\gamma&\alpha&\beta\\\ \beta&\gamma&\alpha\end{pmatrix},$

where $P$ and $D$ are as in (2.3) and with

$\alpha=\bar{\alpha}=\frac{1}{3}\Big{(}q+1+\frac{1}{q}\Big{)},\qquad\beta=\bar{\gamma}=\frac{1}{3}\Big{(}q+\omega+\frac{\omega^{2}}{q}\Big{)}.$
(2.26)

######  Remark 2.3.

If $z=0$, then

$P^{-1}=\frac{1}{3}\begin{pmatrix}1&\omega^{2}&\omega\\\
1&\omega&\omega^{2}\\\ 1&1&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&z^{-1}&0\\\
0&0&z^{-2}\end{pmatrix},$

becomes degenerate, the factor $G$ is non-singular. Moreover, it is
independent of $z$. From (2.23) and (2.26) we derive the following expansion
of $M$ as $z\to 0$:

$\displaystyle M=$ $\displaystyle
G(x,t)\Big{(}I+z\big{\\{}-\frac{u_{x}}{3}\Omega+N\hat{\Lambda}\big{\\}}$
$\displaystyle+z^{2}\big{\\{}-\frac{u}{3}\tilde{\Omega}-\frac{u_{x}}{3}N\omega\hat{\Lambda}+\frac{N^{2}}{2}\hat{\Lambda}^{2}\big{\\}}+\mathcal{O}(z^{3})\Big{)}.$
(2.27)

###  2.4 Scattering data

We consider $z$ on the common boundary of two adjacent domains $\Omega_{\nu}$,
the limiting values of $\tilde{\Phi}=Me^{Q}$ being the solutions of the system
of differential equation (2.5) must be related by a matrix independent of
$(x,t)$. Denote $M_{\pm}(x,t,z)$ as the limiting values of $M(x,t,z^{\prime})$
as $z^{\prime}\to z$ from the positive or negative side of $l_{n}$, then they
are related as follows

$M_{+}(x,t,z)=M_{-}(x,t,z)S(x,t,z),\quad z\in\Sigma,$ (2.28)

where

$S(x,t,z)=e^{Q(x,t,z)}S_{0}e^{-Q(x,t,z)},$ (2.29)

with some matrix $S_{0}$ independent of $(x,t)$ and $\Sigma=l_{1}\cup\dots\cup
l_{6}$ as in (2.12).

The matrix $S_{0}$ has a particular structure[26] and only depends on $z$ and
is completely determined by the initial data $u_{0}$ for the Cauchy problem
(1.1). For example, $z\in l_{1}=\mathbb{R}_{+}$ we have $S_{0}$ have a special
matrix structure

$S_{0}=\begin{pmatrix}1&r_{+}(z)&0\\\ -r_{-}(,z)&1-r_{-}r_{+}(z)&0\\\
0&0&1\end{pmatrix},$ (2.30)

where $r_{\pm}(z)\in L^{\infty}(\mathbb{R}^{\pm})$, and
$r_{\pm}(z)=\mathcal{O}(z^{-1})$ as $z\to\pm\infty$. Using similar arguments
we get exactly the same structure for $z\in l_{4}=\mathbb{R}_{-}$. Now, by
symmetry (S1) from Proposition 2.1, we get that
$\Gamma_{1}\overline{S_{0}(z)}\Gamma_{1}=S_{0}^{-1}(z)$ and thus
$r_{-}(z)=\overline{r_{+}(z)}$, which we acquire $z\in\mathbb{R}=l_{1}\cup
l_{4}$,

$S_{0}=\begin{pmatrix}1&\bar{r}(z)&0\\\ -r(z)&1-|r(z)|^{2}&0\\\
0&0&1\end{pmatrix},$ (2.31)

Here $r(z):=r_{-}(z)$ lies in Schwartz space [27, 28]. And as $z\to 0$, we
have $r(z)=\mathcal{O}(z^{3}),z\in\mathbb{R}$.

In order to finish our following work, we assume our initial data satisfies
that $u_{0}\in\mathcal{S}(\mathbb{R})$ to generic scattering data such that
$M(z)$ at poles lie only on the lines $\rho e^{\frac{i\pi}{6}+\frac{i\pi
m}{3}},\,m=0,\dots,5$ for $\rho>0$.

For initial data $u_{0}\in X_{0}$, the collection
$\mathcal{D}=\\{r(z),\\{z_{k},c_{k}\\}_{k=1}^{6N}\\}$ is called the scattering
data for $u_{0}(x)$ and map $\mathcal{S}:u_{0}\mapsto\mathcal{D}$ is called
the forward scattering map. The essential fact of integrability is that if the
potential $u_{0}(x)$ evolves according to (1.1) then the evolution of the
scattering data $\mathcal{D}$ is trivial

$\mathcal{D}(t)=\\{r(z,t),\\{z_{k}(t),c_{k}(t)\\}_{k=1}^{6N}\\}=\\{r(z)e^{t\Lambda^{-1}},\\{z_{k},c_{k}(t)e^{t\Lambda^{-1}}\\}_{k=1}^{6N}\\}.$

The inverse scattering map $\mathcal{S}^{-1}:\mathcal{D}(t)\mapsto u(x,t)$
seeks to recover the solution of (1.1) from its scattering data.

In what follows, we will assume that $r(z)$ has an analytic extension to a
small neighborhood of the real axis. This is , for example, the case if we
assume that the solution is exponentially decaying as $|x|\to\infty$.
Otherwise one can split $r(z)$ into an analytic part plus a reminder producing
a polynomially decaying in $t$ error term, the decay depending on the rate of
decay of the initial condition $u_{0}(x)$ as $|x|\to\infty$.

###  2.5 Set up of a RH problem

The dependence of $S(x,t,z)$ on the parameters $(x,t)$ justifies the use of
the variable $y=y(x,t)$ in (2.7). The price to pay for this is that the
solution of the initial problem can be given only implicitly, or
perimetrically. It will be given in terms of functions in the new scale,
whereas the original scale will also be given in terms of functions in the new
scale. Indeed, introducing

$M_{\Diamond}(y,t,z):=M(x(y,t),t,z),$ (2.32)

(2.7) can be written in terms of the parameters $(y,t)$ as

$M_{\Diamond+}(y,t,z)=M_{\Diamond-}(y,t,z)S_{\Diamond}(y,t,z),\quad
z\in\Sigma,$ (2.33)

where the jump matrix

$S_{\Diamond}(y,t,z)=e^{y\Lambda(z)+t\Lambda^{-1}(z)}S_{0}(z)e^{-y\Lambda(z)-t\Lambda^{-1}(z)},$

is determined in terms of the initial condition $u(x,0)=u_{0}(x)$ and depends
explicitly on the parameters $(y,t)$.

######  RHP 2.1.

Find a analytic function
$M_{\Diamond}(y,t;z):\mathbb{C}\setminus(\Sigma\cup\mathcal{Z})\to
SL_{3}(\mathbb{C})$ with the following properties:

$\bigstar$_(the normalization condition)_

$M_{\Diamond}(y,t,z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

For each
$z\in\Sigma=\mathbb{R}\,\cup\,\omega\mathbb{R}\,\cup\,\omega^{2}\mathbb{R}$,
the boundary values $M_{\Diamond\pm}$ satisfy the jump relation
$M_{\Diamond+}(y,t;z)=M_{\Diamond-}(y,t;z)S_{\Diamond}(y,t;z),\,\,y\in\Sigma$
where the jump matrix

$S_{\Diamond}(y,t,z)=e^{y\Lambda(z)+t\Lambda^{-1}(z)}S_{0}(z)e^{-y\Lambda(z)-t\Lambda^{-1}(z)},$
(2.34) $\displaystyle S_{\Diamond}(y,t,z)$
$\displaystyle=\begin{pmatrix}1&\bar{r}(z)e^{-2it\theta(\zeta,z)}&0\\\
-r(z)e^{2it\theta(\zeta,z)}&1-|r(z)|^{2}&0\\\
0&0&1\end{pmatrix},\qquad\qquad\,\,\,\,\,z\in\mathbb{R},$
$\displaystyle=\begin{pmatrix}1-|r(\omega^{2}z)|^{2}&0&-r(\omega^{2}z)e^{2it\theta(\zeta,\omega^{2}z)}\\\
0&1&0\\\
\bar{r}(\omega^{2}z)e^{-2it\theta(\zeta,\omega^{2}z)}&0&1\end{pmatrix},\quad
z\in\omega\mathbb{R},$ $\displaystyle=\begin{pmatrix}1&0&0\\\
0&1&\bar{r}(\omega z)e^{-2it\theta(\zeta,\omega z)}\\\ 0&-r(\omega
z)e^{2it\theta(\zeta,\omega z)}&1-|r(\omega
z)|^{2}\end{pmatrix},\qquad\,\,\,\,z\in\omega^{2}\mathbb{R},$

$\bigstar$_(Singularities)_

As $z\to 0$, the limit of $M(z)$ has pole singularities:

$M_{\Diamond}(y,t,0)=\begin{pmatrix}\alpha&\beta&\gamma\\\
\gamma&\alpha&\beta\\\ \beta&\gamma&\alpha\end{pmatrix},$ (2.35)

where $\alpha(y,t)$ and $\beta(y,t)$ have the form

$\alpha=\frac{1}{3}\big{(}\hat{q}+1+\frac{1}{\hat{q}}\big{)},\quad\beta=\frac{1}{3}\big{(}\hat{q}+\omega+\frac{\omega^{2}}{\hat{q}}\big{)},$

with some $\hat{q}(y,t)>0$ such that $\hat{q}\to 1$ as $y\to\infty$.

$\bigstar$_(the residue condition)_

$M_{\Diamond}(y,t,z)$ has simple poles at $\mathcal{Z}$ in $\Omega_{1}$ at
which

$\displaystyle\underset{z=z_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to
z_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&0&0\\\ 0&0&0\\\
0&c_{j}e^{2it\theta(\zeta,z_{j})}&0\\\ \end{pmatrix},$
$\displaystyle\underset{z=\bar{z}_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to\bar{z}_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&0&0\\\
0&0&0\\\ \bar{c}_{j}e^{-2it\theta(\zeta,\bar{z}_{j})}&0&0\\\ \\\
\end{pmatrix},$ $\displaystyle\underset{z=\omega
z_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to\omega
z_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&0&c_{j}e^{2it\theta(\zeta,\omega
z_{j})}\\\ 0&0&0\\\ 0&0&0\\\ \end{pmatrix},$
$\displaystyle\underset{z=\omega\bar{z}_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to\omega\bar{z}_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&0&0\\\
0&0&\bar{c}_{j}e^{-2it\theta(\zeta,\omega\bar{z}_{j})}\\\ \ 0&0&0\\\
\end{pmatrix},$
$\displaystyle\underset{z=\omega^{2}z_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to\omega^{2}z_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&0&0\\\
c_{j}e^{2it\theta(\zeta,\omega^{2}z_{j})}&0&0\\\ 0&0&0\\\ \end{pmatrix},$
$\displaystyle\underset{z=\omega^{2}\bar{z}_{j}}{Res}M_{\Diamond}(y,t,z)=\underset{z\to\omega^{2}\bar{z}_{j}}{lim}M_{\Diamond}(y,t,z)\begin{pmatrix}0&\bar{c}_{j}e^{-2it\theta(\zeta,\omega^{2}\bar{z}_{j})}&0\\\
0&0&0\\\ 0&0&0\\\ \end{pmatrix}.$

######  Remark 2.4.

The exist and uniqueness of the RH problem 2.1 is depend on the $\alpha$ and
$\beta$ as in (2.24), that constitutes an additional condition that must be
imposed on the solution of the RH problem 2.1. This is different from that in
the case of classical integrable equation, on the other hand, it is typical
for the so-called peakon equations, like the CH and DP equation[25, 26].

The solution $u(x,t)$ of OV equation (1.1) can be expressed $z\to 0$ in terms
of the solution $M_{\Diamond}(y,t,z)$ of the matrix RH problem 2.1 and the
expression is given in a parametric form:

$\displaystyle u(x,t)=\hat{u}(y(x,t),t),$ (2.36)

where

$\displaystyle x(y,t)=y+\underset{z\to
0}{\lim}\Big{(}\frac{\sum_{j=1}^{3}M_{\Diamond
j3}(y,t,z)}{\sum_{j=1}^{3}M_{\Diamond j3}(y,t,0)}-1\Big{)}\frac{1}{z},$
$\displaystyle\hat{u}(y,t)=\frac{\partial x(y,t)}{\partial t}.$

###  2.6 Stationary saddle points and decay domains

In this section we first consider the structure of the jump matrix
$S_{\Diamond}(y,t,z)$. By (2.34) its $(j,l)$ entry can be written

$S_{\Diamond}(y,t,z)_{jl}=e^{y(\lambda_{j}-\lambda_{l})+t(\lambda_{j}^{-1}-\lambda_{l}^{-1})}S_{0}(z)_{jl}.$

The exponential factor which is trivial for $j=l$ can otherwise be written as
$e^{t(\zeta\nu+\frac{3}{\nu})}$, where $\zeta=\frac{y}{t}$ and
$\nu=\lambda_{j}(z)-\lambda_{l}(z)=z(\omega^{j}-\omega^{l})$, Since
$(\lambda_{j}(z)-\lambda_{l}(z))^{2}=-3\lambda_{j}(z)\lambda_{l}(z)$ for
$j\neq l$ the exponential in (2.34) can be written as

$e^{y(\lambda_{j}-\lambda_{l})+t(\lambda_{j}^{-1}-\lambda_{l}^{-1})}=e^{y\nu+t\frac{3}{\nu}},$
(2.38)

For $1\leq j,l\leq 3$, we have

$\displaystyle\theta_{jl}(z)$
$\displaystyle=\Big{[}\zeta(\lambda_{j}-\lambda_{l})+\Big{(}\frac{1}{\lambda_{j}}-\frac{1}{\lambda_{l}}\Big{)}\Big{]}=(\zeta\nu+\frac{3}{\nu}),$

and $\nu=\lambda_{j}(z)-\lambda_{l}(z)=z(\omega^{j}-\omega^{l})$. In
particular, we have

$\displaystyle\lambda_{1}(z)-\lambda_{2}(z)=z(\omega-\omega^{2})=zi\sqrt{3},$
$\displaystyle\lambda_{1}(z)-\lambda_{3}(z)=z(\omega-1)=-z\sqrt{3}e^{-i\pi/6},$
$\displaystyle\lambda_{3}(z)-\lambda_{2}(z)=z(1-\omega^{2})=z\sqrt{3}e^{i\pi/6},$

that is for the $(1,2)$ entry one has $\nu=i\sqrt{3}z$ and thus

$e^{t\theta_{12}}=e^{it\sqrt{3}(\zeta
z-\frac{1}{z})}=e^{-2it\theta(\zeta,z)},$

where

$\theta(\zeta,z)=-\frac{\sqrt{3}}{2}\Big{(}\zeta z-\frac{1}{z}\Big{)},$ (2.39)

and we also obtain that for the $(1,3)$ entry one have
$-\theta(\zeta,\omega^{2}z)$ and for $(3,2)$ entry one have
$\theta(\zeta,\omega z)$.

And we have

$\frac{\partial}{\partial
z}\theta(\zeta,z)=-\frac{\sqrt{3}}{2}\big{(}\zeta+\frac{1}{z^{2}}\big{)}=0.$
(2.40)

  1. 1.

For the range $\frac{y}{t}>\epsilon$, there is no stationary point on
$\mathbb{R}$.

  2. 2.

For the range $\frac{y}{t}<-\epsilon$, there are exist two stationary points
$\pm\varkappa$, where $\varkappa\equiv\varkappa(\zeta)=1/\sqrt{|\zeta|}$.

(a) The distribution of stationary phase points for $\zeta>0$

(b) The distribution of stationary phase points for $\zeta=0$

(c) The distribution of stationary phase points for $-2\leq\zeta<0$

(d) The distribution of stationary phase points for $\zeta<-2$

(e) In the case $\zeta>0$, $e^{2it\theta(\zeta,z)}\to 0$ as $|t|\to 0$ in the
red region, $e^{-2it\theta(\zeta,z)}\to 0$ as $|t|\to 0$ in the white region

(f) In the case $\zeta<0$, $e^{2it\theta(\zeta,z)}\to 0$ as $|t|\to 0$ in the
red region, $e^{-2it\theta(\zeta,z)}\to 0$ as $|t|\to 0$ in the white region

(g) In the case $\zeta>0$, $e^{2it\theta(\zeta,\omega z)}\to 0$ as $|t|\to 0$
in the red region, $e^{-2it\theta(\zeta,\omega z)}\to 0$ as $|t|\to 0$ in the
white region

(h) In the case $\zeta<0$, $e^{2it\theta(\zeta,\omega z)}\to 0$ as $|t|\to 0$
in the red region, $e^{-2it\theta(\zeta,\omega z)}\to 0$ as $|t|\to 0$ in the
white region

(i) In the case $\zeta>0$, $e^{2it\theta(\zeta,\omega^{2}z)}\to 0$ as $|t|\to
0$ in the red region, $e^{-2it\theta(\zeta,\omega^{2}z)}\to 0$ as $|t|\to 0$
in the white region

(j) In the case $\zeta<0$, $e^{2it\theta(\zeta,\omega^{2}z)}\to 0$ as $|t|\to
0$ in the red region, $e^{-2it\theta(\zeta,\omega^{2}z)}\to 0$ as $|t|\to 0$
in the white region

##  3 Soliton resolution in region $x/t<\epsilon$

In this section, we give the soliton resolution for the $N$-loop soliton
solutions in the asymptotic region $x/t<\epsilon$.

###  3.1 Conjugation

Our analysis is to introduce a transformation which renormalizes the RH
problem 2.1 such that it is well conditioned for $t\to\infty$ with
$\pm\varkappa$ fixed. In order to arrive at a problem which is well
normalized, we introduce the a new matrix-value function

$M_{\oplus}(y,t,z)=M_{\Diamond}(y,t,0)\tilde{\delta}(z)^{-1}.$ (3.1)

######  RHP 3.1.

Given $\pm\varkappa\in\mathbb{R}$, find a scalar function
$\delta(z)=\delta(z,\varkappa)$, meromorphic for
$z\in\mathbb{C}\setminus(|z|>\varkappa)$ with the following properties:

  1. 1.

$\delta(z)\to 1$ as $z\to\infty$.

  2. 2.

$\delta(z)$ has continuous boundary values
$\delta_{\pm}(z)=\underset{\epsilon\to 0}{lim}\,\delta(z\pm i\epsilon)$ for
$z\in(|z|>\varkappa)$.

  3. 3.

$\delta_{\pm}(z)$ obey the jump relation

$\delta_{+}(z)=\begin{cases}\frac{\delta_{-}(z)}{1-|r(z)|^{2}},\quad|z|>\varkappa,\\\
\delta_{-}(z),\qquad z\in(-\varkappa,\varkappa).\end{cases}$

  4. 4.

$\delta(z)$ has simple pole at $z_{k}\in\mathcal{Z}$.

The RH problem 3.1 has the unique solution

$\delta(z)=\Big{(}\prod_{j=1}^{N}\frac{z-\omega^{2}z_{j}}{z-\omega\bar{z}_{j}}\Big{)}\Big{(}\prod_{j=1}^{N}\frac{z-\bar{z}_{k}}{z-z_{k}}\frac{z-\omega^{2}\bar{z}_{j}}{z-\omega
z_{j}}\Big{)}e^{\chi(z)},$ (3.2)

where

$\displaystyle\chi(z)=-i\Big{(}\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty}\Big{)}\frac{\nu(s)}{s-z}ds,$

where

$\nu=-\frac{1}{2\pi}\log(1-|r(s)|^{2}),$ (3.3)

and

$\displaystyle\delta_{1}(z)$
$\displaystyle=\exp\Big{\\{}i\int_{-\infty}^{-\varkappa}\frac{\nu(s)-\chi_{1}(s)\nu(-\varkappa)(s+\varkappa+1)}{s-z}ds+i\nu(-\varkappa)\int_{-\varkappa-1}^{-\varkappa}\frac{s+\varkappa+1}{s-z}ds\Big{\\}}$
$\displaystyle=i\beta_{1}(z,-\varkappa)+i\nu(-\varkappa)+i\nu(-\varkappa)[(z+\varkappa)\log(z+\varkappa)$
$\displaystyle-(z+\varkappa+1)\log(z+\varkappa+1)]+i\nu(-\varkappa)\log(z+\varkappa)$
$\displaystyle=e^{i\nu(-\varkappa)+i\beta_{1}(z,-\varkappa)}(z+\varkappa)^{i\nu(-\varkappa)}e^{-i\nu(\varkappa)[(z+\varkappa)\log(z+\varkappa)-(z+\varkappa+1)\log(z+\varkappa+1)]},$

$\chi_{1}(s)$ is the eigenfunction on the interval $(-\varkappa-1,-\varkappa)$
and

$\beta_{1}(z,-\varkappa)=\int_{-\infty}^{-\varkappa}\frac{\nu(s)-\chi_{1}(s)\nu(-\varkappa)(s+\varkappa+1)}{s-z}ds,$

and

$\displaystyle\delta_{2}(z)$
$\displaystyle=\exp\Big{\\{}i\int^{\infty}_{\varkappa}\frac{\nu(s)-\chi_{2}(s)\nu(\varkappa)(s-\varkappa-1)}{s-z}ds+i\nu(\varkappa)\int_{\varkappa}^{\varkappa+1}\frac{s-\varkappa-1}{s-z}ds\Big{\\}}$
$\displaystyle=i\beta_{2}(z,\varkappa)+i\nu(\varkappa)+i\nu(\varkappa)[(z-\varkappa-1)\log(z-\varkappa-1)$
$\displaystyle-(z-\varkappa)\log(z-\varkappa)]+i\nu(\varkappa)\log(z-\varkappa)$
$\displaystyle=e^{i\nu(\varkappa)+i\beta_{2}(z,\varkappa)}(z-\varkappa)^{i\nu(\varkappa)}e^{i\nu(\varkappa)[(z-\varkappa-1)\log(z-\varkappa-1)-(z-\varkappa)\log(z-\varkappa)]},$

$\chi_{2}(s)$ is the eigenfunction on the interval $(\varkappa,\varkappa+1)$
and

$\beta_{2}(z,\varkappa)=\int_{-\infty}^{-\varkappa}\frac{\nu(s)-\chi_{1}(s)\nu(-\varkappa)(s+\varkappa+1)}{s-z}ds.$

We obtain that

$\displaystyle e^{\chi(z)}$
$\displaystyle=e^{i\nu(-\varkappa)+i\nu(\varkappa)+i\beta_{1}(z,-\varkappa)+i\beta_{2}(z,\varkappa)}(z+\varkappa)^{i\nu(-\varkappa)}(z-\varkappa)^{i\nu(\varkappa)}$
$\displaystyle
e^{i\nu(-\varkappa)[(z+\varkappa)\log(z+\varkappa)-(z+\varkappa+1)\log(z+\varkappa+1)]+i\nu(\varkappa)[(z-\varkappa-1)\log(z-\varkappa-1)-(z-\varkappa)\log(z-\varkappa)]}.$

Here we have chosen the branch of the logarithm with $-\pi<\arg(z)<\pi$ and we
have

$\displaystyle\|\beta_{1}\|_{L^{\infty}}\leq\frac{c\|r\|_{H^{1}}}{1-\|\rho\|_{L^{\infty}}},\quad\|\beta_{2}\|_{L^{\infty}}\leq\frac{c\|r\|_{H^{1}}}{1-\|\rho\|_{L^{\infty}}},$
$\displaystyle|\beta_{1}(z,-\varkappa)-\beta_{1}(-\varkappa,-\varkappa)|\leq\frac{c\|r\|_{H^{1}}}{1-\|\rho\|_{L^{\infty}}}|z+\varkappa|^{1/2},$
$\displaystyle|\beta_{1}(z,\varkappa)-\beta_{1}(\varkappa,\varkappa)|\leq\frac{c\|r\|_{H^{1}}}{1-\|\rho\|_{L^{\infty}}}|z-\varkappa|^{1/2}.$

######  Proposition 3.1.

The function $\delta(z)$ defined by (3.2) has the following properties:

  1. 1.

For $z\in\mathbb{C}\setminus(z>|\varkappa|)$, we have
$\delta(z)=1/\overline{\delta(\bar{z})}$.

  2. 2.

For $z\in\mathbb{C}\setminus(z>|\varkappa|)$, we have $\delta(z)\leq 1$.

Then we introduce the diagonal factor $\tilde{\delta}(z)$

$\tilde{\delta}(z)=\begin{pmatrix}\delta(z)\delta^{-1}(\omega^{2}z)&0&0\\\
0&\delta(\omega z)\delta^{-1}(z)&0\\\
0&0&\delta(\omega^{2}z)\delta^{-1}(\omega z)\end{pmatrix}.$ (3.4)

The function $M_{\oplus}(y,t,z)$ defined by (3.1) satisfies the following RH
problem 3.2.

######  RHP 3.2.

Find an analytic function $M_{\oplus}(y,t,z):\mathbb{C}\setminus\Sigma\to
SL_{3}(\mathbb{C})$ with the following properties:

$\bigstar$_(the normalization condition)_

$M_{\oplus}(y,t,z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

For each $z\in\Sigma$, the boundary values $M^{\pm}_{\oplus}(y,t,z)$ satisfy
the jump relation
$M^{+}_{\oplus}(y,t,z)=M^{-}_{\oplus}(y,t,z)S^{1}_{\oplus}(y,t,z),\,\,y\in\Sigma$
where

$\displaystyle S^{1}_{\oplus}(y,t,z)$ $\displaystyle=\begin{pmatrix}1&0&0\\\
-r(z)\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta(\zeta,z)}&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&\bar{r}(z)\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta(\zeta,z)}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\qquad\qquad\qquad
z\leq|\varkappa|,$
$\displaystyle=\begin{pmatrix}1&\frac{\bar{r}(z)}{1-|r(z)|^{2}}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\
-\frac{r(z)}{1-|r(z)|^{2}}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\
0&0&1\end{pmatrix},\qquad\qquad\qquad z\geq|\varkappa|,$
$\displaystyle=\begin{pmatrix}1&0&-r(\omega^{2}z)\frac{\delta(z)\delta(\omega
z)}{\delta^{2}(\omega^{2}z)}e^{2it\theta(\zeta,\omega^{2}z)}\\\ 0&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
\bar{r}(\omega^{2}z)\frac{\delta^{2}(\omega^{2}z)}{\delta(\omega
z)\delta(z)}e^{-2it\theta(\zeta,\omega^{2}z)}&0&1\end{pmatrix},z\in\omega\mathbb{R},|z|\leq|\varkappa|,$
$\displaystyle=\begin{pmatrix}1&0&\frac{\bar{r}(\omega^{2}z)}{1-|r(\omega^{2}z)|^{2}}\frac{\delta^{2}(\omega^{2}z)}{\delta(\omega
z)\delta_{-}(z)}e^{-2it\theta(\zeta,\omega^{2}z)}&\\\ 0&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
-\frac{r(\omega^{2}z)}{1-|r(\omega^{2}z|^{2}}\frac{\delta_{-}(z)\delta(\omega
z)}{\delta^{2}(\omega^{2}z)}e^{2it\theta(\zeta,\omega^{2}z)}&0&1\end{pmatrix},z\in\omega\mathbb{R},z\geq|\varkappa|,$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&-r(\omega
z)\frac{\delta(\omega^{2}z)\delta(z)}{\delta^{2}(\omega
z)}e^{2it\theta(\zeta,\omega z)}&1\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&1&\bar{r}(\omega z)\frac{\delta^{2}(\omega
z)}{\delta_{+}(\omega^{2}z)\delta(z)}e^{-2it\theta(\zeta,\omega^{2}z)}\\\
0&0&1\end{pmatrix},z\in\omega^{2}\mathbb{R},z\leq|\varkappa|,$
$\displaystyle=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&\frac{\bar{r}(\omega
z)}{1-|r(\omega z)|^{2}}\frac{\delta^{2}(\omega
z)}{\delta(\omega^{2}z)\delta_{-}(z)}e^{-2it\theta(\zeta,\omega^{2}z)}&1\end{pmatrix}\begin{pmatrix}1&0&0\\\
0&1&-\frac{r(\omega z)}{1-|r(\omega
z)}\frac{\delta(\omega^{2}z)\delta_{+}(z)}{\delta^{2}(\omega
z)}e^{2it\theta(\zeta,\omega z)}\\\
0&0&1\end{pmatrix},z\in\omega^{2}\mathbb{R},z\geq|\varkappa|.$

$\bigstar$_(the residue condition)_

The residue condition of $M_{\oplus}(y,t,z)$ is as same as
$M_{\Diamond}(y,t,z)$ of RH problem 2.1. $\bigstar$_(Singularities)_

As $z\to 0$, the limit of $M_{\oplus}(y,t,z)$ has pole singularities as
$M_{\Diamond}(y,t,z)$.

###  3.2 A mixed $\bar{\partial}$-RH problem

In this section our introduce factorizations of the jump matrix whose factors
admit continuous-but not necessarily analytic-extensions off the real axis
following the ideas [29]. Since the phase function (2.39) has two stationary
points at $\pm\varkappa$, our new contour is $\hat{\Sigma}$ as shown in Fig.
1(a).

(a) Coutour $\hat{\Sigma}$ for $\zeta<0$.

Let $\chi_{\mathcal{Z}}\in C_{0}^{\infty}(\mathbb{C},[0,1])$ be supported near
the discrete spectrum such that

$\chi_{\mathcal{Z}}(z)=\begin{cases}1\quad\mbox{dist}(z,\mathcal{Z})\leq\rho/3,\\\
0\quad\mbox{dist}(z,\mathcal{Z})\geq 2\rho/3,\end{cases}$ (3.5)

where

$\rho=\min\\{\underset{z,z^{\prime}\in\mathcal{Z}}{\min}|z-z^{\prime}|,\mbox{dist}(\mathcal{Z},\mathbb{R})\\}.$

Without loss of generality, we only consider jumps on the real axis and other
case and other cases may be obtained by rotation.

Rez$\varkappa$$-\varkappa$$\Sigma_{1}$$\Sigma_{5}$$\Sigma_{4}$$\Sigma_{8}$$\Sigma_{3}$$\Sigma_{2}$$\Sigma_{7}$$\Sigma_{6}$$\Omega_{1}$$\Omega_{5}$$\Omega_{4}$$\Omega_{8}$$\Omega_{2}$$\Omega_{6}$$\Omega_{3}$$\Omega_{7}$
Figure 3.1: the region $\Omega_{k}$

######  Lemma 3.2.

It is possible to define functions
$R_{j}:\bar{\Omega}_{j}\to\mathbb{C},j=1,2,\dots,8$ with boundary values
satisfying

$\displaystyle
R_{1}(z)=\begin{cases}-\frac{r(z)}{1-|r(z)|^{2}},\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,z\in(-\infty,\varkappa),\\\
f_{1}=-\frac{\tilde{r}_{0}}{1-|\tilde{r}_{0}|^{2}}\frac{\delta_{+}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}(z+\varkappa)^{-2i\nu}(1-\chi_{\mathcal{Z}}(z)),\quad
z\in\Sigma_{1},\end{cases}$ $\displaystyle
R_{2}(z)=\begin{cases}\bar{r}(z),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,z\in(-\varkappa,0),\\\
f_{2}=\tilde{r}_{0}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}(z+\varkappa)^{2i\nu}(1-\chi_{\mathcal{Z}}(z))\quad\qquad\quad\,\,\,\,\,z\in\Sigma_{2},\end{cases}$
$\displaystyle
R_{3}(z)=\begin{cases}\bar{r}(z),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\,z\in(0,\varkappa),\\\
f_{3}=\bar{r}_{0}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}(z-\varkappa)^{2i\nu}(1-\chi_{\mathcal{Z}}(z))\quad\qquad\quad\,\,\,\,\,z\in\Sigma_{3},\end{cases}$
$\displaystyle
R_{4}(z)=\begin{cases}-\frac{r(z)}{1-|r(z)|^{2}},\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,z\in(\varkappa,+\infty),\\\
f_{4}=-\frac{\hat{r}_{0}}{1-|\hat{r}_{0}|^{2}}\frac{\delta_{+}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}(z-\varkappa)^{-2i\nu}(1-\chi_{\mathcal{Z}}(z))\quad\,\,\,z\in\Sigma_{4},\end{cases}$
$\displaystyle
R_{5}(z)=\begin{cases}\frac{\bar{r}(z)}{1-|r(z)|^{2}},\quad\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,z\in(-\infty,\varkappa),\\\
f_{5}=\frac{\tilde{r}_{0}}{1-|\tilde{r}_{0}|^{2}}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{-}^{2}(z)}(z+\varkappa)^{2i\nu}(1-\chi_{\mathcal{Z}}(z))\qquad\quad
z\in\Sigma_{5},\end{cases}$ $\displaystyle
R_{6}(z)=\begin{cases}-r(z),\quad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,z\in(0,\varkappa),\\\
f_{6}=-\tilde{r}_{0}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}(z+\varkappa)^{-2i\nu}(1-\chi_{\mathcal{Z}}(z))\qquad\quad\,\,\,\,z\in\Sigma_{6},\end{cases}$
$\displaystyle
R_{7}(z)=\begin{cases}-r(z),\quad\qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,z\in(0,\varkappa),\\\
f_{7}=-\hat{r}_{0}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}(z-\varkappa)^{-2i\nu}(1-\chi_{\mathcal{Z}}(z))\qquad\quad\,\,\,\,z\in\Sigma_{7},\end{cases}$
$\displaystyle
R_{8}(z)=\begin{cases}\frac{\bar{r}(z)}{1-|r(z)|^{2}},\quad\qquad\qquad\qquad\qquad\qquad\quad\,\,\,\,z\in(\varkappa,+\infty),\\\
f_{8}=\frac{\hat{r}_{0}}{1-|\hat{r}_{0}|^{2}}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{-}^{2}(z)}(z-\varkappa)^{2i\nu}(1-\chi_{\mathcal{Z}}(z))\qquad\quad
z\in\Sigma_{8},\end{cases}$

where

$\displaystyle\hat{r}_{0}=r(\varkappa)e^{-i\nu(-\varkappa)-i\nu(\varkappa)-i\beta_{1}(\varkappa,-\varkappa)+i\beta_{2}(\varkappa,\varkappa)}(2\varkappa)^{i\nu(-\varkappa)},$
$\displaystyle\tilde{r}_{0}=r(-\varkappa)e^{-i\nu(-\varkappa)-i\nu(\varkappa)-i\beta_{1}(-\varkappa,-\varkappa)+i\beta_{2}(-\varkappa,\varkappa)}(-2\varkappa)^{i\nu(\varkappa)}.$

######  Lemma 3.3.

There exist function $R_{i}$ on $\Omega_{i}$ satisfying Lemma 3.2 so that

$|\bar{\partial}R_{i}(z)|\lesssim|p^{\prime}(Re(z))|+|z\pm\varkappa|^{-1/2}+\bar{\partial}(\chi_{\mathcal{Z}}),$
(3.6)

where the implied constant are uniform for $r$ in a bounded subset of $H^{1}$.

###### Proof.

We give the details firstly for $R_{4}(z)$:

$R_{4}(z)=\Big{(}f_{4}+[-\frac{r(Rez)}{1-|r(Rez)|^{2}}-f_{4}(z)]\mathcal{K}(\phi)\Big{)}(1-\chi_{\mathcal{Z}}),$

where $z=\varkappa+\rho e^{i\psi},\rho\geq
0,0\leq\psi\leq\arctan(\frac{0.2}{\varkappa-1.5})$, where
$\phi=\arg(z-\varkappa)$ and $\mathcal{K}$ is a smooth function on
$(0,\arctan(\frac{0.2}{\varkappa-1.5}))$,

$\mathcal{K}(\phi)=\begin{cases}1,\quad z\in[0,\pi/20],\\\ 0,\quad
z\in[\pi/30,\pi/10].\end{cases}$

We have

$\displaystyle\bar{\partial}R_{4}(z)$
$\displaystyle=\Big{(}\frac{1}{2}(-\frac{r(Rez)}{1-|r(Rez)|^{2}})^{\prime}\mathcal{K}(\phi)-\Big{[}-\frac{r(Rez)}{1-|r(Rez)|^{2}}-f_{4}(z)\Big{]}\frac{ie^{i\phi}}{|z-\varkappa|}\mathcal{K}^{\prime}(\phi)\Big{)}$
$\displaystyle\times(1-\chi_{\mathcal{Z}})-(f_{4}(z)+\Big{[}-\frac{r(Rez)}{1-|r(Rez)|^{2}}-f_{4}(z)\Big{]}\mathcal{K}(\phi))\bar{\partial}(\chi_{\mathcal{Z}}).$

It follows that we acquire

$|\bar{\partial}R_{4}(z)|\lesssim\Big{|}(-\frac{r(Rez)}{1-|r(Rez)|^{2}})^{\prime}\Big{|}+|z-\varkappa|^{-1/2}+\bar{\partial}(\chi_{\mathcal{Z}}).$

Then we give the estimate for $R_{2}(z)$. The other cases are easily inferred.
For $z\in\Omega_{3}$, we set

$z=-\varkappa+re^{i\varphi}=(-\varkappa+r\cos\varphi)+ir\sin\varphi.$

Then the continuous extension of $R_{2}(z)$ can be constructed by

$\displaystyle R_{2}(z)$
$\displaystyle=[h\bar{r}(z)+(1-h)f_{2}](1-\chi_{\mathcal{Z}}(z))$
$\displaystyle=f_{2}+h(\bar{r}(z)-f_{2})(1-\chi_{\mathcal{Z}}(z)),$

where

$h=h(r,\varphi)=\frac{d_{2}}{d_{1}+d_{2}},$

where $d_{1,2}$ respectively represent the distance from $z$ to
$(-\varkappa,0)$ and $\Sigma_{2}$, they are defined as

$\displaystyle d_{1}=d_{1}(r,\varphi)=r\cos\varphi,$ $\displaystyle
d_{2}=d_{2}(r,\varphi)=\min(\frac{1}{10}\sin\frac{\varkappa}{\pi}x-r\cos\varphi,\frac{0.2}{\varkappa-1.5}r\cos\varphi+\frac{0.2\varkappa}{\varkappa-1.5})$

Finally, we have

$\displaystyle|\bar{r}(z)(z+\varkappa)^{2i\nu}|$
$\displaystyle=|\bar{r}(z)|e^{-2\nu\arg(z+\varkappa)}$
$\displaystyle\leq|\bar{r}(z|(1-|\bar{r}(z)|^{2}))^{-1}$ $\displaystyle\leq
c_{1}+c_{2}|1+z^{2}|^{-\frac{1}{4}}.$

Utilizing
$\bar{\partial}=\frac{1}{2}e^{i\varphi}(\partial_{r}+ir^{-1}\partial_{\varphi})$,
we have

$\displaystyle\bar{\partial}R_{2}(z)$
$\displaystyle=-[f_{2}+h(\bar{r}(z)-f_{2})]\bar{\partial}\chi_{\mathcal{Z}}(z)$
$\displaystyle+\frac{1}{2}e^{i\varphi}(h_{r}+ir^{-1}h_{\varphi})(\bar{r}(z)-f_{2})(1-\chi_{\mathcal{Z}}(z)),$

and

$\displaystyle|\bar{\partial}R_{11}(z)|\leq
c_{1}|\bar{\partial}\chi_{\mathcal{Z}}(z)|+c_{2}|\bar{(}z)-f_{2}|+c_{3}\frac{|\bar{r}(z)-f_{2}|}{|z+\varkappa|},$

which acquire the estimate of (3.6). ∎

We use the extension in Lemma 3.2 to define a new unknown function

$M^{(2)}_{\oplus}(y,t,z)=M_{\oplus}(y,t,z)\mathcal{R}^{(2)}(z),$ (3.7)
$\displaystyle\mathcal{R}^{(2)}=\begin{cases}\begin{pmatrix}1&0&0\\\
R_{1}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix}^{-1},\quad
z\in\Omega_{1},\quad\begin{pmatrix}1&R_{2}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}^{-1},\quad
z\in\Omega_{2},\\\ \\\
\begin{pmatrix}1&R_{3}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}^{-1},\quad
z\in\Omega_{3},\quad\begin{pmatrix}1&0&0\\\
R_{4}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix}^{-1},\quad
z\in\Omega_{4},\\\ \\\
\begin{pmatrix}1&R_{5}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\qquad
z\in\Omega_{5},\quad\begin{pmatrix}1&0&0\\\
R_{6}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\qquad
z\in\Omega_{6},\\\ \\\ \begin{pmatrix}1&0&0\\\
R_{7}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\qquad
z\in\Omega_{7},\qquad\begin{pmatrix}1&R_{8}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\qquad z\in\Omega_{8},\\\
\\\ \begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\qquad
z\in(\Omega_{9}\cup\Omega_{10}).\end{cases}$

satisfies the following $\bar{\partial}-$RH problem.

######  RHP 3.3.

Find a continuous with sectionally continuous first partial derivatives
function
$M^{(2)}_{\oplus}(y,t,z):\mathbb{C}\setminus(\hat{\Sigma}\cup\Gamma)\to$SL${}_{3}(\mathbb{C})$
with the following properties:

$\bigstar$_(the normalization condition)_

$M^{(2)}_{\oplus}(y,t,z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

For each $z\in\hat{\Sigma}$, the boundary values $M^{(2)}_{\oplus}(y,t,z)$
satisfy the jump relation
$M^{(2+)}_{\oplus}(y,t,z)=M^{(2-)}_{\oplus}(y,t,z)S^{(2)}_{\oplus}(y,t,z),\,\,y\in\hat{\Sigma}$
where the jump matrix

$\displaystyle S^{(2)}_{\oplus}(y,t,z)=\begin{cases}\begin{pmatrix}1&0&0\\\
R_{1}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{1}\quad\begin{pmatrix}1&R_{2}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad z\in\Sigma_{2}\\\ \\\
\begin{pmatrix}1&R_{3}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{3}\quad\begin{pmatrix}1&0&0\\\
R_{4}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{4}\\\ \\\
\begin{pmatrix}1&R_{5}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{5}\quad\begin{pmatrix}1&0&0\\\
R_{6}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix}\quad
z\in\Sigma_{6}\\\ \\\ \begin{pmatrix}1&0&0\\\
R_{7}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\
0&0&1\end{pmatrix},z\in\Sigma_{7}quad\begin{pmatrix}1&R_{8}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{8}\end{cases}$ (3.8)

$\bigstar$_(the residue condition)_

The residue condition of $M^{(2)}_{\oplus}(y,t,z)$ is as same as
$M_{\Diamond}(y,t,z)$ of RH problem 2.1.

$\bigstar$_(Singularities)_

As $z\to 0$, the limit of $M^{(2)}_{\oplus}(y,t,z)$ has pole singularities as
$M_{\Diamond}(y,t,z)$.

$\bigstar$_( $\bar{\partial}$ about the jump matrix)_

$\displaystyle\bar{\partial}S^{(2)}(y,t;z)=\begin{cases}\begin{pmatrix}1&0&0\\\
\bar{\partial}R_{1}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{1}\quad\begin{pmatrix}1&\bar{\partial}R_{2}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad z\in\Sigma_{2}\\\ \\\
\begin{pmatrix}1&\bar{\partial}R_{3}\frac{\delta^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{3}\quad\begin{pmatrix}1&0&0\\\
\bar{\partial}R_{4}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta_{+}^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{4}\\\ \\\
\begin{pmatrix}1&\bar{\partial}R_{5}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{5}\quad\begin{pmatrix}1&0&0\\\
\bar{\partial}R_{6}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\ 0&0&1\end{pmatrix}\quad
z\in\Sigma_{6}\\\ \\\ \begin{pmatrix}1&0&0\\\
\bar{\partial}R_{7}\frac{\delta(\omega^{2}z)\delta(\omega
z)}{\delta^{2}(z)}e^{2it\theta}&1&0\\\
0&0&1\end{pmatrix},z\in\Sigma_{7}\qquad\,\,\,\begin{pmatrix}1&\bar{\partial}R_{8}\frac{\delta_{-}^{2}(z)}{\delta(\omega^{2}z)\delta(\omega
z)}e^{-2it\theta}&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Sigma_{8}\end{cases}$ (3.9)

######  Remark 3.4.

In the $\bar{\partial}$-RH problem 3.3, it is useful to recall how the
extensions $R_{j}(z)$ are defined in Lemma 3.2, particularly in Lemma 3.3.
Through the $\bar{\partial}$ of $M^{(2)}_{\oplus}(y,t,z)$ may seem to that
$M^{(2)}_{\oplus}(y,t,z)$ is non-analytic near points the discrete spectrum,
the $\bar{\partial}$-derivative vanishes in small neighborhoods of each point
of the discrete spectrum so that $M^{(2)}_{\oplus}(y,t,z)$ is analytic in each
neighborhood.

###  3.3 Analysis on the pure RH problem

In this section we build a solution $M_{rhp}^{(2)}(z)$ to the RH problem that
results from RH problem 3.3 for $M^{(2)}_{\oplus}(y,t,z)$ by dropping the
$\bar{\partial}$ component. Specifically: Let $M_{rhp}^{(2)}(z)$ be the
solution of the RH problem 3.3 resulting from stetting
$\bar{\partial}\mathcal{R}^{(2)}=0$. We perform the following factorization of
$M^{(2)}_{\oplus}(y,t,z)$:

$M^{(2)}_{\oplus}(y,t,z)=M^{(3)}(z)M_{rhp}^{(2)}(z),$ (3.10)

where $M^{(3)}(z)$ is a continuously differentiable function satisfying the
following $\bar{\partial}$-problem 3.1.

######  $\bar{\partial}$-RHP 3.1.

Find a continuous with sectionally continuous first partial derivatives
function $M^{(3)}(z):\mathbb{C}\to SL_{3}(\mathbb{C})$ with the following
properties:

$\bigstar$_(the normalization condition)_

$M^{(3)}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the $\bar{\partial}$ condition)_

For $z\in\mathbb{C}$, we have

$\bar{\partial}M^{(3)}(z)=M^{(3)}(z)W^{(3)}(z),$ (3.11)

where
$W^{(3)}(z):=M_{rhp}^{(2)}(z)\bar{\partial}\mathcal{R}^{(2)}(z)M_{rhp}^{(2)}(z)^{-1}$.

In this section focuses on find $M_{rhp}^{(2)}(z)$ which is meromorphic away
from the contour $\hat{\Sigma}$ on which its boundary values satisfy the jump
relation (3.8).

######  RHP 3.4.

Find an analytic function
$M_{rhp}^{(2)}(z):\mathbb{C}\setminus\hat{\Sigma}_{r}\to SL_{3}(\mathbb{C})$
such that

$\bigstar$_(the normalization condition)_

$M_{rhp}^{(2)}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

For $z\in\mathbb{C}\setminus\hat{\Sigma}_{r}$, we have the jump relation
$M_{rhp}^{(2)^{+}}(z)=M_{rhp}^{(2)^{-}}(z)V^{(2)}_{rhp}(z)$, where

$V^{(2)}_{rhp}(z)=S^{(2)}_{\oplus}(y,t,z)|_{\Sigma_{r}}.$

$\hat{\Sigma}_{l}$$\hat{\Sigma}_{r}$$\varkappa$$-\varkappa$$\Sigma_{4}$$\Sigma_{1}$$\Sigma_{8}$$\Sigma_{5}$$\Sigma_{3}$$\Sigma_{2}$$\Sigma_{7}$$\Sigma_{6}$$V^{(2)}$
decays exponentially on red contours  Figure 3.2:
$\hat{\Sigma}=\hat{\Sigma}_{b}\cup\hat{\Sigma}_{r}$, where $\hat{\Sigma}_{b}$
and $\hat{\Sigma}_{r}$ are respectively the black and red portion of
$\hat{\Sigma}$ and $\hat{\Sigma}_{b}=\hat{\Sigma}_{l}\cup\hat{\Sigma}_{r}$

Now we decompose $W^{(2)}=V^{(2)}_{rhp}(z)-I$ into two parts:

$W^{(2)}=W^{e}+W^{\prime},$ (3.12)

where $W^{\prime}=W^{(2)}|_{\hat{\Sigma}_{b}}$ and
$W^{e}=W^{(2)}|_{\hat{\Sigma}_{r}}$. on $\hat{\Sigma}_{l}$ away from
$-\varkappa$, we estimate:

$\displaystyle|R_{1}e^{2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{2}e^{-2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{5}e^{-2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{6}e^{2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$

and on $\hat{\Sigma}_{r}$ away from $\varkappa$, we estimate:

$\displaystyle|R_{3}e^{-2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{4}e^{2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{7}e^{2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$
$\displaystyle|R_{8}e^{-2it\theta}|\leq C_{r}e^{-\frac{1}{2}\epsilon^{2}t},$

with the discussion above we conclude that

$\big{|}W^{e}\big{|}\lesssim e^{-ct}.$ (3.13)

######  Proposition 3.5.

There exists a 2$\times$2 matrix $E_{1}(z)$ with

$E_{1}(z)=I+\mathcal{O}\big{(}\frac{e^{-ct}}{z}\big{)},$ (3.14)

such that

$M_{rhp}^{(2)}(z)=E_{1}(z)M_{rhp}^{(2)^{*}}(z),$ (3.15)

where $M_{rhp}^{(2)^{*}}(z)$ solves the RH problem 3.5 and 3.8 with jump
contour $\Sigma_{r}$ and $\Sigma_{l}$.

###### Proof.

We will later show the existence of $M_{rhp}^{(2)^{*}}(z)$ and
$\big{\|}M_{rhp}^{(2)^{*}}(z)\big{\|}_{L^{\infty}}$ is finite. Assuming that,
it is easy to see that on
$\hat{\Sigma}^{(2)}\setminus(\hat{\Sigma}^{(2)}_{r}\cup\hat{\Sigma}^{(2)}_{l})$,
$E_{1}$ satisfies the following jump condition:

$E_{1+}=E_{1-}\big{(}M_{rhp}^{(2)^{*}}(z)(I+W^{e})\big{(}M_{*}^{LC}\big{)}^{-1}\big{)}.$

Using (3.13) the conclusion follows from solving a small norm RH problem 3.5
and 3.8 see below. ∎

####  3.3.1 A local model near the saddle point

######  RHP 3.5.

Find a 3$\times$3 matrix-valued function $\mathbf{M_{r}}(z)$, analytic on
$\mathbb{C}\setminus\hat{\Sigma}_{r}$, with the following properties:

$\bigstar$_(the normalization condition)_

$\mathbf{M_{r}}(z)=I+\mathcal{O}(z^{-1}),\quad z\to\infty.$

$\bigstar$_(the jump condition)_

$\mathbf{M_{r}}(z)$ is continuous boundary value $\mathbf{M_{r}}^{\pm}(z)$ on
$\hat{\Sigma}_{r}$, we have the jump relation
$\mathbf{M_{r}}^{+}(z)=\mathbf{M_{r}}^{-}(z)\mathbf{V_{r}}(z)$, where

$\mathbf{V_{r}}(z)=S^{(2)}_{\oplus}(y,t,z)|_{\Sigma_{r}}..$

Now set

$\zeta(z)=\sqrt{ct}(z-\varkappa),\quad c=\frac{2\sqrt{3}}{\varkappa^{3}}.$
(3.16)

and

$r_{\xi}=\hat{r}_{0}e^{-\frac{\sqrt{3}}{\varkappa}t}\Big{(}\frac{8\sqrt{3}}{\varkappa}t\Big{)}^{-i\nu/2}.$
(3.17)

Under the change of variables (3.38), the phase $e^{2it\theta}$ identifies to
$e^{-i\zeta^{2}/2}e^{-i\sqrt{3}t/\varkappa}$. The factor $e^{-i\zeta^{2}/2}$
will be following important in the identification of parabolic cylinder
functions.

######  RHP 3.6.

Find a 3$\times$3 matrix-valued function $\mathbf{N^{PC}}(z)$, analytic on
$\mathbb{C}\setminus\hat{\Sigma}_{r}$, with the following properties:

$\bigstar$_(the normalization condition)_

$\mathbf{N^{PC}}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty$

$\bigstar$_(the jump condition)_

$\mathbf{N^{PC}}(z)$ is continuous boundary value $\mathbf{N^{PC}}^{\pm}(z)$
on $\hat{\Sigma}_{r}$, we have the jump relation
$\mathbf{N^{PC}}^{+}(z)=\mathbf{N^{PC}}^{-}(z)\mathbf{V^{(PC)}}(z)$, where

$\mathbf{V^{(PC)}}(z)=S^{(2)}_{\oplus}(y,t,z)|_{\hat{\Sigma}_{r}}$

where

$\mathbf{V^{(PC)}}(z)=\begin{cases}\begin{pmatrix}1&r_{\xi}(-\zeta)^{2i\nu}e^{-i\nu^{2}/2}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\qquad\quad z\in\Sigma_{3},\\\ \\\
\begin{pmatrix}1&0&0\\\
-\frac{r_{\xi}}{1-|r_{\xi}|^{2}}(-\zeta)^{2i\nu}e^{-i\nu^{2}/2}&1&0\\\
0&0&1\end{pmatrix},\quad z\in\Sigma_{4},\\\ \\\ \begin{pmatrix}1&0&0\\\
-r_{\xi}(-\zeta)^{2i\nu}e^{-i\nu^{2}/2}&1&0\\\ 0&0&1\end{pmatrix},\qquad\quad
z\in\Sigma_{7},\\\ \\\
\begin{pmatrix}1&\frac{r_{\xi}}{1-|r_{\xi}|^{2}}(-\zeta)^{2i\nu}e^{-i\nu^{2}/2}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\qquad z\in\Sigma_{8}.\\\ \end{cases}$ (3.18)

It is possible to further reduce the RH problem 3.6 to a model RH problem
3$\times$3 matrix solution is piecewise analytic in the upper and lower
complex plane. In each half-plane, the entries of the matrix satisfy ODEs that
are obtained from analytic properties as well as the large $\zeta$ behavior.
The solution of the ODEs are explicitly calculated in terms of parabolic
cylinder functions. Let

$\mathbf{N^{PC}}(\xi,\zeta)=\Phi(\xi,\zeta)\mathcal{P}(\xi)\begin{pmatrix}e^{i\nu^{2}/4}&0&0\\\
0&e^{-i\nu^{2}/4}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}(-\zeta)^{-i\nu}&0&0\\\
0&(-\zeta)^{i\nu}&0\\\ 0&0&1\end{pmatrix},$ (3.19)

where

$\mathbf{V^{(PC)}}(z)=\begin{cases}\begin{pmatrix}1&r_{\xi}&0\\\ 0&1&0\\\
0&0&1\end{pmatrix},\quad z\in\Omega_{3},\qquad\begin{pmatrix}1&0&0\\\
-\frac{r_{\xi}}{1-|r_{\xi}|^{2}}&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Omega_{4},\\\ \\\ \begin{pmatrix}1&0&0\\\ -r_{\xi}&1&0\\\
0&0&1\end{pmatrix},\quad
z\in\Omega_{7},\quad\,\begin{pmatrix}1&\frac{r_{\xi}}{1-|r_{\xi}|^{2}}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\quad z\in\Omega_{8},\\\ \\\
\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
z\in\Omega_{9}\cup\Omega_{10}.\end{cases}$ (3.20)

By construction, the matrix $\Phi$ is continuous along the rays of
$\hat{\Sigma}_{r}$. Let us set up the RH problem it satisfies and compute its
jumps along the real axis. We have along the real axis

$\displaystyle\Phi_{+}=\Phi_{-}\Bigg{\\{}\mathcal{P}\begin{pmatrix}e^{i\nu^{2}/4}&0&0\\\
0&e^{-i\nu^{2}/4}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}(-\zeta)^{-i\nu}&0&0\\\
0&(-\zeta)^{i\nu}&0\\\ 0&0&1\end{pmatrix}\Bigg{\\}}_{-}$ (3.21)
$\displaystyle\Bigg{\\{}\begin{pmatrix}e^{-i\nu^{2}/4}&0&0\\\
0&e^{i\nu^{2}/4}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}(-\zeta)^{i\nu}&0&0\\\
0&(-\zeta)^{-i\nu}&0\\\ 0&0&1\end{pmatrix}\mathcal{P}^{-1}\Bigg{\\}}_{+}.$
(3.22)

Due to the branch cut of the logarithmic function along the
$(-\infty,-\varkappa)\cup(\varkappa,+\infty)$, we have along the negative real
axis,

$\begin{pmatrix}(-\zeta)^{-i\nu}&0&0\\\ 0&(-\zeta)^{i\nu}&0\\\
0&0&1\end{pmatrix}_{-}\begin{pmatrix}(-\zeta)^{i\nu}&0&0\\\
0&(-\zeta)^{-i\nu}&0\\\
0&0&1\end{pmatrix}_{+}=\begin{pmatrix}e^{-2\nu\pi}&0&0\\\ 0&e^{2\nu\pi}&0\\\
0&0&1\end{pmatrix}.$

while along the $(-\varkappa,\varkappa)$

$\begin{pmatrix}(-\zeta)^{-i\nu}&0&0\\\ 0&(-\zeta)^{i\nu}&0\\\
0&0&1\end{pmatrix}_{-}\begin{pmatrix}(-\zeta)^{i\nu}&0&0\\\
0&(-\zeta)^{-i\nu}&0\\\ 0&0&1\end{pmatrix}_{+}=I.$

This implies that the matrix $\Phi$ has the same (constant) jump matrix along
the

$V^{(0)}=\begin{pmatrix}1&\bar{r}_{\xi}(z)&0\\\
-r_{\xi}(z)&1-|r_{\xi}(z)|^{2}&0\\\ 0&0&1\end{pmatrix}.$ (3.23)

and the 3$\times$3 matrix $\Phi$ satisfies the following model RH problem 3.7.

######  RHP 3.7.

Find a 3$\times$3 matrix-valued function $\Phi(z;\xi)$, analytic on
$\mathbb{C}\setminus\mathbb{R}$, with the following properties:

$\bigstar$_(the boundary value condition)_

$\Phi(z)\sim\begin{pmatrix}e^{-i\nu^{2}/4}&0&0\\\ 0&e^{i\nu^{2}/4}&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}(-\zeta)^{i\nu}&0&0\\\
0&(-\zeta)^{-i\nu}&0\\\ 0&0&1\end{pmatrix}\quad
z\to\infty\quad\mbox{in}\quad\mathbb{C}\setminus\mathbb{R}.$

$\bigstar$_(the jump condition)_ $\Phi(z)$ is analytic for
$z\in\mathbb{C}\setminus\mathbb{R}$ with continuous boundary values
$\Phi_{\pm}$ on $\mathbb{R}$ with the jump relation

$\Phi_{+}(z)=\Phi_{-}(z)V^{(0)}.$ (3.24)

Differentiating (3.24) with respect to $\zeta$, we obtain that

$\Bigg{(}\frac{d\Phi}{d\zeta}+\begin{pmatrix}\frac{1}{2}i\zeta&0&0\\\
0&-\frac{1}{2}i\zeta&0\\\
0&0&1\end{pmatrix}\Phi\Bigg{)}_{+}=\Bigg{(}\frac{d\Phi}{d\zeta}+\begin{pmatrix}\frac{1}{2}i\zeta&0&0\\\
0&-\frac{1}{2}i\zeta&0\\\ 0&0&1\end{pmatrix}\Phi\Bigg{)}_{-}V^{(0)}.$

We know that $\det V^{(0)}=1$, thus $\det\Phi_{+}=\det\Phi_{-}$ and $\det\Phi$
is analytic in the whole complex plane. It is equal to one at infinity, thus
by Liouville theorem, $\det\Phi=1$. It follows that $(\Phi)^{-1}$ exists and
it bounded. The matrix

$\Bigg{(}\frac{d\Phi}{d\zeta}+\begin{pmatrix}\frac{1}{2}i\zeta&0&0\\\
0&-\frac{1}{2}i\zeta&0\\\ 0&0&1\end{pmatrix}\Phi\Bigg{)}\Phi^{-1},$

has no jump along the real line and is therefore an entire function of
$\zeta$. According to (3.19), we have that

$\displaystyle\Bigg{(}\frac{d\Phi}{d\zeta}+\begin{pmatrix}\frac{1}{2}i\zeta&0&0\\\
0&-\frac{1}{2}i\zeta&0\\\
0&0&1\end{pmatrix}\Phi\Bigg{)}\Phi^{-1}=\Bigg{(}\frac{\mathbf{N^{PC}}(z)}{d\zeta}+\frac{\mathbf{N^{PC}}(z)}{\zeta}\begin{pmatrix}i\nu&0&0\\\
0&-i\nu&0\\\ 0&0&1\end{pmatrix}\Bigg{)}\mathbf{N^{PC}}(z)^{-1}$
$\displaystyle+\Bigg{\\{}\begin{pmatrix}\frac{i\zeta}{2}&0&0\\\
0&-\frac{i\zeta}{2}&0\\\
0&0&1\end{pmatrix}\mathbf{N^{PC}}(z)-\mathbf{N^{PC}}(z)\begin{pmatrix}\frac{i\zeta}{2}&0&0\\\
0&-\frac{i\zeta}{2}&0\\\ 0&0&1\end{pmatrix}\Bigg{\\}}\mathbf{N^{PC}}(z)^{-1}.$
(3.25)

The first term in the right hand side of (3.25) tends to $0$ as
$\zeta\to\infty$ while the second term behaves like $\mathcal{O}(1/\zeta)$.
For the last term in the right-hand side of (3.25), we use that

$\mathbf{N^{PC}}(z)\sim(I+\frac{m^{(0)}}{\zeta}).$ (3.26)

Defining

$\displaystyle\beta$ $\displaystyle=\begin{pmatrix}\frac{i}{2}&0&0\\\
0&-\frac{i}{2}&0\\\
0&0&1\end{pmatrix}\mathbf{N^{PC}}(z)-\mathbf{N^{PC}}(z)\begin{pmatrix}\frac{i}{2}&0&0\\\
0&-\frac{i}{2}&0\\\ 0&0&1\end{pmatrix}$ (3.27)
$\displaystyle=\begin{pmatrix}0&im_{12}&0\\\ -im_{21}&0&0\\\
0&0&0\end{pmatrix}.$ (3.28)

Equivalently, $\beta_{12}=im_{12},\beta_{21}=-im_{21}$. Again applying
Liouville’s theorem, the $3\times 3$ matrix $\Phi$ satisfies the ODE:

$\frac{d\Phi}{d\zeta}+\begin{pmatrix}\frac{i\zeta}{2}&0&0\\\
0&-\frac{i\zeta}{2}&0\\\ 0&0&1\end{pmatrix}\Phi=\beta\Phi,$ (3.29)

where $\beta$ is an off-diagonal matrix.

The system (3.29) decouples into two first-order systems for
$(\Phi_{11},\Phi_{21},\Phi_{31})$, $(\Phi_{12},\Phi_{22},\Phi_{32})$ and
$(\Phi_{13},\Phi_{23},\Phi_{33})$,

$\begin{cases}\frac{d\Phi_{11}}{d\zeta}+\frac{i\zeta}{2}\Phi_{11}=\beta_{12}\Phi_{21},\\\
\frac{d\Phi_{21}}{d\zeta}-\frac{i\zeta}{2}\Phi_{21}=\beta_{21}\Phi_{11},\end{cases}$
(3.30)

and

$\begin{cases}\frac{d\Phi_{12}}{d\zeta}+\frac{i\zeta}{2}\Phi_{12}=\beta_{12}\Phi_{22},\\\
\frac{d\Phi_{22}}{d\zeta}-\frac{i\zeta}{2}\Phi_{22}=\beta_{21}\Phi_{12},\end{cases}$
(3.31)

and

$\begin{cases}\frac{d\Phi_{13}}{d\zeta}+\frac{i\zeta}{2}\Phi_{13}=\beta_{12}\Phi_{23},\\\
\frac{d\Phi_{23}}{d\zeta}-\frac{i\zeta}{2}\Phi_{23}=\beta_{21}\Phi_{13}.\end{cases}$
(3.32)

Combining the above equations, one obtains that the entries of $\Phi$ satisfy
following Lemma 3.6.

######  Lemma 3.6.

The entries of $\Phi$ obey the differential equations

$\displaystyle\Phi_{11}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{11}+\frac{i}{2}\Big{)}\Phi_{11}=0,$
$\displaystyle\Phi_{21}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{21}-\frac{i}{2}\Big{)}\Phi_{21}=0,$
$\displaystyle\Phi_{12}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{12}+\frac{i}{2}\Big{)}\Phi_{12}=0,$
$\displaystyle\Phi_{22}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{21}-\frac{i}{2}\Big{)}\Phi_{22}=0,$
$\displaystyle\Phi_{13}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{13}+\frac{i}{2}\Big{)}\Phi_{13}=0,$
$\displaystyle\Phi_{23}^{{}^{\prime\prime}}+\Big{(}\frac{\zeta^{2}}{4}-(\beta_{12}\beta_{21})\Phi_{23}-\frac{i}{2}\Big{)}\Phi_{23}=0.$

The next step is to complement the ODEs with additional conditions taking into
account the conditions at infinity as well as the jump conditions of $\Phi$.
This will determine $\Phi$ uniquely and will identify the coefficients
$\beta_{12},\beta_{21}$. The parabolic cylinder equation is

$y^{\prime\prime}+(-\frac{z^{2}}{4}+a+\frac{1}{2})y=0.$ (3.33)

The parabolic cylinder functions
$D_{a}(z),D_{a}(-z),D_{-a-1}(iz),D_{-a-1}(-iz)$ all satisfy (3.33) and are
entire for all any value $a$.
The large-$z$ behavior of $D_{a}(z)$ is given by the following formulas

$D_{a}(z)\sim\begin{cases}z^{a}e^{-z^{2}/4},\qquad\qquad\qquad\qquad\qquad\qquad|\arg(z)|<\frac{3\pi}{4},\\\
z^{a}e^{-z^{2}/4}-\frac{\sqrt{2\pi}}{\Gamma(-a)}e^{ia\pi}z^{-a-1}e^{z^{2}/4},\qquad\frac{\pi}{4}<\arg(z)<\frac{5\pi}{4},\\\
z^{a}e^{-z^{2}/4}-\frac{\sqrt{2\pi}}{\Gamma(-a)}e^{-ia\pi}z^{-a-1}e^{z^{2}/4},\quad-\frac{5\pi}{4}<\arg(z)<-\frac{\pi}{4}.\end{cases}$
(3.34)

######  Proposition 3.7.

The unique solution to RH Problem 3.7 is given by

$\displaystyle\Phi(\zeta,r)=\begin{cases}\begin{pmatrix}e^{-\frac{3\pi\nu}{4}}D_{i\nu}(e^{-\frac{3i\pi}{4}}\zeta)&-\beta_{12}e^{\frac{\pi}{4}(\nu-i)}D_{-i\nu-1}(e^{-\frac{i\pi}{4}}\zeta)&0\\\
i\beta_{21}e^{-\frac{3\pi}{4}(\nu+i)}D_{i\nu-1}(e^{-\frac{3i\pi}{4}}\zeta)&e^{\frac{\pi\nu}{4}}D_{-i\nu}(e^{-\frac{i\pi}{4}\zeta})&0\\\
0&0&1\end{pmatrix},\quad\zeta\in\mathbb{C}^{+},\\\ \\\
\begin{pmatrix}e^{\frac{\pi\nu}{4}}D_{i\nu}(e^{\frac{i\pi}{4}\zeta})&-\beta_{12}e^{-\frac{3\pi}{4}(\nu-i)}D_{-i\nu-1}(e^{\frac{3i\pi}{4}}\zeta)&0\\\
i\beta_{21}e^{\frac{\pi}{4}(\nu+i)}D_{i\nu-1}(e^{\frac{i\pi}{4}}\zeta)&e^{-\frac{3\pi\nu}{4}}D_{-i\nu}(e^{\frac{3i\pi}{4}}\zeta)&0\\\
0&0&1\end{pmatrix},\quad\quad\,\,\zeta\in\mathbb{C}^{-}.\\\ \end{cases}$

We now impose the jump conditions to find the coefficients $\beta_{12}$ and
$\beta_{21}$, we know that $\beta_{12}\beta_{21}=\nu$, we acquire

$(\Phi_{-})^{-1}\Phi_{+}=V^{(0)}=\begin{pmatrix}1&\bar{r}_{\xi}(z)&0\\\
-r_{\xi}(z)&1-|r_{\xi}(z)|^{2}&0\\\ 0&0&1\end{pmatrix},$

we obtain

$\displaystyle-r_{\xi}$
$\displaystyle=\Phi^{-}_{11}\Phi^{+}_{21}-\Phi^{-}_{21}\Phi^{+}_{11}$
$\displaystyle=e^{\frac{\pi}{4}\nu}D_{i\nu}(e^{i\pi/4}\zeta)\frac{1}{\beta_{12}}e^{-\frac{3\pi}{4}\nu}\Big{(}\partial_{\zeta}(D_{i\nu}(\zeta
e^{-3i\pi/4}))+\frac{i\zeta}{2}D_{i\nu}(\zeta e^{-3i\pi/4})\Big{)}$
$\displaystyle\times\frac{e^{\frac{-\pi\nu}{2}}}{\beta_{12}}Wr\Big{(}D_{i\nu}(e^{i\pi/4}\zeta),D_{i\nu}(\varsigma
e^{-3i\pi/4})\Big{)}$
$\displaystyle=\frac{\sqrt{2\pi}e^{-\frac{\pi\nu}{2}e^{i\pi/4}}}{\beta_{12}\Gamma(-i\nu)},$

we obtain that $\beta_{12}$ and $\beta_{21}$ are the complex constants

$\displaystyle\beta_{12}=-\frac{\sqrt{2\pi}e^{i\pi/4}e^{-\pi\nu/2}}{-r_{\xi}\Gamma(-i\nu)},$
$\displaystyle\beta_{21}=\frac{r_{\xi}\Gamma(i\nu)}{\sqrt{2\pi}e^{-i\pi/4}e^{-\pi\nu/2}}=\frac{\nu}{\beta_{12}}.$
(3.35)

The essential fact for our needs is the asymptotic behavior of the solution,
as is easily verified using the well known asymptotic behavior of $D_{a}(z)$

$\mathbf{N^{PC}}(z)=I+\frac{1}{\zeta}\begin{pmatrix}0&-i\beta_{12}(r)&0\\\
i\beta_{21}(r)&0&0\\\ 0&0&1\end{pmatrix}+\mathcal{O}(\zeta^{-2}).$ (3.36)

######  Lemma 3.8.

Let $c$ be a small but fixed positive number with $0<2c<1$. Then

$\Big{|}(-\zeta)^{\pm 2i\nu}e^{\mp i\nu^{2}/2}-e^{\mp i\zeta^{2}/2}e^{\mp
i\sqrt{3}t/\varkappa}\frac{\zeta^{2}\varkappa^{3}}{2\sqrt{3}t}\Big{|}\leq
ct^{-1/2}.$ (3.37)

######  RHP 3.8.

Find a 3$\times$3 matrix-valued function $\mathbf{M_{l}}(z)$, analytic on
$\mathbb{C}\setminus\hat{\Sigma}_{l}$, with the following properties:

$\bigstar$_(the normalization condition)_

$\mathbf{M_{l}}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

$\mathbf{M_{l}}$ is continuous boundary value $\mathbf{M_{l}}^{\pm}(z)$ on
$\hat{\Sigma}_{l}$, we have the jump relation
$\mathbf{M_{l}}^{+}(z)=\mathbf{M_{l}}^{-}(z)\mathbf{V_{l}}(z)$, where

$\mathbf{V_{l}}(z)=\hat{S}_{1}|_{\hat{\Sigma}_{l}}.$

Now set

$\zeta(z)=\sqrt{ct}(z+\varkappa),\quad c=\frac{2\sqrt{3}}{\varkappa^{3}},$
(3.38)

and

$r_{\xi}=\tilde{r}_{0}e^{-\frac{\sqrt{3}}{\varkappa}t}\Big{(}\frac{8\sqrt{3}}{\varkappa}t\Big{)}^{-i\nu/2}.$
(3.39)

Under the change of variables (3.38), the phase $e^{2it\theta}$ identifies to
$e^{-i\zeta^{2}/2}e^{-i\sqrt{3}t/\varkappa}$. The factor $e^{-i\zeta^{2}/2}$
will be following important in the identification of parabolic cylinder
functions.

we obtain that $\beta_{12}$ and $\beta_{21}$ are the complex constants

$\displaystyle\beta_{12}=\frac{\sqrt{2\pi}e^{i\pi/4}e^{-\pi\nu/2}}{-r_{\xi}\Gamma(-i\nu)},$
$\displaystyle\beta_{21}=\frac{-\sqrt{2\pi}e^{-i\pi/4}e^{-\pi\nu/2}}{r_{\xi}\Gamma(i\nu)}=\frac{\nu}{\beta_{12}}.$
(3.40)

The essential fact for our needs is the asymptotic behavior of the solution,
as is easily verified using the well known asymptotic behavior of $D_{a}(z)$

$\mathbf{N^{PC}}(z)=I+\frac{1}{\zeta}\begin{pmatrix}0&-i\beta_{12}(r)&0\\\
i\beta_{21}(r)&0&0\\\ 0&0&1\end{pmatrix}+\mathcal{O}(\zeta^{-2}).$ (3.41)

For the crosses $M_{rhp}^{(2)}(z)$ centered at $\pm\omega\varkappa$ and
$\pm\omega^{2}\varkappa$, by using the symmetries, which gives the following

$\displaystyle M_{rhp}^{(2)}(z)\sim
I+\frac{1}{\sqrt{ct}(z+\varkappa)}\begin{pmatrix}0&-i\beta_{12}&0\\\
i\beta_{21}&1&0\\\ 0&0&1\end{pmatrix},$ (3.42) $\displaystyle
M_{rhp}^{(2)}(\bar{z})\sim
I+\frac{1}{\sqrt{ct}(\bar{z}+\varkappa)}\begin{pmatrix}0&i\beta_{12}&0\\\
-i\beta_{21}&1&0\\\ 0&0&1\end{pmatrix},$ (3.43) $\displaystyle
M_{rhp}^{(2)}(z\omega)\sim
I+\frac{1}{\sqrt{ct}(-z\omega+\varkappa)}\begin{pmatrix}0&0&0\\\
0&0&i\beta_{12}\\\ 0&-i\beta_{21}&0\end{pmatrix},$ (3.44) $\displaystyle
M_{rhp}^{(2)}(-z\omega)\sim
I+\frac{1}{\sqrt{ct}(z\omega+\varkappa)}\begin{pmatrix}0&0&0\\\
0&0&-i\beta_{12}\\\ 0&i\beta_{21}&0\end{pmatrix},$ (3.45) $\displaystyle
M_{rhp}^{(2)}(z\omega^{2})\sim
I+\frac{1}{\sqrt{ct}(-z\omega^{2}+\varkappa)}\begin{pmatrix}0&0&i\beta_{12}\\\
0&0&0\\\ -i\beta_{21}&0&0\end{pmatrix},$ (3.46) $\displaystyle
M_{rhp}^{(2)}(-z\omega^{2})\sim
I+\frac{1}{\sqrt{ct}(z\omega^{2}+\varkappa)}\begin{pmatrix}0&0&-i\beta_{12}\\\
0&0&0\\\ i\beta_{21}&0&0\end{pmatrix}.$ (3.47)

####  3.3.2 A small-norm RH problem

We will construct the solution $M_{rhp}^{(2)}$ by seeking a solution of the
form

$M_{rhp}^{(2)}(z)=\begin{cases}E(z)\mathbf{M^{(lo)}}(z),\quad|z\pm
z_{0}|>\rho,\\\ E(z)\mathbf{M^{(r)}}(z),\quad\,\,|z-z_{0}|\leq\rho,\\\
E(z)\mathbf{M^{(l)}}(z),\quad\,\,|z+z_{0}|\leq\rho,\end{cases}$ (3.48)

where
$z_{0}=\pm\varkappa,\omega\varkappa,\bar{\omega}\varkappa,\omega^{2}\varkappa,\bar{\omega}^{2}\varkappa$
and $\rho$ be the radius of the circle $C_{r}$ centered at $z_{0}$ and
$\mathbf{M^{(r)}}(z),\mathbf{M^{(l)}}(z),\mathbf{M^{(lo)}}(z)$ solve the RH
problem 3.5, 3.8, 4.2 and the error $E(z)$, the solution of a small-norm RH
problem, we will prove exists and bound it asymptotically.

######  RHP 3.9.

Find a 3$\times$3 matrix-valued function $E(z)$, analytic on
$\mathbb{C}\setminus C_{r}$ with the following properties:

$\bigstar$_(the normalization condition)_

$E(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

$E(z)$ is continuous boundary value $E^{\pm}(z)$ on $C$, we have the jump
relation $E^{+}(z)=E^{-}(z)V^{(E)}(z)$, where

$V^{(E)}(z)=\begin{cases}\mathbf{M^{(lo)}}(z)\mathbf{M^{(r)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad
z\in C_{r}(\varkappa),\\\
\mathbf{M^{(lo)}}(z)\mathbf{M^{(l)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad z\in
C_{l}(-\varkappa).\\\
\mathbf{M^{(lo)}}(z)\mathbf{M^{(r)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad z\in
C_{l}(\omega\varkappa).\\\
\mathbf{M^{(lo)}}(z)\mathbf{M^{(l)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad z\in
C_{l}(\bar{\varkappa}).\\\
\mathbf{M^{(lo)}}(z)\mathbf{M^{(r)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad z\in
C_{l}(\omega^{2}\varkappa).\\\
\mathbf{M^{(lo)}}(z)\mathbf{M^{(l)}}(z)\mathbf{M^{(lo)}}(z)^{-1},\quad z\in
C_{l}(\omega\bar{\varkappa}).\end{cases}$

Starting from (3.42)-(3.47) and using (4.2) the boundedness of
$\mathbf{M^{(lo)}}(z)$, one finds that

$|V^{(E)}(z)-I|\lesssim t^{-1/2},$ (3.49)

and it follows that

$\|C_{V^{(E)}}f\|_{L^{2}}\leq\|f\|_{L^{2}}\|V^{(E)}-I\|_{L^{\infty}}\lesssim
t^{-1/2}.$ (3.50)

This uniformly vanishing bounded on $V^{(E)}-I$ establishes RH problem 3.9 as
a small-norm RH problem, for which there is a well known existence and
uniqueness theorem [30, 31, 32]. In fact, we may write

$E(z)=I+\frac{1}{2\pi i}\int_{C_{r}\cup
C_{l}}\frac{(I+\eta(s))(V^{(E)}-I)}{s-z}ds,$ (3.51)

where $\eta\in L^{2}(C)$ is the unique solution of

$(I-C_{V^{(E)}})\eta=C_{V^{(E)}}I.$ (3.52)

The singular integral operator $C_{V^{(E)}}:L^{2}(C)$ is defined by

$\displaystyle C_{V^{(E)}}f=C_{-}(f(V^{(E)}-I)),$ (3.53) $\displaystyle
C_{-}f(z)=\underset{z\to C}{\lim}\frac{1}{2\pi i}\int_{C}f(s)\frac{ds}{s-z},$
(3.54)

where $C_{-}$ is the well known Cauchy projection operator. It’s well known
that $\|C_{-}\|_{L^{2}}$ is bounded. It then follows from (3.50) and (3.53)
that

$\|C_{V^{(E)}}\|_{L^{2}(C_{r}\cup C_{l})}\lesssim\|C_{-}\|_{L^{2}(C_{r}\cup
C_{l})}\|V^{(E)}-I\|_{C_{r}\cup C_{l}}\lesssim\mathcal{O}(t^{-1/2}),$ (3.55)

which guarantees the existence of the resolvent operator
$(I-C_{V^{(E)}})^{-1}$ and thus of both $\eta$ and $E$.

In order to reconstruct the solution $u(x,t)$ of (1.1) we need consider the
behavior of $z\to 0$ of the solution RH problem 2.1. This will include the
behavior $z\to 0$ expansion of $E$ which we give here. Geometrically expanding
$(s-z)^{-1}$ for $z$ large in (3.51), which is justified by the finiteness of
moments in (3.49), we have

$E(z)=I+\frac{E_{1}}{z}+\mathcal{O}(z^{-2}),$ (3.56)

where

$E_{1}=-\frac{1}{2\pi i}\int_{C_{r}\cup C_{l}}(I+\eta(s))(V^{(E)}-I)ds.$
(3.57)

Then using the bounds on $V^{(E)}-I$ we have

$E_{1}=-\frac{1}{2\pi i}\int_{C_{r}\cup
C_{l}}(V^{(E)}-I)ds+\mathcal{O}(t^{-1}).$

This last integral can be asymptotically computed by residue yield to leading
order

$\displaystyle E_{1}(x,t)$
$\displaystyle=\frac{1}{2i\sqrt{t}}\mathbf{M^{(lo)}}(z)\begin{pmatrix}0&\beta_{12}(r_{\xi})\\\
-\beta_{21}(r_{\xi})&0\end{pmatrix}\mathbf{M^{(lo)}}(z)^{-1}$
$\displaystyle+\frac{1}{2i\sqrt{t}}\mathbf{M^{(lo)}}(z)\begin{pmatrix}0&\beta_{12}(r_{\xi})\\\
-\beta_{21}(r_{\xi})&0\end{pmatrix}\mathbf{M^{(lo)}}(z)^{-1}+\mathcal{O}(t^{-1}).$

####  3.3.3 Analysis on the $N$-loop soliton problem

######  RHP 3.10.

Find a 3$\times$3 matrix-valued function $\mathbf{M^{(lo)}}(z)$ analytic on
$\mathbb{C}\setminus\Gamma$, with the following properties:

$\bigstar$_(the normalization condition)_

$\mathbf{M^{(lo)}}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the residue condition)_

The residue condition of $\mathbf{M^{(lo)}}(z)$ is as same as
$M_{\Diamond}(y,t,z)$ of RH problem 2.1.

$\bigstar$_(Singularities)_

As $z\to 0$, the limit of $\mathbf{M^{(lo)}}(z)$ has pole singularities as
$M_{\Diamond}(y,t,z)$.

According to symmetries of Proposition 2.1, the simplest cases involves $6$
poles. Suppose there exists $N^{\prime}$ poles in $\Omega_{1}$ where
$N=6N^{\prime}$ which is $z=\rho e^{\frac{i\pi}{6}+\frac{i\pi
m}{3}},m=0,\dots,5$ for some $r>0$. Then yield the following behavior as $z\to
0$:

$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j1}(z)=\hat{q}(1+z\omega
F+\frac{1}{2}z^{2}\omega^{2}F^{2})+\mathcal{O}(z^{3}),$ (3.58)
$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j2}(z)=\hat{q}(1+z\omega^{2}F+\frac{1}{2}z^{2}\omega
F^{2})+\mathcal{O}(z^{3}),$ (3.59)
$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j3}(z)=\hat{q}(1+zF+\frac{1}{2}z^{2}F^{2})+\mathcal{O}(z^{3}).$
(3.60)

where for $1\leq i,j\leq 3$, the $\mathbf{M^{(lo)}}_{ij}(z)$ stands for the
$(i,j)$ entry of matrix $\mathbf{M^{(lo)}}(z)$ and
$q(x,t)=\hat{q}(y(x,t),t)=\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j3}(0)$.

On the other hand, the definitions of $y(x,t)$ and $F(y,t)$ as (2.7) and
(2.25) yield the following necessary condition to be satisfied by the
coefficients $F$ and $\hat{q}$ in the expansion (4.3)-(4.5), for small $z$, of
the solution of the RH problem 4.2:

$\frac{\partial F}{\partial y}=\frac{1-\hat{q}}{\hat{q}},$

and $\hat{q}(y,t)$ satisfies the differential equation:

$\Bigg{(}\frac{\hat{q}_{y}}{\hat{q}}\Bigg{)}_{t}=\hat{q}^{2}-\frac{1}{\hat{q}}.$

We yield real-valued $x(y,t)$ as same as $\hat{u}(y,t)$ for all
$\alpha\in\mathbb{C}$:

$\displaystyle x(y,t)=y-\frac{2}{\rho}\frac{\mbox{Re}(\alpha
e^{i\pi/3})}{1+2/\rho\mbox{Im}\alpha},$
$\displaystyle\hat{q}(y,t)=1+\frac{2}{\rho}\mbox{Im}\alpha.$

Moreover, in order to evaluate the asymptotics of $u(x,t)$, we use the
expansion as $z\to 0$,

$\displaystyle u(x,t)^{(lo)}$ $\displaystyle=\hat{u}(y,t)=\frac{\partial
x(y,t)}{\partial t}$
$\displaystyle=\sum_{j=1}^{N^{\prime}}\frac{\hat{e}}{\rho_{j}^{2}}\frac{(\cos(\phi_{j}+\frac{\pi}{3})-\hat{e}+\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2})}{(1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2})^{2}}$
$\displaystyle+\sum_{j=1}^{N^{\prime}}\frac{4}{\hat{e}}\frac{(j-1)\cos^{2}(\phi_{j}+\frac{\pi}{3})+\cos(\phi_{j}-\frac{\pi}{3})+\hat{e}}{(\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2}+\hat{e})^{2}}.$

Where

$\hat{e}(y,t)=\frac{|\gamma|}{2\sqrt{3}\rho}e^{-\sqrt{3}\rho
y-\frac{\sqrt{3}}{\rho}t}=e^{-\sqrt{3}\rho(y+\frac{t}{\rho^{2}}+y_{0})},$

and

$y_{0}=-\frac{1}{\sqrt{3}\rho}\log\frac{|\gamma|}{2\sqrt{3}\rho},$

where $\phi,\rho$ is determined by the constant of residue condition of RH
problem 2.1 and

$\displaystyle
x(y,t)=y+F(y,t)=y+\frac{2\sqrt{3}}{\rho_{j}}\frac{-2\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}{1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}.$

###  3.4 Analysis on the remaining $\bar{\partial}$-problem

$\bar{\partial}$-RH problem 3.1 is equivalent to the integral equation

$M^{(3)}(z)=I-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{\bar{\partial}M^{(3)}(s)}{s-z}dA(s)=I-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{M^{(3)}(s)W^{(3)}(z)}{s-z}dA(s),$
(3.61)

where $dA(s)$ is Lebesgue measure on the plane.

Equation (3.61) can be written using operator notation as

$(I-S)M^{(3)}(z)=I,$ (3.62)

where $S$ is the solid Cauchy operator

$S(f)(z)=-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{f(s)W^{(3)}(s)}{s-z}dA(s).$
(3.63)

The following Proposition 3.9 shows that for sufficiently large $t$ the
operator $S$ is small-norm, so that the resolvent operator $(I-S)^{-1}$ exists
and can be expressed as a Neumann series.

######  Proposition 3.9.

There exist a constant $C$ such that for all $t>0$, the operator (3.63)
satisfies the inequality

$\|S\|_{L^{\infty}\to L^{\infty}}\leq Ct^{-1/2\sqrt{3}}.$ (3.64)

###### Proof.

We detail the case for matrix functions having support in the region
$\Omega_{4}$, the case for the other regions follows similarly. Let $A\in
L^{\infty}(\Omega_{4})$ and $s=u+iv$, then it follows that

$\displaystyle|S(A)(s)|\leq\iint_{\Omega_{4}}\frac{|A(s)M^{(2)}_{rhp}(s)W^{(2)}(s)M^{(2)}_{rhp}(s)^{-1}|}{|s-z|}$
$\displaystyle\lesssim\|A\|_{L^{\infty}(\Omega_{4})}\|M^{(2)}_{rhp}\|_{L^{\infty}(\Omega_{4}^{*})}\|M^{(2)^{-1}}_{rhp}\|_{L^{\infty}(\Omega_{4}^{*})}\iint_{\Omega_{4}}\frac{|\bar{\partial}R_{4}(s)|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}}{|s-z|},$ (3.65)

where $\Omega_{4}^{*}:=\Omega_{4}\cap\mbox{supp}(1-\chi_{\mathcal{Z}})$ is
bounded away from the pole $z_{k}$ of $M^{(2)}_{RHP}$, so that
$\|(M^{(2)}_{rhp})^{\pm 1}\|_{L^{\infty}(\Omega_{4}^{*})}$ are finite.

Using (3.6) the result follows Proposition 3.10 using the estimates to bound
the finial integral term of (3.65):

$\|S\|_{L^{\infty}\to L^{\infty}}\leq C(I_{1}+I_{2}+I_{3})\leq
Ct^{-1/2\sqrt{3}},$ (3.66)

where

$\displaystyle
I_{1}=\iint_{\Omega_{4}}\frac{|\bar{\partial}(\chi_{\mathcal{Z}})|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}}{|s-z|}dA(s),$ (3.67) $\displaystyle
I_{2}=\iint_{\Omega_{4}}\frac{|p^{\prime}(Re(z))|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}}{|s-z|}dA(s),$ (3.68) $\displaystyle
I_{3}=\iint_{\Omega_{4}}\frac{|z\pm\varkappa|^{-1/2}e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}}{|s-z|}dA(s).$ (3.69)

∎

######  Proposition 3.10.

There exist constants $c_{1},c_{2},$ and $c_{3}$ such that for all $t>0$, the
integrals $I_{j},j=1,2,3$, defined by (3.67)-(3.69) satisfy the bounds

$|I_{j}|\leq c_{j}t^{-1/2\sqrt{3}},\quad j=1,2,3.$ (3.70)

###### Proof.

Our proof follows that found in [33]. Let $s=u+iv$ and $z=\alpha+i\beta$.
Throughout we use the elementary fact

$\displaystyle\Big{\|}\frac{1}{s-z}\Big{\|}_{L^{2}(v+\varkappa)}^{2}$
$\displaystyle=\Big{(}\int_{v+\xi}^{\infty}\frac{1}{(u-\alpha)^{2}+(v-\beta)^{2}}du\Big{)}^{1/2}$
$\displaystyle\leq\int_{\mathbb{R}}\frac{1}{u^{2}+(v-\beta)^{2}}du=\frac{\pi}{|v-\beta|},$

to shown that

$\displaystyle|I_{1}|$
$\displaystyle\leq\int_{0}^{\infty}\int_{v+\varkappa}^{\infty}\frac{|\chi_{\mathcal{Z}}(s)|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}}{|s-z|}dudv\lesssim\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}\int_{v+\varkappa}^{\infty}\frac{|\chi_{\mathcal{Z}}(s)|}{|s-z|}dudv$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}\|\chi_{\mathcal{Z}}(s)\|_{L^{2}(v+\varkappa,\infty)}\Big{\|}\frac{1}{s-z}\Big{\|}_{L^{2}(v+\varkappa,\infty)}dv$
$\displaystyle\lesssim
c_{1}t^{-1/2\sqrt{3}}\int_{\mathbb{R}}\frac{e^{-4\omega^{2}}}{|\omega|^{1/2}}\leq
c_{1}t^{-1/2\sqrt{3}}.$

The bound for $I_{2}$ is similar to $I_{1}$ that is

$\displaystyle|I_{1}|$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}\int_{v+\varkappa}^{\infty}\frac{|r^{\prime}(u)|}{|s-z|}dudv$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}\Big{\|}\frac{1}{s-z}\Big{\|}_{L^{2}(v+\varkappa,\infty)}dv\leq
c_{2}t^{-1/2\sqrt{3}}.$

For $I_{3}$ choose $p>2$ and $q$ Hölder conjugate to $p$, then

$\displaystyle|I_{3}|$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}\|(s-\varkappa)^{-1/2}\|_{L^{p}(v+\varkappa,\infty)}\|(s-z)^{-1}\|_{L^{q}(v+\xi,\infty)}dv$
$\displaystyle\lesssim
c_{p}\int_{0}^{\infty}e^{-2\sqrt{3}tv^{2}}v^{1/p-1/2}|v-\beta|^{1/q-1}dv\leq
c_{3}t^{-1/2\sqrt{3}}.$

The result is confirmed. ∎

To recover the long-time asymptotic behavior of $u(x,t)$ it is necessary to
determine the asymptotic behavior of the coefficient of the $z^{-1}$ term in
the Laurent expansion of $M^{(3)}(z)$ at infinity. An integral representation
of this term is given by the expansion

$\displaystyle M^{(3)}(z)$
$\displaystyle=I-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{M^{(3)}(s)W^{(3)}(s)}{s-z}dA(s)$
$\displaystyle=I+\frac{M_{1}}{z}+\frac{1}{\pi}\iint_{\mathbb{C}}\frac{sM^{(3)}(s)W^{(3)}(s)}{z(s-z)}dA(s),$
(3.71)

where

$M_{1}^{(3)}=\frac{1}{\pi}\iint_{\mathbb{C}}M^{(3)}(s)W^{(3)}dA(s).$ (3.72)

######  Proposition 3.11.

For all $t>0$ there exists a constant $c$ such that

$|M_{1}^{(3)}|\leq ct^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}.$ (3.73)

###### Proof.

Recalling that the set
$\Omega_{4}^{*}=\Omega_{4}\cap\mbox{supp}(1-\chi_{\mathcal{Z})}$ is bounded
away from the poles of $M^{({2})}_{rhp}$, we have

$\displaystyle|M_{1}^{(3)}|$
$\displaystyle\leq\frac{1}{\pi}\iint_{\Omega_{4}}|M^{(3)}(s)M^{(2)}_{rhp}(s)W^{(2)}(s)M^{(2)}_{rhp}(s)^{-1}|dA$
$\displaystyle\leq\frac{1}{\pi}\|M^{(3)}\|_{L^{\infty}(\Omega_{4})}\|M^{(2)}_{rhp}\|_{L^{\infty}(\Omega^{*})}\|M^{(2)^{-1}}_{rhp}\|_{L^{\infty}(\Omega^{*})}\iint_{\Omega_{4}}|\bar{\partial}R_{4}e^{2it\theta}|dA$
$\displaystyle\leq C(I_{4}+I_{5}+I_{6}),$

where

$\displaystyle
I_{4}=\iint_{\Omega_{4}}|\bar{\partial}(\chi_{\mathcal{Z})}|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}dA(s),$ (3.74) $\displaystyle
I_{5}=\iint_{\Omega_{4}}|p^{\prime}(Re(z))|e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}dA(s),$ (3.75) $\displaystyle
I_{6}=\iint_{\Omega_{4}}|z\pm\varkappa|^{-1/2}e^{-2\sqrt{3}t\zeta
uv-\frac{\sqrt{3}tv}{u^{2}+v^{2}}}dA(s).$ (3.76)

We bound $I_{4}$ by applying the Cauchy-Schwartz inequality:

$\displaystyle|I_{4}|$
$\displaystyle\lesssim\int_{0}^{\infty}\|\chi_{\mathcal{Z}}\|_{L^{2}(v+\varkappa,\infty)}\Big{(}\int_{v}^{\infty}e^{-2\sqrt{3}t\zeta
uv}\Big{)}^{1/2}dv$ $\displaystyle\lesssim
ct^{-1/2}\int_{0}^{\infty}\frac{e^{-\sqrt{3}tv^{2}}}{\sqrt{v}}dv\lesssim
ct^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}$

The bound for $I_{5}$ follows in the same manner as for $I_{4}$. For $I_{6}$
we proceed as with $I_{3}$ applying Hölder’s inequality with

$\displaystyle|I_{6}|\leq
c\int_{0}^{\infty}v^{1/p-1/2}\Big{(}\int_{v}^{\infty}e^{-2\sqrt{3}t\zeta
uv}\Big{)}^{1/q}dv\lesssim ct^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}$

∎

###  3.5 The proof of Theorem 1.1 in region $x/t<\epsilon$

In this region we have the solution of RH problem 2.1 is given by

$M_{\Diamond}(y,t,z)=M^{(3)}(z)E(z)M^{(2)}_{rhp}(z)\mathcal{R}^{(2)}(z)\tilde{\delta}(z),$
(3.77)

Now

$\displaystyle u(x,t)$ $\displaystyle=u^{(lo)}(x,t)+\Big{(}\underset{z\to
0}{\lim}\Big{(}\frac{\sum_{j=1}^{3}f(y,t,z)}{\sum_{j=1}^{3}f(y,t,0)}-1\Big{)}\frac{1}{z}\Big{)}_{t}+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$
(3.78)

where

$\displaystyle f(x,t,z)$
$\displaystyle\sim\wp(z)\Big{(}\frac{\delta(\omega^{2}z)}{\delta(\omega
z)}+\frac{i\beta_{12}}{\sqrt{ct}(-z\omega+\varkappa)}+\frac{-i\beta_{21}}{\sqrt{ct}(z\bar{\omega}^{2}+\varkappa)}$
$\displaystyle+\frac{i\beta_{12}}{\sqrt{ct}(-z\bar{\omega}+\varkappa)}+\frac{-i\beta_{21}}{\sqrt{ct}(z\omega^{2}+\varkappa)}+1\Big{)},$

and

$\wp(z)=1+z\Delta+\mathcal{O}(z^{2}),$

and

$\Delta=\Delta(\varkappa(\zeta))=\frac{\sqrt{3}}{\pi}\int_{\varkappa}^{\infty}\frac{\log(1-|r(s)|^{2})}{s^{2}}ds,$

We have

$\frac{f(x,t,z)-f(0)}{f(0)z}=\Delta+\frac{2}{\varkappa^{2}\sqrt{ct}}\big{(}\mbox{Re}(i\beta_{12})-\mbox{Re}(-i\beta_{21})\big{)}+\frac{\delta(\omega^{2}z)-\delta(\omega
z)}{z\delta(\omega z)\varkappa^{2}\sqrt{ct}}$ (3.79)

By using (3.16), (3.35), (3.40) we obtain

$\displaystyle\mbox{Re}(i\beta_{12})=\sqrt{\nu}\cos\Bigg{(}\frac{2\sqrt{3}}{\varkappa}t+\nu\log
t+\phi_{1}\Bigg{)},$
$\displaystyle-\mbox{Re}(i\beta_{21})=\sqrt{\nu}\cos\Bigg{(}\frac{2\sqrt{3}}{\varkappa}t+\nu\log
t+\phi_{2}\Bigg{)},$

where for $j=1,2$, we obtain

$\phi_{j}=\arg\Gamma(-i\nu)+\arg
r((-1)^{-1}\varkappa)+\nu\log\frac{8\sqrt{3}}{\varkappa}+\frac{\pi}{4}+(-1)^{j}\frac{2\pi}{3}$

Now we asymptotics of $u(x,t)$ can be calculated by differentiating (3.79)
with respect to $t$ and taking into account the change of variables $y\mapsto
x$, we acquire that

$u(x,t)=u^{(lo)}(x,t)+\frac{k_{1}}{\sqrt{t}}\cos(\frac{2\sqrt{3}}{\varkappa}t+\nu\log
t+k_{2})-\frac{\delta(\omega^{2}z)-\delta(\omega z)}{2z\delta(\omega
z)\varkappa^{2}ct^{3/2}}+\mathcal{O}(t^{-\frac{\sqrt{3}}{3}-\frac{1}{2}}),$
(3.80)

where

$\displaystyle
k_{1}=-2^{3/2}3^{1/4}\sqrt{\frac{\nu}{\varkappa}}\sin\Big{(}\frac{\arg
r(\varkappa)-\arg r(-\varkappa)}{2}-\frac{2\pi}{3}\Big{)},$ $\displaystyle
k_{2}=\nu\log\frac{8\sqrt{3}}{\varkappa}+\frac{\arg r(\varkappa)+\arg
r(-\varkappa)}{2}+\arg\Gamma(-i\nu)+\frac{\pi}{4}$
$\displaystyle\qquad+\frac{3\varkappa}{\pi}\int_{\varkappa}^{\infty}\frac{\log(1-|r(s)|^{2})}{s^{2}}ds+\frac{1}{\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\log|\varkappa-s|d\log(1-|r(s)|^{2})$
$\displaystyle\qquad+\frac{1}{2\pi}(\int_{-\infty}^{-\varkappa}+\int_{\varkappa}^{\infty})\frac{\log(1-|r(s)|^{2})(2s+\varkappa)}{s^{2}+s\varkappa+\varkappa^{2}}$

where $\Gamma$ is the Euler Gamma function and $\nu$ is define in (3.3).

##  4 Soliton resolution in region $x/t>\epsilon$

###  4.1 Conjugation

In order to move the oscillatory terms, in the jump relation, into regions
where they are decaying, the signature table 2(j) suggests the deformation of
the original RH problem 2.1 according to trigonal factorizations of the jump
matrix of type (2.34):

$\displaystyle S_{\Diamond}(y,t;z)$ $\displaystyle=\begin{pmatrix}1&0&0\\\
-r(z)e^{2it\theta(\zeta,z)}&1&0\\\
0&0&1\end{pmatrix}\begin{pmatrix}1&\bar{r}(z)e^{-2it\theta(\zeta,z)}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\qquad\qquad\qquad z\in\mathbb{R},$
$\displaystyle=\begin{pmatrix}1&0&-r(\omega^{2}z)e^{2it\theta(\zeta,\omega^{2}z)}\\\
0&1&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
\bar{r}(\omega^{2}z)e^{-2it\theta(\zeta,\omega^{2}z)}&0&1\end{pmatrix},\qquad
z\in\omega\mathbb{R},$ $\displaystyle=\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&-r(\omega z)e^{2it\theta(\zeta,\omega
z)}&1\end{pmatrix}\begin{pmatrix}1&0&0\\\ 0&1&\bar{r}(\omega
z)e^{-2it\theta(\zeta,\omega^{2}z)}\\\
0&0&1\end{pmatrix},\qquad\quad\,\,z\in\omega^{2}\mathbb{R}.$

###  4.2 Analysis on the pure RH problem

In order to transform the triangular factors into the new RH problem, we
define $M^{(1)}_{\Diamond}(y,t,z)$ with the contour $\hat{\Sigma}$ shown in
Fig. 4.1 as following:

$M_{\Diamond}^{(1)}(y,t,z)=\begin{cases}M_{\Diamond}(y,t,z)\begin{pmatrix}1&0&0\\\
-r(z)e^{2it\theta(\zeta,z)}&1&0\\\
0&0&1\end{pmatrix},\qquad\,\,z\in\mbox{yellow part},\\\ \\\
M_{\Diamond}(y,t,z)\begin{pmatrix}1&-\bar{r}(z)e^{-2it\theta(\zeta,z)}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\qquad\,\,z\in\mbox{gray part},\\\ \\\
M_{\Diamond}(y,t,z)\begin{pmatrix}1&0&-r(\omega^{2}z)e^{2it\theta(\zeta,\omega^{2}z)}\\\
0&1&0\\\ 0&0&1\end{pmatrix},\quad z\in\mbox{violet part},\\\ \\\
M_{\Diamond}(y,t,z)\begin{pmatrix}1&0&0\\\ 0&1&0\\\
-\bar{r}(\omega^{2}z)e^{-2it\theta(\zeta,\omega^{2}z)}&0&1\end{pmatrix},\quad
z\in\mbox{orange part},\\\ \\\ M_{\Diamond}(y,t,z)\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&-r(\omega z)e^{2it\theta(\zeta,\omega
z)}&1\end{pmatrix},\qquad\,\,\,z\in\mbox{pink part},\\\ \\\
M_{\Diamond}(y,t,z)\begin{pmatrix}1&0&0\\\ 0&1&\bar{r}(\omega
z)e^{-2it\theta(\zeta,\omega^{2}z)}\\\ 0&0&1\end{pmatrix},\qquad
z\in\mbox{green part},\\\ \\\
M_{\Diamond}(y,t,z),\quad\mbox{everywhere\,\,else}\end{cases}$

According this transformation, we obtain the following RH problem.

######  RHP 4.1.

Find a analytic function
$M^{(1)}_{\Diamond}(y,t;z):\mathbb{C}\setminus(\hat{\Sigma}\cup\Gamma)\to
SL_{3}(\mathbb{C})$ with the following properties:

$\bigstar$_(the normalization condition)_

$M^{(1)}_{\Diamond}(y,t,z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the jump condition)_

For each $z\in\hat{\Sigma}$, the boundary values $M^{(1)}_{\Diamond\pm}$
satisfy the jump relation
$M^{(1)}_{\Diamond+}(y,t;z)=M^{(1)}_{\Diamond-}(y,t;z)S^{(1)}_{{\dagger}}(y,t;z),\,\,y\in\hat{\Sigma}$
where

$\displaystyle
S^{(1)}_{{\dagger}}(y,t;z)=\begin{cases}\begin{pmatrix}1&\bar{r}(z)e^{-2it\theta(\zeta,z)}&0\\\
0&1&0\\\ 0&0&1\end{pmatrix},\quad z\in l_{11}\cup l_{12},\\\ \\\
\begin{pmatrix}1&0&0\\\ -r(z)e^{2it\theta(\zeta,z)}&1&0\\\
0&0&1\end{pmatrix},\quad z\in l_{41}\cup l_{42},\\\ \\\
\begin{pmatrix}1&0&0\\\ 0&1&\bar{r}(\omega
z)e^{-2it\theta(\zeta,\omega^{2}z)}\\\ 0&0&1\end{pmatrix},\quad z\in
l_{21}\cup l_{51},\\\ \\\ \begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&-r(\omega
z)e^{2it\theta(\zeta,\omega z)}&1\end{pmatrix},\quad z\in l_{22}\cup
l_{52},\\\ \\\ \begin{pmatrix}1&0&0\\\ 0&1&\bar{r}(\omega
z)e^{-2it\theta(\zeta,\omega^{2}z)}\\\ 0&0&1\end{pmatrix},\quad z\in
l_{31}\cup l_{61},\\\ \\\ \begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&-r(\omega
z)e^{2it\theta(\zeta,\omega z)}&1\end{pmatrix},\quad z\in l_{32}\cup
l_{62},\\\ \end{cases}$

$\bigstar$_(the residue condition)_

The residue condition of $M^{(1)}_{\Diamond}(y,t,z)$ is as same as
$M_{\Diamond}(y,t,z)$ .

$\bigstar$_(Singularities)_

As $z\to 0$, the limit of $M^{(1)}_{\Diamond}$ has pole singularities:

$M^{(1)}_{\Diamond}(y,t,0)=\begin{pmatrix}\alpha&\beta&\gamma\\\
\gamma&\alpha&\beta\\\ \beta&\gamma&\alpha\end{pmatrix},$ (4.1)

where $\alpha(y,t)$ and $\beta(y,t)$ have the form

$\alpha=\frac{1}{3}\big{(}\hat{q}+1+\frac{1}{\hat{q}}\big{)},\quad\beta=\frac{1}{3}\big{(}\hat{q}+\omega+\frac{\omega^{2}}{\hat{q}}\big{)},$

with some $\hat{q}(y,t)>0$ such that $\hat{q}\to 1$ as $y\to\infty$.

$l_{11}$$l_{12}$$l_{41}$$l_{42}$$l_{22}$$l_{21}$$l_{31}$$l_{32}$$l_{51}$$l_{52}$$l_{62}$$l_{61}$
Figure 4.1: Coutour $\hat{\Sigma}=\sum_{j}l_{j1(2)}$ for $j=1,\dots,6$.

We notice that the jump matrix $S^{(1)}_{{\dagger}}(y,t;z)$ is exponentially
decaying in $t$ to the identity matrix, the solution to this problem decay to
$I$ and consequently $\hat{u}(y,t)$ decay fast to $0$ uniformly in this
region. From the signature table of Fig.2(j) and the triangulates of the jump
matrices, we observe that along the characteristics line $x=vt$, by choosing
the radius of each element of $\Gamma$ small enough, we have for

$|S^{(1)}_{{\dagger}}(y,t;z)|\lesssim e^{-ct}.$ (4.2)

###  4.3 Analysis on the $N$-loop soliton problem

Next we consider the contribution of discrete spectrum to the solution of the
RH problem. The exact $N$ loop soliton solution to the OV equation (1.1) is
found in implicit form by means of a transformation back to the original
independent variables. The shifts that occurs when the solitons interact are
found. For solving this problem we firstly consider the following RH problem
4.2 with $r=0$.

######  RHP 4.2.

Find a 3$\times$3 matrix-valued function $\mathbf{M^{(lo)}}(z)$ analytic on
$\mathbb{C}\setminus\Gamma$, with the following properties:

$\bigstar$_(the normalization condition)_

$\mathbf{M^{(lo)}}(z)=I+\mathcal{O}(z^{-1})\quad z\to\infty.$

$\bigstar$_(the residue condition)_

The residue condition of $\mathbf{M^{(lo)}}(z)$ is as same as
$M_{\Diamond}(y,t,z)$ of RH problem 2.1.

$\bigstar$_(Singularities)_

As $z\to 0$, the limit of $\mathbf{M^{(lo)}}(z)$ has pole singularities as
$M_{\Diamond}(y,t,z)$.

According to symmetries of Proposition 2.1, the simplest cases involves $6$
poles. Suppose there exists $N^{\prime}$ poles in $\Omega_{1}$ where
$N=6N^{\prime}$ which is $z=\rho e^{\frac{i\pi}{6}+\frac{i\pi
m}{3}},m=0,\dots,5$ for some $r>0$. Then yield the following behavior as $z\to
0$:

$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j1}(z)=\hat{q}(1+z\omega
F+\frac{1}{2}z^{2}\omega^{2}F^{2})+\mathcal{O}(z^{3}),$ (4.3)
$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j2}(z)=\hat{q}(1+z\omega^{2}F+\frac{1}{2}z^{2}\omega
F^{2})+\mathcal{O}(z^{3}),$ (4.4)
$\displaystyle\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j3}(z)=\hat{q}(1+zF+\frac{1}{2}z^{2}F^{2})+\mathcal{O}(z^{3}).$
(4.5)

where for $1\leq i,j\leq 3$, the $\mathbf{M^{(lo)}}_{ij}(z)$ stands for the
$(i,j)$ entry of matrix $\mathbf{M^{(lo)}}(z)$ and
$q(x,t)=\hat{q}(y(x,t),t)=\sum_{i=1}^{3}\mathbf{M^{(lo)}}_{j3}(0)$.

On the other hand, the definitions of $y(x,t)$ and $F(y,t)$ as (2.7) and
(2.25) yield the following necessary condition to be satisfied by the
coefficients $F$ and $\hat{q}$ in the expansion (4.3)-(4.5), for small $z$, of
the solution of the RH problem 4.2:

$\frac{\partial F}{\partial y}=\frac{1-\hat{q}}{\hat{q}},$

and $\hat{q}(y,t)$ satisfies the differential equation:

$\Bigg{(}\frac{\hat{q}_{y}}{\hat{q}}\Bigg{)}_{t}=\hat{q}^{2}-\frac{1}{\hat{q}}.$

We yield real-valued $x(y,t)$ as same as $\hat{u}(y,t)$ for all
$\alpha\in\mathbb{C}$:

$\displaystyle x(y,t)=y-\frac{2}{\rho}\frac{\mbox{Re}(\alpha
e^{i\pi/3})}{1+2/\rho\mbox{Im}\alpha},$
$\displaystyle\hat{q}(y,t)=1+\frac{2}{\rho}\mbox{Im}\alpha.$

Moreover, in order to evaluate the asymptotics of $u(x,t)$, we use the
expansion as $z\to 0$,

$\displaystyle u^{(lo)}(x,t)$ $\displaystyle=\hat{u}(y,t)=\frac{\partial
x(y,t)}{\partial t}$
$\displaystyle=\sum_{j=1}^{N^{\prime}}\frac{\hat{e}}{\rho_{j}^{2}}\frac{(\cos(\phi_{j}+\frac{\pi}{3})-\hat{e}+\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2})}{(1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2})^{2}}$
$\displaystyle+\sum_{j=1}^{N^{\prime}}\frac{4}{\hat{e}}\frac{(j-1)\cos^{2}(\phi_{j}+\frac{\pi}{3})+\cos(\phi_{j}-\frac{\pi}{3})+\hat{e}}{(\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2}+\hat{e})^{2}}.$

Where

$\hat{e}(y,t)=\frac{|\gamma|}{2\sqrt{3}\rho}e^{-\sqrt{3}\rho
y-\frac{\sqrt{3}}{\rho}t}=e^{-\sqrt{3}\rho(y+\frac{t}{\rho^{2}}+y_{0})},$

and

$y_{0}=-\frac{1}{\sqrt{3}\rho}\log\frac{|\gamma|}{2\sqrt{3}\rho},$

where $\phi,\rho$ is determined by the constant of residue condition of RH
problem 2.1 and

$\displaystyle
x(y,t)=y+F(y,t)=y+\frac{2\sqrt{3}}{\rho_{j}}\frac{-2\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}{1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}.$

###  4.4 The proof of Theorem 1.1 in region $x/t>\epsilon$

######  Proposition 4.1.

For $z=\rho e^{i\theta}$ and $\rho\to\infty$, and $j=1,2,3$ the function

$\displaystyle u(x,t)=$ $\displaystyle u^{(lo)}+R(x,t)$
$\displaystyle=\sum_{j=1}^{N^{\prime}}\frac{12}{\rho_{j}^{2}}\frac{\hat{e}(\cos(\phi_{j}+\frac{\pi}{3})-\hat{e}+\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2})}{(1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2})^{2}}$
$\displaystyle+\sum_{j=1}^{N^{\prime}}\frac{4}{\hat{e}}\frac{(j-1)\cos^{2}(\phi_{j}+\frac{\pi}{3})+\cos^{2}(\phi_{j}-\frac{\pi}{3})+\hat{e}}{(\cos(\phi_{j}+\frac{\pi}{3})\hat{e}^{2}+\hat{e})^{2}}+\mathcal{O}(t^{-n}),$

where $\phi,\rho$ is determined by the constant of residue condition of RH
problem 2.1 and

$\displaystyle
x(y,t)=y+F(y,t)=y+\frac{2\sqrt{3}}{\rho_{j}}\frac{-2\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}{1-4\cos(\phi_{j}+\frac{\pi}{3})\hat{e}+\hat{e}^{2}}.$

Acknowledgements This work is supported by the National Natural Science
Foundation of China (Grant No. 12271104, 51879045).

Data Availability Statements

The data that supports the findings of this study are available within the
article.

Conflict of Interest

The authors have no conflicts to disclose.

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