11footnotetext:  School of Mathematical Sciences and Key Laboratory for
Nonlinear Science, Fudan University, Shanghai 200433, P.R. China.

#  On the asymptotic stability of $N$-soliton solutions of the three-wave
resonant interaction equation

Yiling YANG1 and Engui FAN1 Email address<EMAIL_ADDRESS>Corresponding author and email address<EMAIL_ADDRESS>
###### Abstract

The three-wave resonant interaction (three-wave) equation not only possesses
$3\times 3$ matrix spectral problem, but also being absence of stationary
phase points, which give rise to difficulty on the asymptotic analysis with
stationary phase method or classical Deift-Zhou steepest descent method. In
this paper, we study the long time asymptotics and asymptotic stability of
$N$-soliton solutions of the initial value problem for the three-wave equation
in the solitonic region

$\displaystyle
p_{ij,t}-n_{ij}p_{ij,x}+\sum_{k=1}^{3}(n_{kj}-n_{ik})p_{ik}p_{kj}=0,$ (0.1)
$\displaystyle p_{ij}(x,0)=p_{ij,0}(x),\quad x\in\mathbb{R},\ t>0,\
i,j,k=1,2,3,$ $\displaystyle for\ i\neq j,\ p_{ij}=-\bar{p}_{ji},\
n_{ij}=-n_{ji},$ (0.2)

where $n_{ij}$ are constants. The study makes crucial use of the inverse
scattering transform as well as of the $\overline{\partial}$ generalization of
Deift-Zhou steepest descent method for oscillatory Riemann-Hilbert (RH)
problems. Based on the spectral analysis of the Lax pair associated with the
three-wave equation and scattering matrix, the solution of the Cauchy problem
is characterized via the solution of a RH problem. Further we derive the
leading order approximation to the solution $p_{ij}(x,t)$ for the three-wave
equation in the solitonic region of any fixed space-time cone. The asymptotic
expansion can be characterized with an $N(I)$-soliton whose parameters are
modulated by a sum of localized soliton-soliton interactions as one moves
through the region; the residual error order $\mathcal{O}(t^{-1})$ from a
$\overline{\partial}$ equation. Our results provide a verification of the
soliton resolution conjecture and asymptotic stability of N-soliton solutions
for three-wave equation.
Keywords: three-wave resonant interaction equation; Riemann-Hilbert problem,
$\overline{\partial}$ steepest descent method, long time asymptotics,
asymptotic stability, soliton resolution.
AMS: 35Q51; 35Q15; 37K15; 35C20.

###### Contents

  1. 1 Introduction
  2. 2 The spectral analysis and the RH problem
  3. 3 The scattering maps
  4. 4 The deformation of RH problem
  5. 5 A mixed $\bar{\partial}$-RH problem and decomposition
  6. 6  Asymptotic analysis on soliton solutions
  7. 7 The small norm RH problem for error function
  8. 8  Asymptotic analysis on the pure $\bar{\partial}$-Problem
  9. 9 Asymptotic stability of N-soliton solutions

##  1 Introduction

In this paper, we consider the initial value problem for the three-wave
resonant interaction (three-wave) equation

$\displaystyle
p_{ij,t}-n_{ij}p_{ij,x}+\sum_{k=1}^{3}(n_{kj}-n_{ik})p_{ik}p_{kj}=0,$ (1.1)
$\displaystyle p_{ij}(x,0)=p_{ij,0}(x),\quad x\in\mathbb{R},\ t>0,\
i,j,k=1,2,3,$ $\displaystyle for\ i\neq j,\ p_{ij}=-\bar{p}_{ji},\
n_{ij}=-n_{ji}$ (1.2)

where the wave speeds $n_{ij}$ are a positive constant, and
$p_{ij}=p_{ij}(x,t)$ are a complex-valued functions of $x$ and $t$. Without
loss of generality we will assume that

$\displaystyle n_{23}>n_{13}>n_{12}.$

The presence of resonant triads, in which two wave modes conspire to generate
a third mode that grows until the small-amplitude assumption is violated, is a
primary obstacle to the effective description of small-amplitude dispersive
waves by linear theory. In [1], they derive a weakly nonlinear model for this
process, that is the complex amplitudes of these modes satisfying the three-
wave resonant interaction equations. The three-wave equations have a wide
variety of physical applications, originating from the fact that resonant wave
coupling is such a basic nonlinear phenomenon, such as buckling of cylindrical
shells [2], capillary-gravity waves [3], waves in plasmas [4, 5] and Rossby
waves [6]. It also has a variety of applications to nonlinear optics like
information storage and processing [7], resonant Bragg reflection [8]. So
(1.1) has been found numerous applications in physics and attracted the
attention of scientific community over the last few decades. The three-wave
equation can be solved through the inverse scattering method because it admits
a Lax representation [9, 10]. This integrability give us mathematical tools to
investigate several problems such as numerically solving the direct spectral
problem for both vanishing and non vanishing boundary values [11], the
initial-boundary value problem [14], semiclassical soliton ensembles [1], some
explicit solutions [12, 13], algebro-geometric quasi-periodic solutions [15],
qualitative results [17], finite-dimensional integrable system [16], series
approach (avoiding the inverse-scattering machinery) [18]. Moreover, it was
shown that if the signs of $\sum_{k=1}^{3}(n_{kj}-n_{ik})\neq 0$ and do not
all have the same sign, then the Cauchy problem for (1.1) with initial value
$p_{ij}(x,0)\in L^{2,s}(\mathbb{R})$ for all $s>0$ has a unique global
solution in which each field is a $C^{s}$ function of time with values
$L^{2,s}(\mathbb{R})$ for all $s>0$ (see Theorem 9.2.3) [19].

The inverse scattering transform (IST) procedure, as one of the most powerful
tool to investigate solitons of nonlinear integrable models, was first
discovered by Gardner, Green, Kruskal and Miura [20]. The development of the
IST formalism affects many fields of mathematics. The modern version of IST is
based on the dressing method proposed by Zakharov and Shabat, first in terms
of the factorization of integral operators on a line into a product of two
Volterra integral operators [21] and then using the Riemann-Hilbert (RH)
problem [22]. The most powerful version of the dressing method incorporates
the $\bar{\partial}$ problem formalism. The $\bar{\partial}$ problem was put
forward by Beals and Coifman as a generalization of the RH problem and was
applied to the study of first-order one-dimensional spectral problems [23,
24]. In general, the initial value problems of integrable systems only can be
solved by suing IST or RH method in the case of refectioness potentials. So a
natural idea is to study the asymptotic behavior of solutions to integrable
systems. The study on the long-time behavior of nonlinear wave equations was
first carried out by Manakov in 1974 [25]. Later, Zakharov and Manakov gave
the first result on the large-time asymptotic of solutions for the NLS
equation with decaying initial value [26] by this method. The inverse
scattering method also worked for long-time behavior of integrable systems
such as KdV, Landau-Lifshitz and the reduced Maxwell-Bloch system [27, 28,
29]. In 1993, Deift and Zhou developed a nonlinear steepest descent method to
rigorously obtain the long-time asymptotics behavior of the solution for the
MKdV equation by deforming contours to reduce the original RH problem to a
model one whose solution is calculated in terms of parabolic cylinder
functions [30]. Since then this method has been widely applied to the focusing
NLS equation, KdV equation, Camassa-Holm equation, Degasperis-Procesi, Fokas-
Lenells equation, Sasa-Satuma equation, short-pulse equation etc. [31, 32, 33,
34, 35, 36, 37, 38, 39, 40].

In recent years, McLaughlin and Miller further extended Deift-Zhou steepest
descent method to a $\bar{\partial}$ steepest descent method, which combine
steepest descent with $\bar{\partial}$-problem rather than the asymptotic
analysis of singular integrals on contours to analyze asymptotic of orthogonal
polynomials with non-analytical weights [41, 42]. When it is applied to
integrable systems, the $\bar{\partial}$ steepest descent method also has
displayed some advantages, such as avoiding delicate estimates involving
$L^{p}$ estimates of Cauchy projection operators, and leading the non-
analyticity in the RH problem reductions to a $\bar{\partial}$-problem in some
sectors of the complex plane. And its result can accommodate many situations
at once. In particular by considering small cones instead of fixed frames
which of reference it is able to account for uncertainties in the computation
(or measurement) of the spectral data and thus speed of the resulting
solitons. Moreover, for focusing NLS equation, this description of Long-time
asymptotic behavior to solution should also be useful to study non-integrable
perturbations where the discrete spectra would no longer be stationary. Dieng
and McLaughin used it to study the defocusing NLS equation under essentially
minimal regularity assumptions on finite mass initial data [43]; This method
was also successfully applied to prove asymptotic stability of N-soliton
solutions to focusing NLS equation [44]; Jenkins et.al studied soliton
resolution for the derivative nonlinear NLS equation for generic initial data
in a weighted Sobolev space [45]. For finite density initial data, Cussagna
and Jenkins improved $\bar{\partial}$ steepest descent method to study the
asymptotic stability for defocusing NLS equation with non-zero boundary
conditions [46]. Recently $\bar{\partial}$ steepest descent method has been
successfully used to study the short pulse, modifed Camassa-Holm and Fokas-
Lenells equations [47, 48, 49].

When various methods above are used to the nonlinear evolution equations
related to the higher order matrix spectral problems, the analysis process
becomes very difficult for both construction of exact solutions and asymptotic
analysis of solutions. Up to now, the RH methods have been extended to
construct exact solutions for integrable nonlinear evolution equations
associated with the $3\times 3$ matrix spectral problem, such as Sasa-Satuma
equation, Degasperis-Procesi, good Boussinesq, bad Boussinesq, three-wave,
Novikov equations [50, 51, 52, 53, 54]. However, among the these integrable
systems, only the Degasperis-Procesi equation, coupled nonlinear Schroinger
equation, Sasa-Satuma equation, cmKdV equation have been studied for long-time
asymptotic properties [35, 55, 56, 39].

Compared with other integrable systems, the three-wave equation exhibits some
different characteristics, for example, it not only possesses $3\times 3$
matrix spectral problem, but also involves three phase functions in its
corresponding RH problem. However, these three phase functions are absence of
stationary phase points. To the best of our knowledge, there is not any result
on asymptotics for the three-wave equation by Deift and Zhou method or
$\bar{\partial}$ steepest descent method. In this paper, we study the apply
$\overline{\partial}$ steepest descent method to study the asymptotic
stability of $N$-soliton solutions of the initial value problem for the three-
wave equation (1.1). This result is also a verification of the soliton
resolution conjecture for the three-wave equation.

This paper is arranged as follows. To make our presentation easy to understand
and self-contained, we recall some main results on the construction process of
RH problem with respect to the initial problem of the three-wave equation
(1.1) in section 2 ( for example, see [11, 14] in details), which will be used
to analyze long-time asymptotics of the three-wave equation in our paper. In
section 3, we establish the scattering maps from initial data $p_{ij,0}(x)\in
H^{1,2}(\mathbb{R})$ to the reflection coefficient $r_{j}(z)\in
H^{1,1}(\mathbb{R})$. In section 4, the function $T(z)$ is introduced to
define a new RH problem for $M^{(1)}(z)$, which admits a regular discrete
spectrum and two triangular decompositions of the jump matrix near original
point. In section 5, a mixed $\bar{\partial}$-RH problem for $M^{(2)}(z)$ is
obtained by continuous extension to $M^{(1)}(z)$ via introducing a matrix-
valued function $R^{(2)}(z)$. We further decompose $M^{(2)}(z)$ into a model
RH problem for $M^{rhp}(z)$ and a pure $\bar{\partial}$ Problem for
$M^{(3)}(z)$. The $M^{rhp}(z)$ can be obtained via a modified reflectionless
RH problem $M^{sol}(z)$ for the soliton components which is solved in Section
6. In section 7, the error function $E(z)$ between $M^{rhp}(z)$ and
$M^{sol}(z)$ can be computed with a small-norm RH problem. In Section 8, we
analyze the $\bar{\partial}$-problem for $M^{(3)}$. Finally, in Section 9,
based on the result obtained above, a relation formula is found

$\displaystyle
M(z)=M^{(3)}(z)E(z)M^{sol}(z)R^{(2)}(z)^{-1}T(z)^{-\sigma_{3}},$

from which we then obtain the long-time asymptotic behavior and asymptotic
stability for the three-wave equation (1.1) via reconstruction formula.

##  2 The spectral analysis and the RH problem

At the beginning of this section, we fix some notations used this paper. If
$I$ is an interval on the real line $\mathbb{R}$, and $X$ is a Banach space,
then $C^{0}(I,X)$ denotes the space of continuous functions on $I$ taking
values in $X$. It is equipped with the norm

$\|f\|_{C^{0}(I,X)}=\sup_{x\in I}\|f(x)\|_{X}.$

Moreover, denote $C^{0}_{B}(X)$ as a space of bounded continuous functions on
$X$.

If the elements $f_{1}$ and $f_{2}$ are in space $X$, then we call vector
$\vec{f}=(f_{1},f_{2})^{T}$ is in space $X$ with
$\parallel\vec{f}\parallel_{X}\triangleq\parallel f_{1}\parallel_{X}+\parallel
f_{2}\parallel_{X}$. Similarly, if every entries of matrix $A$ are in space
$X$, then we call $A$ is also in space $X$.

We introduce the following normed spaces:
The weighted $L^{p}(\mathbb{R})$ space is defined by

$L^{p,s}(\mathbb{R})=\left\\{f(x)\in L^{p}(\mathbb{R})|\hskip
2.84544pt|x|^{s}f(x)\in L^{p}(\mathbb{R})\right\\};$

The Sobolev space is defined by

$W^{k,p}(\mathbb{R})=\left\\{f(x)\in L^{p}(\mathbb{R})|\hskip
2.84544pt\partial^{j}f(x)\in L^{p}(\mathbb{R})\text{ for
}j=0,1,...,k\right\\};$

The weighted Sobolev space is defined by

$H^{k,s}(\mathbb{R})=\left\\{f(x)\in L^{2}(\mathbb{R})|\hskip
2.84544pt(1+|x|^{s})f(x),\partial^{j}f\in L^{2}(\mathbb{R}),\text{ for
}j=1,...,k\right\\}.$

And the norm of $f(x)\in L^{p}(\mathbb{R})$ and $g(x)\in L^{p,s}(\mathbb{R})$
are abbreviated to $\parallel f\parallel_{p}$, $\parallel g\parallel_{p,s}$
respectively.

The three-wave equation (1.1) admits the Lax pair [9, 10]

$\Phi_{x}=\left(izA+P\right)\Phi,\hskip
14.22636pt\Phi_{t}=\left(izB+Q\right)\Phi,$ (2.1)

while $\Phi(x,t,z)$ is a common 3-dim vector solution. $A$ and $B$ are real
diagonal constant matrices given by

$A={\rm diag}\\{a_{1},a_{2},a_{3}\\},\ \ B={\rm diag}\\{b_{1},b_{2},b_{3}\\}$

satisfying tr($A$)=tr($B$)=0. $P(x,t)$, $Q(x,t)$ are matrix valued functions
given by

$P=\left(\begin{array}[]{ccc}0&p_{12}&p_{13}\\\ -\bar{p}_{12}&0&p_{23}\\\
-\bar{p}_{13}&-\bar{p}_{23}&0\end{array}\right),\
Q=\left(\begin{array}[]{ccc}0&n_{12}p_{12}&n_{13}p_{13}\\\
n_{12}\bar{p}_{12}&0&n_{23}p_{23}\\\
n_{13}\bar{p}_{13}&n_{23}\bar{p}_{23}&0\end{array}\right),$

where $n_{ij}=\frac{b_{i}-b_{j}}{a_{i}-a_{j}}$. Besides, since
$a_{1},a_{2},a_{3}$ and $n_{23},n_{13},n_{12}$ are real, we assume that
$a_{1}>a_{2}>a_{3},\ n_{23}>n_{13}>n_{12}$ without loss of generality. We
first recall some main results on the construction process of RH problem.
Making transformation

$\Phi_{\pm}=\mu_{\pm}e^{iz(xA+tB)},$ (2.2)

then

$\mu_{\pm}\sim I,\hskip 14.22636ptx\rightarrow\pm\infty,$

and the system (2.1) then becomes

$\displaystyle(\mu_{\pm})_{x}=iz[A,\mu_{\pm}]+P\mu_{\pm},\hskip 14.22636pt$
(2.3) $\displaystyle(\mu_{\pm})_{t}=iz[B,\mu_{\pm}]+Q\mu_{\pm},\hskip
14.22636pt$ (2.4)

which leads to two Volterra type integrals

$\mu_{\pm}=I+\int_{\pm\infty}^{x}e^{iz\widehat{A}(x-y)}P(y)\mu_{\pm}(y)dy.$
(2.5)

The Able formula gives that $\det(\mu_{\pm})=\det(\Phi_{\pm})=1$. Denote

$\mu_{\pm}=(\mu_{\pm,ij})_{3\times
3}=\left(\left[\mu_{\pm}\right]_{1},\left[\mu_{\pm}\right]_{2},\left[\mu_{\pm}\right]_{3}\right),$

where $\left[\mu_{\pm}\right]_{i}$ for $i=1,2,3$ are the $i$-th columns of
$\mu_{\pm}$ respectively. Then from (2.5), we can show that
$\left[\mu_{-}\right]_{3}$ and $\left[\mu_{+}\right]_{1}$ are analytical in
$\mathbb{C}^{+}$; $\left[\mu_{+}\right]_{3}$ and $\left[\mu_{-}\right]_{1}$
are analytical in $\mathbb{C}^{-}$. Denote $X^{A}$ is the cofactor matrix of a
$3\times 3$ matrix $X$. It follows from (2.3) that the conjugate eigenfunction
$\mu^{A}$ satisfies the Lax pair:

$\displaystyle(\mu^{A}_{\pm})_{x}=-iz[A,\mu_{\pm}^{A}]-P^{T}\mu_{\pm}^{A},$
(2.6)
$\displaystyle(\mu^{A}_{\pm})_{t}=-iz[B,\mu_{\pm}^{A}]-Q^{T}\mu_{\pm}^{A},$
(2.7)

which leads to two Volterra type integrals:

$\mu^{A}_{\pm}=I-\int_{\pm\infty}^{x}e^{-iz\widehat{A}(x-y)}P^{T}(y)\mu^{A}_{\pm}(y)dy.$
(2.8)

$\bar{P}^{T}=-P$ and $\bar{Q}^{T}=-Q$ imply the following symmetry:

$\displaystyle\mu_{\pm}(z)=\overline{\mu^{A}_{\pm}(\bar{z})}.$ (2.9)

Since $\Phi_{\pm}(z;x,t)$ are two fundamental matrix solutions of the Lax pair
(2.1), there exists a linear relation between $\Phi_{+}(z;x,t)$ and
$\Phi_{-}(z;x,t)$, namely,

$\displaystyle\Phi_{-}(z;x,t)=\Phi_{+}(z;x,t)S(z),\hskip
14.22636ptz\in\mathbb{R},$ (2.10) $\displaystyle S(z)=(s_{ij}(z))_{3\times
3},\hskip 14.22636pt\det S(z)=1,$ (2.11)

where $S(z)$ is called scattering matrix and only depends on $z$. And combing
with (2.2), above equation is changed into

$\displaystyle\mu_{+}(z)=\mu_{-}(z)e^{iz(x\widehat{A}+t\widehat{B})}S(z).$
(2.12)

Consider the cofactor matrix of $S(z)$

$\displaystyle\mu^{A}_{+}(z)=\mu^{A}_{-}(z)e^{-iz(x\widehat{A}+t\widehat{B})}S^{A}(z).$
(2.13)

Then $S(z)$ and $S^{A}(z)$ admit symmetry reduction as

$S(z)=\overline{S^{A}(\bar{z})}.$ (2.14)

Moreover, $s_{11}(z)$, $s^{A}_{33}(z)$ are analysis in $\mathbb{C}^{+}$, while
$s_{33}(z)$, $s^{A}_{11}(z)$ are analysis in $\mathbb{C}^{-}$ with
$s_{11}(z)=\overline{s^{A}_{11}(\bar{z})}$ and
$s_{33}(z)=\overline{s^{A}_{33}(\bar{z})}$. The reflection coefficients are
defined by

$r_{1}(z)=\frac{s_{12}(z)}{s_{11}(z)},\ r_{2}(z)=\frac{s_{31}(z)}{s_{33}(z)},\
r_{3}(z)=\frac{s_{32}(z)}{s_{33}(z)},\ r_{4}(z)=\frac{s_{13}(z)}{s_{11}(z)},$
(2.15)

with $r_{4}+r_{1}\bar{r}_{3}+\bar{r}_{2}=0$. In addition, $\mu_{\pm}(z)$
admits the asymptotics

$\displaystyle\mu_{\pm}(z)=I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty,$ (2.16)

with reconstruction formula

$p_{ij}=-i(a_{i}-a_{j})\lim_{z\to\infty}[z\mu(z)]_{ij}.$ (2.17)

And the scattering matrix satisfy

$\displaystyle S(z)=I+\mathcal{O}(z^{-1}),\
S^{A}(z)=I+\mathcal{O}(z^{-1}),\hskip 14.22636ptz\rightarrow\infty.$ (2.18)

The zeros of $s_{11}(z)$ and $s^{A}_{33}(z)$ on $\mathbb{R}$ are known to
occur and they correspond to spectral singularities. They are excluded from
our analysis in the this paper. Recall the main result in [57] by Beals and
Coifman:

######  Lemma 1.

There exists a dense open set $P_{0}\subset L^{1}(\mathbb{R})$ such that if
$p_{ij,0}(x)\in P_{0}$, then $s_{11}(z)$ and $s^{A}_{33}(z)$ only has finite
number of simple zeros.

In next section 3, we establish the relationship between initial data
$p_{ij,0}(x)\in H^{1,2}(\mathbb{R})$ to the reflection coefficient
$r_{j}(z)\in H^{1,1}(\mathbb{R})$. To deal with our following work, we assume
our initial data satisfy this assumption.

######  Assumption 1.

The initial data $p_{ij,0}(x)\in H^{1,2}(\mathbb{R})\cap P_{0}$ and it
generates generic scattering data which satisfy that $s_{11}(z)$ and
$s^{A}_{33}(z)$ has no zeros on $\mathbb{R}$.

In fact, since scattering data $s_{11}(z),\ s^{A}_{33}(z)$ are analytical in
$\mathbb{C}^{+}$ and $s_{11}(z),s^{A}_{33}(z)\rightarrow 1,\
z\rightarrow\infty$, we can deduce that $s_{11}(z),s^{A}_{33}(z)$ have finite
zeros in $\mathbb{C}^{+}$. And suppose that $s_{11}(z)$ has $N_{1}$ simple
zeros $z_{1},...,z_{N_{1}}$ on $\mathbb{C}^{+}$, and $s^{A}_{33}(z)$ has
$N_{2}$ simple zeros $z_{N_{1}+1},...,z_{N_{1}+N_{2}}$ on $\mathbb{C}^{+}$.
The symmetries (2.14) imply that $\bar{z}_{1},...,\bar{z}_{N_{1}}$ and
$\bar{z}_{N_{1}+1},...,\bar{z}_{N_{1}+N_{2}}$ are the simple zeros of
$s^{A}_{11}(z)$ and $s_{33}(z)$ respectively. Denote the discrete spectrum as

$\mathcal{Z}=\left\\{z_{n},\ \bar{z}_{n}\right\\}_{n=1}^{N_{1}+N_{2}}.$ (2.19)

The distribution of $\mathcal{Z}$ on the $z$-plane is shown in Figure 1.

Re$z$Im$z$$z_{1}$$\bar{z}_{1}$$z_{2}$$\bar{z}_{2}$$z_{3}$$\bar{z}_{3}$$z_{4}$$\bar{z}_{4}$
Figure 1: Distribution of the discrete spectrum $\mathcal{Z}$.

Define a sectionally meromorphic matrix

$\displaystyle
M_{+}(z)=\left(\begin{array}[]{ccc}\mu_{+,11}&\frac{1}{s_{11}}(\mu^{A}_{-,31}\mu^{A}_{+,23}-\mu^{A}_{-,21}\mu^{A}_{+,33})&\frac{\mu_{-,13}}{s^{A}_{33}}\\\
\mu_{+,21}&\frac{1}{s_{11}}(\mu^{A}_{-,11}\mu^{A}_{+,33}-\mu^{A}_{-,31}\mu^{A}_{+,13})&\frac{\mu_{-,23}}{s^{A}_{33}}\\\
\mu_{+,31}&\frac{1}{s_{11}}(\mu^{A}_{-,21}\mu^{A}_{+,13}-\mu^{A}_{-,11}\mu^{A}_{+,23})&\frac{\mu_{-,33}}{s^{A}_{33}}\end{array}\right),\text{as
}z\in\mathbb{C}^{+};$ (2.23) $\displaystyle
M_{-}(z)=\left(\begin{array}[]{ccc}\frac{\mu_{-,11}}{s^{A}_{11}}&\frac{1}{s_{33}}(\mu^{A}_{+,31}\mu^{A}_{-,23}-\mu^{A}_{+,21}\mu^{A}_{-,33})&\mu_{+,13}\\\
\frac{\mu_{-,12}}{s^{A}_{11}}&\frac{1}{s_{33}}(\mu^{A}_{+,11}\mu^{A}_{-,33}-\mu^{A}_{+,31}\mu^{A}_{-,13})&\mu_{+,23}\\\
\frac{\mu_{-,13}}{s^{A}_{11}}&\frac{1}{s_{33}}(\mu^{A}_{+,21}\mu^{A}_{-,13}-\mu^{A}_{+,11}\mu^{A}_{-,23})&\mu_{+,33}\end{array}\right),\text{as
}z\in\mathbb{C}^{-},$ (2.27)

with their cofactor matrix:

$\displaystyle
M_{+}^{A}(z)=\left(\begin{array}[]{ccc}\frac{\mu^{A}_{-,11}}{s_{11}}&\frac{1}{s^{A}_{33}}(\mu_{+,31}\mu_{-,23}-\mu_{+,21}\mu_{-,33})&\mu^{A}_{+,13}\\\
\frac{\mu^{A}_{-,12}}{s_{11}}&\frac{1}{s^{A}_{33}}(\mu_{+,11}\mu_{-,33}-\mu_{+,31}\mu_{-,13})&\mu^{A}_{+,23}\\\
\frac{\mu^{A}_{-,13}}{s_{11}}&\frac{1}{s^{A}_{33}}(\mu_{+,21}\mu_{-,13}-\mu_{+,11}\mu_{-,23})&\mu^{A}_{+,33}\end{array}\right),\text{as
}z\in\mathbb{C}^{+};$ (2.31) $\displaystyle
M_{-}^{A}(z)=\left(\begin{array}[]{ccc}\mu^{A}_{+,11}&\frac{1}{s^{A}_{11}}(\mu_{-,31}\mu_{+,23}-\mu_{-,21}\mu_{+,33})&\frac{\mu^{A}_{-,13}}{s_{33}}\\\
\mu^{A}_{+,21}&\frac{1}{s^{A}_{11}}(\mu_{-,11}\mu_{+,33}-\mu_{-,31}\mu_{+,13})&\frac{\mu^{A}_{-,23}}{s_{33}}\\\
\mu^{A}_{+,31}&\frac{1}{s^{A}_{11}}(\mu_{-,21}\mu_{+,13}-\mu_{-,11}\mu_{+,23})&\frac{\mu^{A}_{-,33}}{s_{33}}\end{array}\right),\text{as
}z\in\mathbb{C}^{-}.$ (2.35)

######  Proposition 1.

$M_{\pm}(z)$ can be construct in another way by $\mu_{\pm}(z)$ as

$\displaystyle
M_{+}(z)=\mu_{+}e^{iz(x\widehat{A}+t\widehat{B})}\left(\begin{array}[]{ccc}1&-\frac{s_{12}}{s_{11}}&\frac{s^{A}_{31}}{s^{A}_{33}}\\\
0&1&\frac{s^{A}_{32}}{s^{A}_{33}}\\\
0&0&1\end{array}\right)=\mu_{-}e^{iz(x\widehat{A}+t\widehat{B})}\left(\begin{array}[]{ccc}s_{11}&0&0\\\
s_{21}&\frac{s^{A}_{33}}{s_{11}}&0\\\
s_{31}&\frac{s^{A}_{23}}{s_{11}}&\frac{1}{s^{A}_{33}}\end{array}\right),$
(2.42) $\displaystyle
M_{-}(z)=\mu_{+}e^{iz(x\widehat{A}+t\widehat{B})}\left(\begin{array}[]{ccc}1&0&0\\\
\frac{s^{A}_{12}}{s^{A}_{11}}&1&0\\\
\frac{s^{A}_{13}}{s^{A}_{11}}&-\frac{s_{32}}{s_{33}}&1\end{array}\right)=\mu_{-}e^{iz(x\widehat{A}+t\widehat{B})}\left(\begin{array}[]{ccc}\frac{1}{s^{A}_{11}}&\frac{s^{A}_{21}}{s_{33}}&s_{13}\\\
0&\frac{s^{A}_{11}}{s_{33}}&s_{23}\\\ 0&0&s_{33}\end{array}\right).$ (2.49)

We determine the residue conditions at these zeros. Denote norming constants

$\displaystyle c_{n}=-\frac{s_{12}(z_{n})}{s_{11}^{\prime}(z_{n})},\ {\rm
for}\ n=1,...,N_{1};\ \ c_{n}=\frac{s^{A}_{23}(z_{n})}{s^{\prime
A}_{33}(z_{n})s_{11}(z_{n})},\ {\rm for}\ n=N_{1}+1,...,N_{1}+N_{2},$
$\displaystyle\tilde{c}_{n}=\frac{s_{33}(\bar{z}_{n})}{s^{\prime
A}_{11}(\bar{z}_{n})s^{A}_{21}(\bar{z}_{n})},\ {\rm for}\ n=1,...,N_{1};\ \
\tilde{c}_{n}=-\frac{s_{32}(\bar{z}_{n})}{s_{33}^{\prime}(\bar{z}_{n})},\ {\rm
for}\ n=N_{1}+1,...,N_{1}+N_{2}.$

And the collection
$\sigma_{d}=\left\\{z_{n},c_{n}\right\\}^{N_{1}+N_{2}}_{n=1}$ is called the
_scattering data_. Denote the phase functions

$\displaystyle\theta_{ij}=(a_{i}-a_{j})\xi+(b_{i}-b_{j}),\hskip
14.22636pti,j=1,2,3,\hskip 14.22636pt\xi=\frac{x}{t},$ (2.50)

with $\theta_{ij}=-\theta_{ji}$ Then we have the following RH problem.

######  RHP 1.

Find a matrix-valued function $M(z)=M(z;x,t)$ which satisfies

$\blacktriangleright$ Analyticity: $M(z)$ is meromorphic in
$\mathbb{C}\setminus\mathbb{R}$ and has single poles on
$\mathcal{Z}\cup\bar{\mathcal{Z}}$;

$\blacktriangleright$ Symmetry: $M(z)=\overline{M^{A}(\bar{z})}$;

$\blacktriangleright$ Jump condition: $M$ has continuous boundary values
$M_{\pm}$ on $\mathbb{R}$ and

$M^{+}(z)=M^{-}(z)V(z),\hskip 14.22636ptz\in\mathbb{R},$ (2.51)

where

$V(z)=e^{iz(x\widehat{A}+t\widehat{B})}\left(\begin{array}[]{ccc}1&-r_{1}&\bar{r}_{2}\\\
-\bar{r}_{1}&1+|r_{1}|^{2}&\bar{r}_{3}-\bar{r}_{1}\bar{r}_{2}\\\
r_{2}&r_{3}-r_{1}r_{2}&1+|r_{3}|^{2}+|r_{2}|^{2}\end{array}\right);$ (2.52)

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M(z)=I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (2.53)

$\blacktriangleright$ Residue conditions: $M(z)$ has simple poles at each
point in $\mathcal{Z}\cup\bar{\mathcal{Z}}$ with

For $n=1,...,N_{1}$:

$\displaystyle\operatorname*{Res}_{z=z_{n}}M(z)=\lim_{z\to
z_{n}}M(z)\left(\begin{array}[]{ccc}0&c_{n}e^{iz_{n}t\theta_{12}}&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$ (2.57)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M(z)=\lim_{z\to\bar{z}_{n}}M(z)\left(\begin{array}[]{ccc}0&0&0\\\
\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{12}}&0&0\\\ 0&0&0\end{array}\right);$
(2.61)

For $n=N_{1}+1,...,N_{1}+N_{2}$,

$\displaystyle\operatorname*{Res}_{z=z_{n}}M(z)=\lim_{z\to
z_{n}}M(z)\left(\begin{array}[]{ccc}0&0&0\\\
0&0&c_{n}e^{iz_{n}t\theta_{23}}\\\ 0&0&0\end{array}\right),$ (2.65)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M(z)=\lim_{z\to\bar{z}_{n}}M(z)\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{23}}&0\end{array}\right).$
(2.69)

From the asymptotic behavior of the functions $\mu_{\pm}(z)$, $S(z)$ and their
cofactor matrix, we have following reconstruction formula

$p_{ij}(x,t)=-i(a_{i}-a_{j})\lim_{z\to\infty}[zM(z)]_{ij},\ \ i,j=1,2,3.$
(2.70)

##  3 The scattering maps

We consider the $x$-part of the Lax pair (2.1) to analyze the initial value
problem and give the proof of proposition 3 in this section. In fact, taking
account of $t$-part of the Lax pair and though the standard direct scattering
transform method, then it deduce that $S(z)$ have time evolution:
$S(z,t)=e^{iz\widehat{B}t}S(z,0)$. So $|r_{i}(z,t)|=|r_{i}(z,0)|$ for
$i=1,2,3,4$. In this section, we abbreviate $\mu_{\pm}(x,0,z)$,
$\mu^{A}_{\pm}(x,0,z)$ as $\mu_{\pm}(x,z)$, $\mu^{A}_{\pm}(x,z)$ respectively.

To find the relationship of initial value and reflection coefficients, we
recall two Volterra integral equations in (2.5) and (2.8)

$\displaystyle\mu_{\pm}(x)=I+\int_{\pm\infty}^{x}e^{iz\widehat{A}(x-y)}P(y)\mu_{\pm}(y)dy,$
(3.1)
$\displaystyle\mu^{A}_{\pm}(x)=I-\int_{\pm\infty}^{x}e^{-iz\widehat{A}(x-y)}P^{T}(y)\mu^{A}_{\pm}(y)dy.$
(3.2)

We need estimates on the $L^{2}$-integral property of $\mu_{\pm}(z)$
$\mu^{A}_{\pm}(z)$ and their derivatives. And we abbreviate
$C_{B}^{0}(\mathbb{R}_{x}^{\pm}\times\mathbb{R}_{z})$,
$C^{0}(\mathbb{R}_{x}^{\pm},L^{2}(\mathbb{R}_{z}))$,
$L^{2}(\mathbb{R}_{x}^{\pm}\times\mathbb{R}_{z})$ to $C_{B}^{0}$, $C^{0}$,
$L^{2}_{xz}$ respectively. The following result is useful in the analysis of
direct scattering map, namely estimates for Volterra-type integral equations
above.

######  Lemma 2.

Suppose that $F$ is a three factorial square matrix and $g$ is a column
vector, then

$|Fg|\leq|F||g|.$

######  Lemma 3.

For $\psi(\eta)\in L^{2}(\mathbb{R})$, $f(x)\in L^{2,1/2}(\mathbb{R})$,
following inequality hold:

$\displaystyle\bigg{|}\int_{\mathbb{R}}\int_{x}^{\pm\infty}f(y)e^{-\frac{i}{2}\eta(p(x)-p(y))}\psi(\eta)dyd\eta\bigg{|}$
$\displaystyle=\bigg{|}\int_{x}^{\pm\infty}f(y)\psi(\frac{1}{2}(p(x)-p(y)))dy\bigg{|}\lesssim\left(\int_{x}^{\pm\infty}|f(y)|^{2}dy\right)^{1/2}\parallel\psi\parallel_{2};$
(3.3)
$\displaystyle\int_{0}^{\pm\infty}\int_{\mathbb{R}}\bigg{|}\int_{x}^{\pm\infty}f(y)e^{-\frac{i}{2}\eta(p(x)-p(y))}dy\bigg{|}^{2}d\eta
dx\lesssim\parallel f\parallel_{2,1/2}^{2}.$ (3.4)

The proof of above lemmas are trivial, so we omit it. From the symmetry
reduction (2.9), we will only consider $\mu_{\pm}(x,z)$. And we just give the
detail for the first column of $\mu_{\pm}(x,z)$. For the sake of brevity, we
denote

$\displaystyle f^{(j)}(x,y,z)=e^{-iz(a_{1}-a_{j})(x-y)},j=2,3,;\ \
[\mu_{\pm}]_{1}(x,z)-e_{1}\triangleq n_{\pm}(x,z),$ (3.5)

where $e_{1}$ is identity vector $(1,0,0)^{T}$. Introduce the integral
operator

$\displaystyle T_{\pm}(f)(x,z)=\int_{x}^{\pm\infty}K_{\pm}(x,y,z)f(y,z)dy,$
(3.6)

where integral kernel $K_{\pm}(x,y,z)$ is

$\displaystyle K_{\pm}(x,y,z)=\left(\begin{array}[]{ccc}0&p_{12}&p_{13}\\\
-f^{(2)}\bar{p}_{12}&0&f^{(2)}p_{23}\\\
-f^{(3)}\bar{p}_{13}&-f^{(3)}\bar{p}_{23}&0\end{array}\right).$ (3.10)

Then (3.1) is changed into

$\displaystyle n_{\pm}=T_{\pm}(e_{1})+T_{\pm}(n_{\pm})\triangleq
n^{0}_{\pm}+T_{\pm}(n_{\pm}).$ (3.11)

Differentiating above equation with respect to $z$ yields

$\displaystyle[n_{\pm}]_{z}=n_{\pm}^{1}+T_{\pm}([n_{\pm}]_{z}),\hskip
14.22636ptn_{\pm}^{1}=[n^{0}_{\pm}]_{z}+[T_{\pm}]_{z}(n_{\pm}).$ (3.12)

$[T_{\pm}]_{z}$ is also a integral operator with integral kernel
$[K_{\pm}]_{z}(x,y,z)$:

$\displaystyle[K_{\pm}]_{z}(x,y,z)=(x-y)\left(\begin{array}[]{ccc}0&0&0\\\
i(a_{1}-a_{2})f^{(2)}\bar{p}_{12}&0&-i(a_{1}-a_{2})f^{(2)}p_{23}\\\
i(a_{1}-a_{3})f^{(3)}\bar{p}_{13}&i(a_{1}-a_{3})f^{(3)}\bar{p}_{23}&0\end{array}\right).$
(3.16)

######  Lemma 4.

For the integral operators $T_{\pm}$ and $[T_{\pm}]_{z}$ defined above, then
$n^{0}_{\pm}(x,z)=T_{\pm}(e_{1})(x,z)$ and $[n^{0}_{\pm}]_{z}(x,z)$ are in
$C_{B}^{0}\cap C^{0}\cap L^{2}_{xz}$.

###### Proof.

$n^{0}_{\pm}(x,z)$ is given by

$\displaystyle
n^{0}_{\pm}(x,z)=T_{\pm}(e_{1})(x,z)=\int_{x}^{\pm\infty}\left(\begin{array}[]{ccc}0\\\
-f^{(2)}(x,y,z)\bar{p}_{12}(y)\\\
-f^{(3)}(x,y,z)\bar{p}_{13}(y)\end{array}\right)dy,$ (3.20)

with

$\displaystyle|n^{0}_{\pm}(x,z)|$
$\displaystyle\leq\bigg{|}\int_{x}^{\pm\infty}f^{(2)}(x,y,z)\bar{p}_{12}(y)dy\bigg{|}+\bigg{|}\int_{x}^{\pm\infty}f^{(3)}(x,y,z)\bar{p}_{13}(y)dy\bigg{|}$
(3.21) $\displaystyle\leq\parallel p_{12}(x)\parallel_{1}+\parallel
p_{13}(x)\parallel_{1}\leq\parallel p_{12}(x)\parallel_{2,1}+\parallel
p_{13}(x)\parallel_{2,1}.$ (3.22)

Then from $p_{ij}(x)\in H^{1,2}$, by lemma 3, it immediately derives to

$\displaystyle\parallel n^{0}_{\pm}\parallel_{C^{0}}\lesssim\parallel
p_{12}(x)\parallel_{2}+\parallel p_{13}(x)\parallel_{2},$ (3.23)
$\displaystyle\parallel n^{0}_{\pm}\parallel_{L^{2}_{xz}}\lesssim\parallel
p_{12}(x)\parallel_{2,1/2}+\parallel p_{13}(x)\parallel_{2,1/2}.$ (3.24)

And $[n^{0}_{\pm}]_{z}(x,z)(x,z)$ is given by

$\displaystyle[n^{0}_{\pm}]_{z}(x,z)=[T_{\pm}]_{z}(e_{1})(x,z)=$
$\displaystyle\int_{x}^{\pm\infty}\left(\begin{array}[]{ccc}0\\\
i(a_{1}-a_{2})(x-y)f^{(2)}(x,y,z)\bar{p}_{12}(y)\\\
i(a_{1}-a_{3})(x-y)f^{(3)}(x,y,z)\bar{p}_{13}(y)\end{array}\right)dy.$ (3.28)

Similarly from lemma 3, it achieves that

$\displaystyle|[n^{0}_{\pm}]_{z}|\leq\parallel
p_{12}(x)\parallel_{1,1}+\parallel p_{13}(x)\parallel_{1,1}\leq\parallel
p_{12}(x)\parallel_{2,2}+\parallel p_{13}(x)\parallel_{2,2}$ (3.29)
$\displaystyle\parallel[n^{0}_{\pm}]_{z}\parallel_{C^{0}}\lesssim\parallel
p_{12}(x)\parallel_{2,1}+\parallel p_{13}(x)\parallel_{2,1},$ (3.30)
$\displaystyle\parallel[n^{0}_{\pm}]_{z}\parallel_{L^{2}_{xz}}\lesssim\parallel
p_{12}(x)\parallel_{2,3/2}+\parallel p_{13}(x)\parallel_{2,3/2}.$ (3.31)

∎

The operator $T_{\pm}$ and $[T_{\pm}]_{z}$ induce linear mappings given in
next lemma.

######  Lemma 5.

The integral operator $T_{\pm}$ and its $z$-derivative $[T_{\pm}]_{z}$ map
$C_{B}^{0}\cap C^{0}\cap L^{2}_{xz}$ to itself. Moreover,
$(I-T^{\pm})^{-1}$exists as a bounded operator on $C_{B}^{0}\cap C^{0}\cap
L^{2}_{xz}$.

###### Proof.

In fact, (3.10) leads to

$\displaystyle|K_{\pm}(x,y,z)|=|P(y)|.$ (3.32)

For any $f(x,z)\in C_{B}^{0}\cap C^{0}\cap L^{2}_{xz}$, by Lemma 2, we have

$\displaystyle|T_{\pm}(f)(x,z)|\leq\int_{x}^{\pm\infty}|P(y)|dy\parallel
f\parallel_{C_{B}^{0}}.$ (3.33)

Moreover,

$\displaystyle\left(\int_{\mathbb{R}}|T_{\pm}(f)(x,z)|^{2}dz\right)^{\frac{1}{2}}$
$\displaystyle=\left(\int_{\mathbb{R}}\bigg{|}\int_{x}^{\pm\infty}K_{\pm}(x,y,z)f(y,z)dy\bigg{|}^{2}dz\right)^{\frac{1}{2}}$
$\displaystyle\leq\bigg{|}\int_{x}^{\pm\infty}\left(\int_{\mathbb{R}}|K_{\pm}(x,y,z)|^{2}|f(y,z)|^{2}dz\right)^{\frac{1}{2}}dy\bigg{|}$
$\displaystyle=\bigg{|}\int_{x}^{\pm\infty}|P(y)|\parallel
f(y,z)\parallel_{L^{2}_{z}}dy\bigg{|}\leq\parallel P\parallel_{1}\parallel
f\parallel_{C^{0}},$ (3.34)

which derives to

$\displaystyle\left(\int_{\mathbb{R}^{\pm}}\int_{\mathbb{R}}|T_{\pm}(f)(x,z)|^{2}dzdx\right)^{\frac{1}{2}}$
$\displaystyle\leq\bigg{|}\int_{\mathbb{R}^{\pm}}\left(\int_{x}^{\pm\infty}|P(y)|\parallel
f(y,z)\parallel_{L^{2}_{z}}dy\right)^{2}dx\bigg{|}^{\frac{1}{2}}$
$\displaystyle\leq\bigg{|}\int_{\mathbb{R}^{\pm}}\left(\int_{0}^{y}|P(y)|^{2}\int_{\mathbb{R}}|f(y,z)|^{2}dzdx\right)dy\bigg{|}^{\frac{1}{2}}$
$\displaystyle\leq\parallel P\parallel_{2,1/2}\parallel
f\parallel_{L^{2}_{xz}}.$ (3.35)

Denote $K^{n}_{\pm}$ is the integral kernel of Volterra operator
$[T_{\pm}]^{n}$ as

$\displaystyle
K^{n}_{\pm}(x,y_{n},z)=\int_{x}^{y_{n}}...\int_{x}^{y_{2}}K_{\pm}(x,y_{1},z)K_{\pm}(y_{1},y_{2},z)...K_{\pm}(y_{n-1},y_{n},z)dy_{1}...dy_{n-1},$
(3.36)

with

$\displaystyle|K^{n}_{\pm}(x,y,z)|\leq\frac{1}{(n-1)!}\left(\int_{x}^{\pm\infty}|P(y)|dy\right)^{n-1}|P(y)|.$
(3.37)

Analogously, $[T_{\pm}]^{n}$ admits that

$\displaystyle\parallel[T_{\pm}]^{n}\parallel_{\mathcal{B}(C_{B}^{0})}\leq\frac{\parallel
P\parallel_{1}^{n}}{(n-1)!},\ \
\parallel[T_{\pm}]^{n}\parallel_{\mathcal{B}(C^{0})}\leq\frac{\parallel
P\parallel_{1}^{n}}{(n-1)!},$
$\displaystyle\parallel[T_{\pm}]^{n}\parallel_{\mathcal{B}(L^{2}_{xz})}\leq\frac{\parallel
P\parallel_{1}^{(n-1)}}{(n-1)!}\parallel P\parallel_{2,1/2}.$

Then the standard Volterra theory gives the following operator norm:

$\displaystyle\parallel(I-T_{\pm})^{-1}\parallel_{\mathcal{B}(C_{B}^{0})}\leq
e^{\parallel P\parallel_{1}}\parallel P\parallel_{1},$
$\displaystyle\parallel(I-T_{\pm})^{-1}\parallel_{\mathcal{B}(C^{0})}\leq
e^{\parallel P\parallel_{1}}\parallel P\parallel_{1},$
$\displaystyle\parallel(I-T_{\pm})^{-1}\parallel_{\mathcal{B}(L^{2}_{xz})}\leq
e^{\parallel P\parallel_{1}}\parallel P\parallel_{2,1/2}.$

As for the $z$-derivative $[T_{\pm}]_{z}$, from

$\displaystyle|[K_{\pm}]_{z}|\lesssim|x-y||P(y)|,$

it analogously leads to

$\displaystyle\parallel[T_{\pm}]_{z}\parallel_{\mathcal{B}(C_{B}^{0})}\leq\parallel
P\parallel_{1,1},\
\parallel[T_{\pm}]_{z}\parallel_{\mathcal{B}(C^{0})}\leq\parallel
P\parallel_{1,1},\
\parallel[T_{\pm}]_{z}\parallel_{\mathcal{B}(L^{2}_{xz})}\leq\parallel
P\parallel_{2,3/2}.$

∎

By using above lemma, we can show that $[T_{\pm}]_{z}(n_{\pm})\in
C_{B}^{0}\cap C^{0}\cap L^{2}_{xz}$, which implies that $n_{\pm}^{1}\in
C_{B}^{0}\cap C^{0}\cap L^{2}_{xz}$. Since the operator $(I-T_{\pm})^{-1}$
exist, the equations (3.11)-(3.12) are solvable with

$\displaystyle n_{\pm}(x,z)=(I-T_{\pm})^{-1}(n_{\pm}^{0})(x,z),$ (3.38)
$\displaystyle[n^{\pm}(x,z)]_{z}=(I-T_{\pm})^{-1}(n_{\pm}^{1})(x,z).$ (3.39)

Combining above Lemmas and the definition of $n_{\pm}$ (3.5), we immediately
obtain the following property of $\mu_{\pm}(x,z)$.

######  Proposition 2.

Suppose that $p_{ij0}(x)\in H^{1,2}(\mathbb{R})$ for $i,j=1,2,3$, then
$\mu_{\pm}(0,z)-I$ and its $z$-derivative $[\mu_{\pm}(0,z)]_{z}$ belong in
$C_{B}^{0}(\mathbb{R})\cap L^{2}(\mathbb{R})$.

Then we begin to prove proposition 3. we rewrite (2.12) and (2.13) at $t=0$:

$\displaystyle\mu_{+}(z)=\mu_{-}(z)e^{izx\widehat{A}}s(z),$ (3.40)
$\displaystyle\mu^{A}_{+}(z)=\mu^{A}_{-}(z)e^{-izx\widehat{A}}s^{A}(z).$
(3.41)

It is requisite to shown $r_{i}(z),\hskip 8.5359ptr^{\prime}_{i}(z),\hskip
8.5359ptzr_{i}(z)\text{ in }L^{2}(\mathbb{R})$ for $i=1,2,3,4$. We only give
the proof for $i=1$, the others are analogously. Denote
$m_{\pm}^{ij}(z)=[\mu_{\pm}(0,z)-I]_{ij}$, $i,j=1,2,3$ for concise, then it
belongs in $C_{B}^{0}(\mathbb{R})\cap L^{2}(\mathbb{R})$. (3.40) leads to

$\displaystyle s_{11}(z)=$
$\displaystyle\det\left([\mu_{-}]_{1}(0,z),[\mu_{-}]_{2}(0,z),[\mu_{+}]_{1}(0,z)\right)$
$\displaystyle=$
$\displaystyle(m_{-}^{11}+1)(m_{-}^{22}+1)m_{+}^{31}-(m_{-}^{11}+1)m_{-}^{32}m_{+}^{21}-m_{-}^{21}m_{-}^{12}m_{+}^{31}$
$\displaystyle+m_{-}^{21}m_{-}^{32}(m_{+}^{11}+1)+m_{-}^{31}m_{-}^{12}m_{+}^{21}-m_{-}^{31}(m_{-}^{22}+1)(m_{+}^{11}+1),$
(3.42) $\displaystyle s_{12}(z)=$
$\displaystyle\det\left([\mu_{+}]_{2}(0,z),[\mu_{-}]_{2}(0,z),[\mu_{-}]_{3}(0,z)\right)$
$\displaystyle=$
$\displaystyle(m_{-}^{33}+1)(m_{-}^{22}+1)m_{+}^{12}-(m_{+}^{22}+1)m_{-}^{13}m_{-}^{32}-m_{-}^{23}m_{+}^{12}m_{-}^{32}$
$\displaystyle+m_{-}^{13}m_{+}^{32}(m_{-}^{22}+1)+m_{-}^{23}m_{-}^{12}m_{+}^{32}-m_{-}^{12}(m_{+}^{22}+1)(m_{-}^{33}+1).$
(3.43)

Then proposition 2 gives the boundedness of $s_{11}(z)$, $s_{11}^{\prime}(z)$,
$s_{12}(z)$, $s_{12}^{\prime}(z)$ and the $L^{2}$-integrability of
$s_{12}(z)$, $s_{12}^{\prime}(z)$. So we just need to show $zs_{12}(z)\in
L^{2}(\mathbb{R})$. From (3.1), we obtain

$\displaystyle[\mu_{+}]_{2}$
$\displaystyle(0,z)-e_{2}=(m_{+}^{12}(z),m_{+}^{22}(z),m_{+}^{32}(z))^{T}=\int_{0}^{+\infty}\left(\begin{array}[]{ccc}e^{-iz(a_{1}-a_{2})y}p_{12}\\\
0\\\ -e^{iz(a_{1}-a_{3})y}\bar{p}_{23}\end{array}\right)dy$ (3.47)
$\displaystyle+\int_{0}^{+\infty}\left(\begin{array}[]{ccc}0&e^{-iz(a_{1}-a_{2})y}p_{12}&e^{-iz(a_{1}-a_{2})y}p_{13}\\\
-\bar{p}_{12}&0&p_{23}\\\
-e^{iz(a_{1}-a_{3})y}\bar{p}_{13}&-e^{iz(a_{1}-a_{3})y}\bar{p}_{23}&0\end{array}\right)\left(\begin{array}[]{ccc}m_{+}^{12}\\\
m_{+}^{22}\\\ m_{+}^{32}\end{array}\right)dy.$ (3.54)

So

$\displaystyle zm_{+}^{12}=$ $\displaystyle
z\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}p_{12}dy$
$\displaystyle+z\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}p_{12}m_{+}^{22}+z\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}p_{13}m_{+}^{32}dy$
$\displaystyle\triangleq H_{1}+H_{2}+H_{3}.$ (3.55)

By integration by parts and (2.3), we can further calculate

$\displaystyle-i(a_{1}-a_{2})H_{1}$
$\displaystyle=p_{12}(0)-\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}p^{\prime}_{12}dy,$
$\displaystyle-i(a_{1}-a_{2})H_{2}$
$\displaystyle=p_{12}(0)m_{+}^{22}(0,z)-\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}\left(p_{12}^{\prime}m_{+}^{22}-|p_{12}|^{2}m_{+}^{12}+p_{12}p_{23}m_{+}^{32}\right)dy$
$\displaystyle-i(a_{1}-a_{2})H_{3}$
$\displaystyle=p_{13}(0)m_{+}^{32}(0,z)+i(a_{2}-a_{3})H_{3}$
$\displaystyle-\int_{0}^{+\infty}e^{-iz(a_{1}-a_{2})y}\left(p_{13}^{\prime}m_{+}^{32}-|p_{13}|^{2}m_{+}^{12}-p_{13}\bar{p}_{23}(m_{+}^{22}+1)\right)dy.$
(3.56)

By Lemma 3 and proposition 2, $zm_{+}^{12}$ can be expressed as

$\displaystyle zm_{+}^{12}(z)=\frac{ip_{12}(0)}{(a_{1}-a_{2})}+H_{+}^{12}(z),$
(3.57)

where $H_{+}^{12}(z)$ is a $L^{2}$-integrable and bounded function on
$\mathbb{R}$. In fact, we claim that
$zm_{\pm}^{ij}(z)=\frac{ip_{ij}(0)}{(a_{i}-a_{j})}+H_{\pm}^{ij}(z)$ with a
$L^{2}$-integrable and bounded function $H_{\pm}^{ij}(z)$, which give the
$L^{2}$-integrability of $zs_{12}(z)$ on $\mathbb{R}$. Using above results, we
then finally obtain the following proposition.

######  Proposition 3.

If the initial data $p_{ij}(x)\in H^{1,2}(\mathbb{R})$, then $r_{j}(z)\in
H^{1,1}(\mathbb{R}),\ j=1,2,3,4$.

##  4 The deformation of RH problem

The long-time asymptotic of RHP 1 is affected by the growth and decay of the
exponential function $e^{\pm 2itz\theta_{ij}}$ appearing in both the jump
relation and the residue conditions. So we need control the real part of $\pm
itz\theta_{ij}$. In this section, we introduce a new transform $M(z)\to
M^{(1)}(z)$, which make that the $M^{(1)}(z)$ is well behaved as $t\to\infty$
along any characteristic line. The growth and decay properties of
$e^{itz\theta_{ij}}$ as $t\to\infty$ is determined by

$\displaystyle\text{Re}(itz\theta_{ij})=-t\text{Im}(z)\theta_{ij}.$ (4.1)

So when $z\in\mathbb{C}^{+}$, the asymptotic behavior of $e^{itz\theta_{ij}}$
only depends on the sign of $\theta_{ij}=(a_{i}-a_{j})\xi+(b_{i}-b_{j})$,
namely, the sign of $\xi+n_{ij}$. And the jump matrix $V(z)$ in (2.52) needs
to be restricted according to the sign of $\text{Re}(itz\theta_{ij})$. Unlike
the case of $2\times 2$ jump matrix such as NLS equation [32], it is
complicated to divide a $3\times 3$ jump matrix into a product of upper and
downer triangular matrix. At present paper, we only consider

${\rm Case\ I.}\ \ \xi>-n_{12}>-n_{13}>-n_{23};\ \ {\rm Case\ II.}\ \
-n_{23}<-n_{13}<\xi<-n_{12}.$

In these two cases, the jump matrix $V(z)$ can be split into the product of
two simple upper and downer triangular matrices

$\displaystyle
e^{-iz(x\widehat{A}+t\widehat{B})}V(z)=\left(\begin{array}[]{ccc}1&0&0\\\
-\bar{r}_{1}e^{-iz\theta_{12}}&1&0\\\
r_{2}e^{-iz\theta_{13}}&r_{3}e^{-iz\theta_{23}}&1\end{array}\right)\left(\begin{array}[]{ccc}1&-r_{1}e^{iz\theta_{12}}&\bar{r}_{2}e^{iz\theta_{13}}\\\
0&1&\bar{r}_{3}e^{iz\theta_{23}}\\\ 0&0&1\end{array}\right)$ (4.8)
$\displaystyle=\left(\begin{array}[]{ccc}1&\dfrac{-r_{1}e^{iz\theta_{12}}}{1+|r_{1}|^{2}}&0\\\
0&1&0\\\
-\bar{r}_{4}e^{-iz\theta_{13}}&\dfrac{r_{3}-r_{1}r_{2}}{1+|r_{1}|^{2}}e^{-iz\theta_{23}}&1\end{array}\right)(1+|r_{1}|^{2})^{\sigma_{3}}\left(\begin{array}[]{ccc}1&0&-r_{2}e^{iz\theta_{13}}\\\
-\frac{\bar{r}_{1}e^{-iz\theta_{12}}}{1+|r_{1}|^{2}}&1&\frac{\bar{r}_{3}-\bar{r}_{1}\bar{r}_{2}}{1+|r_{1}|^{2}}e^{iz\theta_{23}}\\\
0&0&1\end{array}\right),$ (4.15)

where $\sigma_{3}=$diag$(-1,1,0)$. In fact, the other cases($\xi<-n_{13}$)
also have analogous factorizations, but they are complicated and calumniation
will be tedious. So we do not dicuss it in this paper. We will utilize these
factorizations to deform the jump contours, so that the oscillating factor
$e^{\pm it\theta_{ij}}$ are decaying in corresponding region, respectively.
For brevity, we denote

$\mathcal{N}\triangleq\left\\{1,...,N_{1}+N_{2}\right\\},\ \
\mathcal{N}_{1}\triangleq\left\\{1,...,N_{1}\right\\},\ \
\mathcal{N}_{2}\triangleq\left\\{N_{1}+1,...,N_{1}+N_{2}\right\\},$

and define partitions $\Delta(\xi)$ and $\nabla(\xi)$

$\displaystyle\Delta(\xi)=\left\\{\begin{array}[]{ll}\emptyset,&\text{as
}\xi>-n_{12};\\\\[12.0pt] \left\\{1,...,N_{1}\right\\},&\text{as
}-n_{13}<\xi<-n_{12};\\\
\end{array}\right.,\nabla(\xi)=\mathcal{N}\setminus\nabla.$ (4.18)

For $z_{n}$ with $n\in\Delta(\xi)$, the residue of $M(z)$ at $z_{n}$ in (2.57)
grows without bound as $t\to\infty$. Similarly, for $z_{n}$ with $n\in\nabla$,
the residue are bounded or approaching to be $0$. Denote a small positive
constant

$\rho_{0}=\min_{1\leq i<j\leq 3}|\theta_{ij}|>0.$ (4.19)

Note that, $\theta_{12}$ has different identities for $\xi>-n_{12}$ and
$-n_{13}<\xi<-n_{12}$. Namely, the functions which will be used following
depend on $\xi$. Denote

$\displaystyle I(\xi)=\left\\{\begin{array}[]{ll}\emptyset,&\text{as
}\xi>-n_{12},\\\\[12.0pt] \mathbb{R},&\text{as }-n_{13}<\xi<-n_{12}.\\\
\end{array}\right.$ (4.22)

Define functions

$\displaystyle\delta(z)=\delta(z,\xi)=\exp\left(i\int_{I(\xi)}\dfrac{\nu(s)}{s-z}ds\right),\
\nu(s)=-\frac{1}{2\pi}\log(1+|r_{1}(s)|^{2}),$ (4.23) $\displaystyle
T(z)=T(z,\xi)=\prod_{n\in\Delta(\xi)}\dfrac{z-z_{n}}{z-\bar{z}_{n}}\delta(z,\xi).$
(4.24)

In the above formulas, we choose the principal branch of power and logarithm
functions. Obviously, for the Case I. $\xi>-n_{12}$, we have $T(z,\xi)\equiv
1$.

######  Proposition 4.

The function $T(z,\xi)$ defined by (4.24) in Case $-n_{13}<\xi<-n_{12}$ has
following properties:
(a) $T$ is meromorphic in $\mathbb{C}\setminus\mathbb{R}$, and for each
$n\in\Delta(\xi)$, $T(z)$ has a simple pole at $\bar{z}_{n}$ and a simple zero
at $z_{n}$;
(b) For $z\in I(\xi)$, as $z$ approaching the real axis from above and below,
$T$ has boundary values $T_{\pm}$, which satisfy:

$T_{+}(z)=(1+|r_{1}(z)|^{2})T_{-}(z),\hskip 14.22636ptz\in I(\xi);$ (4.25)

(c) $\lim_{z\to\infty}T(z)=1$, and when $|\arg(z)|\leq c<\pi$,

$\displaystyle T(z,\xi)=1+iT_{1}(\xi)\frac{1}{z}+\mathcal{O}(z^{-2});$ (4.26)

where

$T_{1}(\xi)=2\sum_{n\in\Delta(\xi)}{\rm Im}(z_{n})-\int_{I(\xi)}\nu(s)ds.$

(d) As $z\to 0$, along $z=\rho e^{i\psi}$, $\rho>0$, $|\psi|\leq c<\pi$

$|T(z,\xi)-T(0,\xi)|\leq|z|^{1/2}.$ (4.27)

###### Proof.

Properties (a) can be obtain by simple calculation from the definition of
$T(z)$ in (4.24). And (b) follows from the Plemelj formula. By the Laurent
expansion (c) can be obtained immediately. For brevity, we omit computation.
As for (d),

$\displaystyle|T(z,\xi)-T(0,\xi)|\leq\Bigg{|}\prod_{n\in\Delta(\xi)}\dfrac{z-z_{n}}{z-\bar{z}_{n}}\prod_{n\in\Delta(\xi)}\dfrac{\bar{z}_{n}}{z_{n}}\exp\left(i\int_{I(\xi)}\nu(s)(\dfrac{1}{s-z}-\frac{1}{s})ds\right)-1\Bigg{|}\bigg{|}\prod_{n\in\Delta(\xi)}\dfrac{z_{n}}{\bar{z}_{n}}\bigg{|}$
$\displaystyle\lesssim|\int_{I(\xi)}\nu(s)(\dfrac{1}{s-z}-\frac{1}{s})ds|\lesssim\parallel\nu^{\prime}\parallel_{2}|z|^{1/2}\lesssim\parallel
r^{\prime}_{1}\parallel_{2}|z|^{1/2}\lesssim|z|^{1/2}.$

∎

We define a new matrix-valued function $M^{(1)}(z)$ by

$M^{(1)}(z)=M(z)T(z)^{\sigma_{3}},$ (4.28)

which then satisfies the following RH problem.

######  RHP 2.

Find a matrix-valued function $M^{(1)}(z)=M^{(1)}(z;x,t)$ which satisfies:

$\blacktriangleright$ Analyticity: $M^{(1)}(z)$ is meromorphic in
$\mathbb{C}\setminus\mathbb{R}$;

$\blacktriangleright$ Jump condition: $M^{(1)}(z)$ has continuous boundary
values $M^{(1)}_{\pm}(z)$ on $\mathbb{R}$ and

$M^{(1)}_{+}(z)=M^{(1)}_{-}(z)V^{(1)}(z),\hskip 14.22636ptz\in\mathbb{R},$
(4.29)

where when $z\in\mathbb{R}\setminus I(\xi)$,

$V^{(1)}(z)=\left(\begin{array}[]{ccc}1&0&0\\\
-\bar{r}_{1}e^{-iz\theta_{12}}&1&0\\\
r_{2}e^{-iz\theta_{13}}&r_{3}e^{-iz\theta_{23}}&1\end{array}\right)\left(\begin{array}[]{ccc}1&-r_{1}e^{iz\theta_{12}}&\bar{r}_{2}e^{iz\theta_{13}}\\\
0&1&\bar{r}_{3}e^{iz\theta_{23}}\\\ 0&0&1\end{array}\right),$ (4.30)

when $z\in I(\xi)$,

$V^{(1)}(z)=\left(\begin{array}[]{ccc}1&\dfrac{-r_{1}e^{iz\theta_{12}}}{1+|r_{1}|^{2}}&0\\\
0&1&0\\\
-\bar{r}_{4}e^{-iz\theta_{13}}&\dfrac{r_{3}-r_{1}r_{2}}{1+|r_{1}|^{2}}e^{-iz\theta_{23}}&1\end{array}\right)\left(\begin{array}[]{ccc}1&0&-r_{2}e^{iz\theta_{13}}\\\
-\frac{\bar{r}_{1}e^{-iz\theta_{12}}}{1+|r_{1}|^{2}}&1&\frac{\bar{r}_{3}-\bar{r}_{1}\bar{r}_{2}}{1+|r_{1}|^{2}}e^{iz\theta_{23}}\\\
0&0&1\end{array}\right);$ (4.31)

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(1)}(z)=$ $\displaystyle I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (4.32)

$\blacktriangleright$ Residue conditions: $M^{(1)}$ has simple poles at each
point $z_{n}$ and $\bar{z}_{n}$ for $n\in\mathcal{N}$ with:

$\displaystyle\operatorname*{Res}_{z=z_{n}}M^{(1)}(z)=\lim_{z\to
z_{n}}M^{(1)}(z)\Gamma_{n}(\xi),$ (4.33)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M^{(1)}(z)=\lim_{z\to\bar{z}_{n}}M^{(1)}(z)\tilde{\Gamma}_{n}(\xi),$
(4.34)

where $\Gamma_{n}(\xi)$ and $\tilde{\Gamma}_{n}(\xi)$ matrix defined by:

for or $n=1,...,N_{1}$,

$\displaystyle\Gamma_{n}(\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&c_{n}e^{iz_{n}t\theta_{12}}&0\\\
0&0&0\\\ 0&0&0\end{array}\right),&\text{ for }\xi>-n_{12};\\\
\left(\begin{array}[]{ccc}0&0&0\\\
c_{n}^{-1}e^{-iz_{n}t\theta_{12}}(T^{-1})^{\prime}(z_{n})^{-2}&0&0\\\
0&0&0\end{array}\right),&\text{ for }-n_{13}<\xi<-n_{12};\end{array}\right.,$
(4.43)
$\displaystyle\tilde{\Gamma}_{n}(\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&0&0\\\
\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{12}}&0&0\\\
0&0&0\end{array}\right),&\text{ for }\xi>-n_{12};\\\
\left(\begin{array}[]{ccc}0&\tilde{c}_{n}^{-1}e^{i\bar{z}_{n}t\theta_{12}}T^{\prime}(\bar{z}_{n})^{-2}&0\\\
0&0&0\\\ 0&0&0\end{array}\right),&\text{ for
}-n_{13}<\xi<-n_{12};\end{array}\right.,$ (4.52)

for $n=N_{1}+1,...,N_{1}+N_{2}$,

$\displaystyle\Gamma_{n}(\xi)=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&c_{n}e^{iz_{n}t\theta_{23}}T(z_{n},\xi)\\\ 0&0&0\end{array}\right),$
(4.56)
$\displaystyle\tilde{\Gamma}_{n}(\xi)=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\
0&\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{23}}T^{-1}(\bar{z}_{n},\xi)&0\end{array}\right).$
(4.60)

###### Proof.

The analyticity and symmetry of $M^{(1)}(z)$ is directly from its definition,
the Proposition 4 and the identities of $M$.Then by simple calculation we can
obtain the residues condition and jump condition from (2.57), (2.61) (2.52)
and (4.28). As for asymptotic behaviors, from Proposition 4 (c), we obtain the
asymptotic behaviors of $M^{(1)}(z)$. ∎

##  5 A mixed $\bar{\partial}$-RH problem and decomposition

In this section, we make continuous extension for the jump matrix $V^{(1)}(z)$
to remove the jump from $\mathbb{R}$. Besides, the new problem is hoped to
takes advantage of the decay/growth of $e^{2itz\theta_{ij}}$ for
$z\notin\mathbb{R}$. For this purpose, we introduce new contours as follow:

$\displaystyle\Sigma_{k}=(-1)^{(k-1)/2}i\varrho+e^{(k-1)i\pi/2+\varphi}R_{+},\hskip
14.22636ptk=1,3;$ (5.1)
$\displaystyle\Sigma_{k}=(-1)^{k/2+1}i\varrho+e^{ki\pi/2-\varphi}R_{+},\hskip
14.22636ptk=2,4,$ (5.2)

where and $\frac{\pi}{4}>\varphi>0$ is an fixed sufficiently small angle
achieving that $\Omega_{i}$ for $i=1,3,4,6$ don’t intersect any of
$\mathbb{D}(z_{n},\varrho)$ or $\mathbb{D}(\bar{z}_{n},\varrho)$. These
contours together with $\mathbb{R}$ are the boundary of new six regions
$\Omega_{1}$,…,$\Omega_{6}$. And $\Omega_{i}=\Omega_{i0}\cup\Omega_{i1}$ for
$i=1,3,4,6$ with

$\displaystyle\Omega_{10}=\left\\{z|\text{Re}z>0,0<\text{Im}z<\varrho\right\\},\
\Omega_{30}=\left\\{z|\text{Re}z<0,0<\text{Im}z<\varrho\right\\},$
$\displaystyle\Omega_{11}=\left\\{z|0<\arg(z-i\varrho)<\varphi\right\\},\
\Omega_{31}=\left\\{z|\pi-\varphi<\arg(z-i\varrho)<\pi\right\\},$
$\displaystyle\Omega_{2}=\left\\{z|\varphi<\arg(z-i\varrho)<\pi-\varphi\right\\},$

and $\Omega_{5}$, $\Omega_{60}$, $\Omega_{40}$, $\Omega_{61}$, $\Omega_{41}$
is the conjugate of $\Omega_{2}$, $\Omega_{10}$, $\Omega_{30}$, $\Omega_{11}$,
$\Omega_{31}$ respectively. In addition, let

$\displaystyle\Omega=\underset{k=1,3,4,6}{\cup}\Omega_{k},\ \
\Sigma^{(2)}=\Sigma_{1}\cup\Sigma_{2}\cup\Sigma_{3}\cup\Sigma_{4},$ (5.3)

which are shown in Figure 2.

$\Sigma_{1}$$\Sigma_{2}$$\Sigma_{3}$$\Sigma_{4}$Re$z$Im$z$$\Omega_{10}$$\Omega_{41}$$\Omega_{30}$$\Omega_{61}$$0$$\Omega_{11}$$\Omega_{2}$$\Omega_{31}$$\Omega_{40}$$\Omega_{5}$$\Omega_{60}$$z_{1}$$\bar{z}_{1}$$z_{2}$$\bar{z}_{2}$$z_{3}$$\bar{z}_{3}$$z_{4}$$\bar{z}_{4}$
Figure 2: The green region is $\Omega$, red lines are two critical lines
Im$z=\pm\varrho$, which divide $\Omega_{i}$, $i=1,3,4,6$ into two part
$\Omega_{i0}$ and $\Omega_{i1}$ respectively.

Introduce following functions for brief:

$\displaystyle
q_{11}(z,\xi)=\left\\{\begin{array}[]{lll}\dfrac{\bar{r}_{1}}{1+|r_{1}|^{2}},\text{
for }-n_{13}<\xi<-n_{12};\\\ r_{1}(z),\text{ for
}\xi>-n_{12};\end{array}\right.,$ (5.6) $\displaystyle
q_{12}(z,\xi)=\left\\{\begin{array}[]{lll}\dfrac{\bar{r}_{3}-\bar{r}_{1}\bar{r}_{2}}{1+|r_{1}|^{2}},\text{
for }-n_{13}<\xi<-n_{12};\\\ \bar{r}_{3},\text{ for
}\xi>-n_{12};\end{array}\right.,$ (5.9) $\displaystyle
q_{13}(z,\xi)=\left\\{\begin{array}[]{lll}-r_{2},\text{ for
}-n_{13}<\xi<-n_{12};\\\ \bar{r}_{2},\text{ for
}\xi>-n_{12};\end{array}\right.,$ (5.12) $\displaystyle
q_{21}(z,\xi)=\left\\{\begin{array}[]{lll}\dfrac{-r_{1}}{1+|r_{1}|^{2}},\text{
for }-n_{13}<\xi<-n_{12};\\\ -\bar{r}_{1},\text{ for
}\xi>-n_{12};\end{array}\right.,$ (5.15) $\displaystyle
q_{22}(z,\xi)=\left\\{\begin{array}[]{lll}-\bar{r}_{4},\text{ for
}-n_{13}<\xi<-n_{12};\\\ r_{2},\text{ for }\xi>-n_{12};\end{array}\right.,$
(5.18) $\displaystyle
q_{23}(z,\xi)=\left\\{\begin{array}[]{lll}\dfrac{r_{3}-r_{1}r_{2}}{1+|r_{1}|^{2}},\text{
for }-n_{13}<\xi<-n_{12};\\\ r_{3},\text{ for
}\xi>-n_{12};\end{array}\right..$ (5.21)

Besides, from $r_{i}\in W^{2,2}(\mathbb{R})$, it also has that
$q_{11}^{\prime}(z)$, $q_{12}^{\prime}(z)$, $q_{21}^{\prime}(z)$ and
$q_{23}^{\prime}(z)$ exist and are in $L^{2}(\mathbb{R})\cup
L^{\infty}(\mathbb{R})$. And $\parallel
q_{11}^{\prime}(z)\parallel_{p}\lesssim\parallel r^{\prime}(z)\parallel_{p}$
for $p=2,\infty$. Then the next step is to construct a matrix function
$R^{(2)}$. We need to remove jump on $\mathbb{R}$, and have some mild control
on $\bar{\partial}R^{(2)}$ sufficient to ensure that the
$\bar{\partial}$-contribution to the long-time asymptotics of $p_{ij}(x,t)$ is
negligible.

Let

$\varrho=\frac{1}{3}\min\left\\{\min_{j\neq
i\in\mathcal{N}}|z_{i}-z_{j}|,\min_{j\in\mathcal{N}}\left\\{|\text{Im}z_{j}|\right\\}\right\\},$
(5.22)

and Introduce two functions $\mathcal{X}_{1}(x)\in C_{0}^{\infty}$ and
$\mathcal{X}_{2}(x)\in C^{\infty}$ with

$\displaystyle\mathcal{X}_{1}(x)=\left\\{\begin{array}[]{lll}1,\ x\leq 0;\\\
0,\ x\geq\varrho;\end{array}\right.,\ \
\mathcal{X}_{2}(x)=\left\\{\begin{array}[]{lll}1,\ \ |x|\leq 1;\\\ x^{-1},\
|x|\geq 2.\end{array}\right.$ (5.27)

In addtion, $|\mathcal{X}_{1}(x)|$, $|\mathcal{X}_{2}(x)|\leq 1$. Note that
$\theta_{12}(z)$ has different property in the cases of $\xi>-n_{12}$ and
$-n_{13}<\xi<-n_{12}$, so the construction of $R^{(2)}(z)$ depend on $\xi$.
Then we choose $R^{(2)}(z,\xi)$ as:

Case I: for $\xi=\frac{x}{t}>-n_{12}$,

$R^{(2)}(z,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}1&R_{11}(z,\xi)e^{it\theta_{12}}&R_{13}(z,\xi)e^{it\theta_{13}}\\\
0&1&R_{12}(z,\xi)e^{it\theta_{23}}\\\
0&0&1\end{array}\right),&z\in\Omega_{j},j=1,3;\\\ \\\
\left(\begin{array}[]{ccc}1&0&0\\\ R_{21}(z,\xi)e^{-it\theta_{12}}&1&0\\\
R_{22}(z,\xi)e^{-it\theta_{13}}&R_{23}(z,\xi)e^{-it\theta_{23}}&1\end{array}\right),&z\in\Omega_{j},j=6,4;\\\
\\\ I,&elsewhere;\\\ \end{array}\right.$ (5.28)

Case II: for $\xi=\frac{x}{t}\in(-n_{13},-n_{12})$,

$R^{(2)}(z,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}1&0&R_{13}(z,\xi)e^{it\theta_{13}}\\\
R_{11}(z,\xi)e^{-it\theta_{12}}&1&R_{12}(z,\xi)e^{it\theta_{23}}\\\
0&0&1\end{array}\right),&z\in\Omega_{j},j=1,3;\\\ \\\
\left(\begin{array}[]{ccc}1&R_{21}(z,\xi)e^{it\theta_{12}}&0\\\ 0&1&0\\\
R_{22}(z,\xi)e^{-it\theta_{13}}&R_{23}(z,\xi)e^{-it\theta_{23}}&1\end{array}\right),&z\in\Omega_{j},j=6,4;\\\
\\\ I,&elsewhere;\\\ \end{array}\right.$ (5.29)

where the functions $R_{ij}$, $i=1,2$, $j=1,2,3$, are defined in following
proposition.

######  Proposition 5.

$R_{1j}$: $\bar{\Omega}_{1}\cup\bar{\Omega}_{3}\to\mathbb{C}$ and $R_{2j}$:
$\bar{\Omega}_{4}\cup\bar{\Omega}_{6}\to\mathbb{C}$, $j=1,2,3$ have boundary
values as follow:

$\displaystyle
R_{1j}(z,\xi)=\Bigg{\\{}\begin{array}[]{ll}q_{1j}(z,\xi)T_{+}(z)^{m_{1j}}&z\in\mathbb{R},\\\
q_{1j}(0,\xi)T(0)^{m_{1j}}\mathcal{X}_{2}(|z-i\varrho|)&z\in\Sigma_{1}\cup\Sigma_{2},\\\
\end{array},$ (5.32) $\displaystyle
R_{2j}(z,\xi)=\Bigg{\\{}\begin{array}[]{ll}q_{2j}(z,\xi)T_{-}(z)^{m_{2j}}&z\in\mathbb{R},\\\
q_{2j}(0,\xi)T(0)^{m_{2j}}\mathcal{X}_{2}(|z-i\varrho|)&z\in\Sigma_{3}\cup\Sigma_{4},\\\
\end{array},$ (5.35)

with $m_{11}=-m_{21}=2$, $m_{12}=-m_{22}=1$, $m_{13}=-m_{23}=-1$. And
$R_{ij}(z,\xi)$ have following property:

(1)

$\displaystyle|R_{ij}(z,\xi)|\lesssim\parallel q_{ij}\parallel_{\infty}.$
(5.36)

(2) For $z\in\Omega_{k_{i}0}$, $k_{1}=1,3$, $k_{2}=4,6$

$\displaystyle|\bar{\partial}R_{ij}(z,\xi)|\lesssim|q_{ij}^{\prime}(\text{Re}z)|+|q_{ij}(\text{Re}z)|.$
(5.37)

(3) For $z\in\Omega_{k_{i}1}$, $k_{1}=1,3$, $k_{2}=4,6$

$\displaystyle|\bar{\partial}R_{ij}(z,\xi)|\lesssim|q_{ij}^{\prime}(|z-i\varrho|)|+|\mathcal{X}_{2}^{\prime}(|z-i\varrho|)|+|z-i\varrho|^{-1/2}.$
(5.38)

(4) For $z\in\Omega_{2}\cup\Omega_{5}$,

$\bar{\partial}R^{(2)}(z)=0,$ (5.39)

###### Proof.

We only give the detail proof for $R_{11}(z)$ as an example. The extensions of
$R_{11}(z)$ can be constructed by:

1\. for $z\in\Omega_{10}$,

$\displaystyle
R_{11}(z,\xi)=q_{11}(\text{Re}z,\xi)T(z,\xi)^{2}\mathcal{X}_{1}(\text{Im}z)+(1-\mathcal{X}_{1}(\text{Im}z))q_{11}(\text{Re}z,\xi)\mathcal{X}_{2}(\text{Re}z)T(z,\xi)^{2};$
(5.40)

2\. for $z\in\Omega_{11}$,

$\displaystyle R_{11}(z,\xi)=$ $\displaystyle
q_{11}(|z-i\varrho|,\xi)\mathcal{X}_{2}(|z-i\varrho|)T(z,\xi)^{2}\cos[k_{0}\arg(z-i\varrho)]$
$\displaystyle+\\{1-\cos[k_{0}\arg(z-i\varrho)]\\}q_{11}(0)T(0,\xi)^{2}\mathcal{X}_{2}(|z-i\varrho|),$
(5.41)

where $k_{0}$ is a positive constant defined by $k_{0}=\frac{\pi}{2\varphi}$.
The other cases are easily inferred. Obviously, $R_{11}$ is bounded and admit
(5.36). For $z\in\Omega_{10}$, denote $z=s+yi$, $s,y\in\mathbb{R}$ with
$\bar{\partial}=\frac{1}{2}(\partial_{s}+\partial_{y}i)$. Then the
$\bar{\partial}$-derivative of (5.40) becomes:

$\displaystyle 2\bar{\partial}R_{11}(z,\xi)=$ $\displaystyle
T(z,\xi)^{2}q_{11}^{\prime}(s,\xi)\mathcal{X}_{1}(y)+T(z,\xi)^{2}q_{11}^{\prime}(s,\xi)(1-\mathcal{X}_{1}(y))\mathcal{X}_{2}(s)$
$\displaystyle\mathcal{X}_{2}^{\prime}(s)q_{11}(s,\xi)(1-\mathcal{X}_{1}(y))T(z,\xi)^{2}+\mathcal{X}_{1}^{\prime}(y)q_{11}(s,\xi)(1-\mathcal{X}_{2}(s))T(z,\xi)^{2}i,$
(5.42)

which immediately leads to (5.37). And for $z\in\Omega_{11}$, denote
$z=\zeta+i\varrho$ with $\zeta=le^{i\phi}$. Then we have
$\bar{\partial}_{z}=\bar{\partial}_{\zeta}=\frac{1}{2}e^{i\phi}\left(\partial_{l}+\frac{i}{l}\partial_{\phi}\right)$.
So

$\displaystyle\bar{\partial}R_{1}(z)=$
$\displaystyle\frac{e^{i\phi}}{2}\left[q_{11}^{\prime}(l)T^{2}(z,\xi)\mathcal{X}_{2}(l)+\left(q_{11}(l)T^{2}(z,\xi)-q_{11}(0)T^{2}(0,\xi)\right)\mathcal{X}_{2}^{\prime}(l)\right]\cos(k_{0}\phi)$
$\displaystyle-\frac{e^{i\phi}i}{2l}k_{0}\sin(k_{0}\phi)\left(q_{11}(l)T^{2}(z,\xi)-q_{11}(0)T^{2}(0,\xi)\right)\mathcal{X}_{2}(l).$
(5.43)

So

$\displaystyle|\bar{\partial}R_{ij}(z,\xi)|\lesssim|q_{ij}^{\prime}(|z-i\varrho|)|+|\mathcal{X}_{2}^{\prime}(|z-i\varrho|)|+\frac{|\mathcal{X}_{2}(|z-i\varrho|)||q_{ij}(|z-i\varrho|)-q_{ij}(0)|}{|s-i\varrho|}.$
(5.44)

By Cauchy-Schwarz inequality, we obtain

$|q_{11}(l)|=|q_{11}(l)-q_{11}(0)|=|\int_{0}^{l}q_{11}^{\prime}(s)ds|\leq\parallel
q_{11}^{\prime}(s)\parallel_{L^{2}}l^{1/2}\lesssim l^{1/2}.$ (5.45)

And note that $T(z)$ is a bounded function in $\bar{\Omega}_{1}$ with
estimation (4.27). Then (5.38) follows immediately. ∎

We now use $R^{(2)}$ to define the new transformation

$M^{(2)}(z)=M^{(1)}(z)R^{(2)}(z),$ (5.46)

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

######  RHP 3.

Find a matrix valued function $M^{(2)}(z)=M^{(2)}(z;x,t)$ with following
properties:

$\blacktriangleright$ Analyticity: $M^{(2)}(z)$ is continuous in $\mathbb{C}$,
sectionally continuous first partial derivatives in
$\mathbb{C}\setminus\left(\Sigma^{(2)}\cup\left\\{z_{n},\bar{z}_{n}\right\\}_{n\in\mathcal{N}}\right)$
and meromorphic out $\bar{\Omega}$;

$\blacktriangleright$ Jump condition: $M^{(2)}(z)$ has continuous boundary
values $M^{(2)}_{\pm}(z)$ on $\Sigma^{(2)}$ and

$M^{(2)}_{+}(z)=M^{(2)}_{-}(z)V^{(2)}(z),\hskip 14.22636ptz\in\Sigma^{(2)},$
(5.47)

where

$V^{(2)}(z)=\left\\{\begin{array}[]{ll}R^{(2)}(z,\xi)|_{\Omega_{1}\cup\Omega_{3}},&\text{as
}z\in\Sigma_{1}\cup\Sigma_{2};\\\\[12.0pt]
R^{(2)}(z,\xi)^{-1}|_{\Omega_{4}\cup\Omega_{6}},&\text{as
}z\in\Sigma_{3}\cup\Sigma_{4};\\\ \end{array}\right.;$ (5.48)

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(2)}(z)=$ $\displaystyle I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (5.49)

$\blacktriangleright$ $\bar{\partial}$-Derivative: For $z\in\mathbb{C}$ we
have

$\displaystyle\bar{\partial}M^{(2)}(z)=M^{(2)}(z)\bar{\partial}R^{(2)}(z,\xi),$
(5.50)

where Case I: for $\xi=\frac{y}{t}>-n_{12}$,

$\bar{\partial}R^{(2)}(z,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&\bar{\partial}R_{11}(z,\xi)e^{2it\theta_{12}}&\bar{\partial}R_{13}(z,\xi)e^{2it\theta_{13}}\\\
0&0&\bar{\partial}R_{12}(z,\xi)e^{2it\theta_{23}}\\\
0&0&0\end{array}\right),&z\in\Omega_{j},j=1,3;\\\ \\\
\left(\begin{array}[]{ccc}0&0&0\\\
\bar{\partial}R_{21}(z,\xi)e^{-2it\theta_{12}}&0&0\\\
\bar{\partial}R_{22}(z,\xi)e^{-2it\theta_{13}}&\bar{\partial}R_{23}(z,\xi)e^{-2it\theta_{23}}&0\end{array}\right),&z\in\Omega_{j},j=4,6;\\\
\\\ 0,&elsewhere;\\\ \end{array}\right.$ (5.51)

Case II: for $\xi=\frac{y}{t}\in(-n_{13},-n_{12})$,

$\bar{\partial}R^{(2)}(z,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&0&\bar{\partial}R_{13}(z,\xi)e^{2it\theta_{13}}\\\
\bar{\partial}R_{11}(z,\xi)e^{-2it\theta_{12}}&0&\bar{\partial}R_{12}(z,\xi)e^{2it\theta_{23}}\\\
0&0&0\end{array}\right),&z\in\Omega_{j},j=1,3;\\\ \\\
\left(\begin{array}[]{ccc}0&\bar{\partial}R_{21}(z,\xi)e^{2it\theta_{12}}&0\\\
0&0&0\\\
\bar{\partial}R_{22}(z,\xi)e^{-2it\theta_{13}}&\bar{\partial}R_{23}(z,\xi)e^{-2it\theta_{23}}&0\end{array}\right),&z\in\Omega_{j},j=6,4;\\\
\\\ 0,&elsewhere;\\\ \end{array}\right.$ (5.52)

$\blacktriangleright$ Residue conditions: $M^{(2)}$ has simple poles at each
point $z_{n}$ and $\bar{z}_{n}$ for $n\in\mathcal{N}$ with:

$\displaystyle\operatorname*{Res}_{z=z_{n}}M^{(2)}(z)=\lim_{z\to
z_{n}}M^{(2)}(z)\Gamma_{n}(\xi),$ (5.53)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M^{(2)}(z)=\lim_{z\to\bar{z}_{n}}M^{(2)}(z)\tilde{\Gamma}_{n}(\xi),$
(5.54)

where $\Gamma_{n}(\xi)$ and $\tilde{\Gamma}_{n}(\xi)$ are given in
(4.43)-(4.60).

To solve RHP 2, we decompose it into a model RH problem for $M^{rhp}(z)$ with
$\bar{\partial}R^{(2)}(z)\equiv 0$ and a pure $\bar{\partial}$-Problem with
nonzero $\bar{\partial}$-derivatives. First we establish a RH problem for the
$M^{rhp}(z)$ as follows.

######  RHP 4.

Find a matrix-valued function $M^{rhp}(z)=M^{rhp}(z;x,t)$ with following
properties:

$\blacktriangleright$ Analyticity: $M^{rhp}(z)$ is meromorphic in
$\mathbb{C}\setminus\Sigma^{(2)}$;

$\blacktriangleright$ Jump condition: $M^{rhp}(z)$ has continuous boundary
values $M^{rhp}_{\pm}(z)$ on $\Sigma^{(2)}$ and

$M^{rhp}_{+}(z)=M^{rhp}_{-}(z)V^{(2)}(z),\hskip 14.22636ptz\in\Sigma^{(2)};$
(5.55)

$\blacktriangleright$ $\bar{\partial}$-Derivative:
$\bar{\partial}R^{(2)}(z)=0$, for $z\in\mathbb{C}$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{rhp}(z)=$ $\displaystyle I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (5.56)

$\blacktriangleright$ Residue conditions: $M^{rhp}(z)$ has simple poles at
each point $z_{n}$ and $\bar{z}_{n}$ for $n\in\mathcal{N}$ with:

$\displaystyle\operatorname*{Res}_{z=z_{n}}M^{rhp}(z)=\lim_{z\to
z_{n}}M^{rhp}(z)\Gamma_{n}(\xi),$ (5.57)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M^{rhp}(z)=\lim_{z\to\bar{z}_{n}}M^{rhp}(z)\tilde{\Gamma}_{n}(\xi),$
(5.58)

where $\Gamma_{n}(\xi)$ and $\tilde{\Gamma}_{n}(\xi)$ are given in
(4.43)-(4.60).

The unique existence and asymptotic of $M^{rhp}(z)$ will shown in section 6.
We now use $M^{rhp}(z)$ to construct a new matrix function

$M^{(3)}(z)=M^{(2)}(z)M^{rhp}(z)^{-1}.$ (5.59)

which removes analytical component $M^{rhp}(z)$ to get a pure
$\bar{\partial}$-problem.

RHP 5. Find a matrix-valued function $M^{(3)}(z)=M^{(3)}(z;x,t)$ with
following identities:

$\blacktriangleright$ Analyticity: $M^{(3)}(z)$ is continuous and has
sectionally continuous first partial derivatives in $\mathbb{C}$.

$\blacktriangleright$ Asymptotic behavior:

$\displaystyle M^{(3)}(z)\sim I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (5.60)

$\blacktriangleright$ $\bar{\partial}$-equation:

$\bar{\partial}M^{(3)}(z)=M^{(3)}(z)W^{(3)}(z),\ \ z\in\mathbb{C},$

where

$W^{(3)}=M^{rhp}(z)\bar{\partial}R^{(2)}(z)M^{rhp}(z)^{-1}.$ (5.61)

The unique existence and asymptotic of $M^{(3)}(z)$ will shown in section 7.
And in the RHP 4, its jump matrix $V^{(2)}(z)$ admits the following estimates.

######  Proposition 6.

For the jump matrix $V^{(2)}(z)$, we have the following estimate

$\displaystyle\parallel
V^{(2)}(z)-I\parallel_{L^{\infty}(\Sigma^{(2)})}=\mathcal{\mathcal{O}}(e^{-t\varrho\rho_{0}}),$
(5.62)

where $\rho_{0}$, $\varrho$ is defined in (4.19) and (5.22).

###### Proof.

We prove (5.62) for $z\in\Sigma_{1}$, other cases can be shown in a similar
way. By using definition of $V^{(2)}(z)$ and (5.36), we have

$\displaystyle\parallel
V^{(2)}(z)-I\parallel_{L^{\infty}(\Sigma_{1})}\leq\sum_{j=1}^{3}\parallel
R_{1j}e^{-t\text{Im}z\rho_{0}}\parallel_{L^{\infty}(\Sigma_{1})}\lesssim
e^{-t\varrho\rho_{0}}.$ (5.63)

∎

######  Corollary 1.

For $1\leq p\leq+\infty$, the jump matrix $V^{(2)}(z)$ satisfies

$\parallel V^{(2)}(z)-I\parallel_{L^{p}(\Sigma^{(2)})}\leq
K_{p}e^{-t\varrho\rho_{0}},$ (5.64)

for some constant $K_{p}\geq 0$ depending on $p$.

This proposition means that the jump matrix $V^{(2)}(z)$ uniformly goes to $I$
on $\Sigma^{(2)}$, so there is only exponentially small error (in t) by
completely ignoring the jump condition of $M^{rhp}(z)$. This proposition
inspire us to construct the solution $M^{rhp}(z)$ of the RHP 4 in following
form

$M^{rhp}(z)=E(z)M^{sol}(z).$ (5.65)

This decomposition splits $M^{rhp}(z)$ into two parts: $M^{sol}(z)$ solves a
model RHP given following obtained by ignoring the jump conditions of RHP 4,
which will be solved in next Section 6; And $E(z)$ is a error function, which
is a solution of a small-norm RH problem and we discuss it in Section 7.

######  RHP 5.

Find a matrix-valued function $M^{sol}(z)=M^{sol}(z;x,t)$ with following
properties:

$\blacktriangleright$ Analyticity: $M^{sol}(z)$ is analytical in
$\mathbb{C}\setminus(\mathcal{Z}\cup\bar{\mathcal{Z}})$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{sol}(z)\sim I+\mathcal{O}(z^{-1}),\hskip
14.22636ptz\rightarrow\infty;$ (5.66)

$\blacktriangleright$ Residue conditions: $M^{sol}(z)$ has simple poles at
each point in $\mathcal{Z}\cup\bar{\mathcal{Z}}$ satisfying:

$\displaystyle\operatorname*{Res}_{z=z_{n}}M^{sol}(z)=\lim_{z\to
z_{n}}M^{sol}(z)\Gamma_{n}(\xi),$ (5.67)
$\displaystyle\operatorname*{Res}_{z=\bar{z}_{n}}M^{sol}(z)=\lim_{z\to\bar{z}_{n}}M^{sol}(z)\tilde{\Gamma}_{n}(\xi),$
(5.68)

where $\Gamma_{n}(\xi)$ and $\tilde{\Gamma}_{n}(\xi)$ are given in
(4.43)-(4.60).

##  6  Asymptotic analysis on soliton solutions

We will build the reflectionless case of RHP 3 as RHP 5 to show that
approximated with a finite sum of soliton solutions in this section. Based on
the original RHP 1, we show the existence and uniqueness of solution of above
RHP 5.

######  Proposition 7.

For $M^{sol}(z)$ denoting the solution of the RHP 5 with scattering data
$\mathcal{D}=\left\\{\vec{r}(z)=(r_{1}(z),...,r_{4}(z)),\left\\{z_{n},c_{n}\right\\}_{n\in\mathcal{N}}\right\\}$,
$M^{sol}(z)$ exists unique. By an explicit transformation, $M^{sol}(z)$ is
equivalent to a reflectionless solution of the original RHP 1 with modified
scattering data
$\widetilde{\mathcal{D}}=\left\\{0,\left\\{z_{n},\widetilde{c}_{n}\right\\}_{n\in\mathcal{N}}\right\\}$,
where

$\widetilde{c}_{n}=c_{n}\exp\left\\{-\frac{1}{i\pi}\int_{I(\xi)}\frac{\log(1+|r_{1}(s)|^{2})}{s-z_{n}}ds\right\\}.$
(6.1)

###### Proof.

To transform $M^{sol}(z)$ to the soliton-solution of RHP 1, we reverses the
triangularity effected in (4.28) and (5.46)

$N(z)=M^{sol}(z)\left(\prod_{n\in\Delta(\xi)}\dfrac{z-z_{n}}{z-\bar{z}_{n}}\right)^{-\sigma_{3}}.$
(6.2)

then we can verify $N(z)$ with scattering data $\widetilde{\mathcal{D}}$
satisfying RHP 1. The transformation (6.2) preserves the normalization
conditions at the origin and infinity obviously. Comparing with (4.28), this
transformation restores the influence by $T(z,\xi)$ on residue condition in
(4.43)-(4.60), and convert them back into (2.57)-(2.69). Its analyticity
follows from the Proposition of $M^{rhp}(z)$ and $T(z)$ immediately. Then
$N(z)$ is solution of RHP 1 with absence of reflection, whose unique exact
solution exists and can be obtained as described similarly in [46] Appendix A.
And the uniqueness and existences of $M^{sol}(z)$ come from (6.2). ∎

From (2.70), denote $p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}})$ is the soliton
solution of a reflectionless scattering data $\widetilde{\mathcal{D}}$ to the
original RHP 1 with

$p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}})=-i(a_{i}-a_{j})\lim_{z\to\infty}[zN(z;\widetilde{\mathcal{D}})]_{ij}.$

Then by (6.2), it also has

$\displaystyle
p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}})=-i(a_{i}-a_{j})\lim_{z\to\infty}[zM^{sol}(z;\mathcal{D})]_{ij}.$
(6.3)

Although $M^{sol}(z)$ has uniqueness and existence, not all discrete spectra
have contribution as $t\to\infty$. Give pairs points $x_{1}\leq
x_{2}\in\mathbb{R}$ and velocities $v_{1}\leq v_{2}\in\mathbb{R}$, we define a
cone

$C(x_{1},x_{2},v_{1},v_{2})=\left\\{(x,t)\in R^{2}:x=x_{0}+vt,\
x_{0}\in[x_{1},x_{2}]\text{, }v\in[v_{1},v_{2}]\right\\}.$ (6.4)

and denote

$\displaystyle I=\left\\{v:\ v_{1}<\text{Re}v<v_{2}\right\\},\ \
\Delta(I)=\left\\{n\in\mathcal{N}:\ v_{z_{k}}\in I\right\\}$
$\displaystyle\mathcal{Z}(I)=\left\\{z_{k}\in\mathcal{Z}:\ v_{z_{k}}\in
I\right\\},\hskip 59.75095ptN(I)=|\mathcal{Z}(I)|,$ (6.5)

where $v_{z_{k}}$ is velocity of the soliton solution corresponding to pole
$z_{k}$ with for $k=1,...,N_{1}$, $v_{z_{k}}=-n_{12}$ and for
$k=N_{1},...,N_{1}+N_{2}$, $v_{z_{k}}=-n_{23}$. We can show the following
proposition.

x$x=v_{2}t+x_{2}$$x=v_{2}t+x_{1}$$x=v_{1}t+x_{1}$$x=v_{1}t+x_{2}$$x_{1}$$x_{2}$C

$\mathbb{R}$$v_{1}$$v_{2}$$-n_{12}$$-n_{23}$$-n_{13}$I

Figure 3: (a) In the example here, $-n_{23}<v_{1}<-n_{13}<v_{2}<-n_{12}$, so
$\mathcal{Z}(I)=\emptyset$; (b) The cone $C(x_{1},x_{2},v_{1},v_{2})$

Let $\Delta^{\pm}(I)$ as a partition of $\mathcal{N}$,
$\Delta^{\pm}(I)=\Delta_{1}^{\pm}(I)\cup\Delta_{2}^{\pm}(I)$ with
$\Delta_{i}^{+}(I)=\left\\{n\in\mathcal{N}_{i}:\ v_{z_{n}}<v_{1}\right\\}$,
and $\Delta_{i}^{-}(I)=\left\\{n\in\mathcal{N}_{i}:\
v_{z_{n}}>v_{2}\right\\}$, $i=1,2$ , then
$\mathcal{N}=\Delta(I)\cup\Delta^{+}(I)\cup\Delta^{-}(I)$. Denote

$\displaystyle
N^{\Delta(I)}(z;\tilde{\mathcal{D}})=N(z;\tilde{\mathcal{D}})\Pi_{1}(z)\Pi_{2}(z),$
(6.6)
$\displaystyle\Pi_{1}(z)=\prod_{n\in\Delta_{1}^{-}(I)}\left(\dfrac{z-z_{n}}{z-\bar{z}_{n}}\right)^{-\sigma_{3}},\
\Pi_{2}(z)=\prod_{n\in\Delta_{2}^{-}(I)}\left(\dfrac{z-z_{n}}{z-\bar{z}_{n}}\right)^{-\sigma_{2}},$
(6.7)

with $\sigma_{2}=$diag$\left\\{0,1,-1\right\\}$. From the residue condition of
$N(z;\tilde{\mathcal{D}})$ in (2.57)-(2.69) with replacing $c_{n}$ by
$\mathring{c}_{n}$, $N^{\Delta(I)}(z;\tilde{\mathcal{D}})$ has residue as
$\operatorname*{Res}_{z=z_{n}}N^{\Delta(I)}(z;\tilde{\mathcal{D}})=\lim_{z\to
z_{n}}N^{\Delta(I)}(z;\tilde{\mathcal{D}})\Gamma_{n}^{\Delta(I)}$ and
$\operatorname*{Res}_{z=\bar{z}_{n}}N^{\Delta(I)}(z;\tilde{\mathcal{D}})=\lim_{z\to\bar{z}_{n}}N^{\Delta(I)}(z;\tilde{\mathcal{D}})\tilde{\Gamma}_{n}^{\Delta(I)}$.
Here,

$\displaystyle\Gamma_{n}^{\Delta(I)}=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&c_{n}e^{iz_{n}t\theta_{12}}\Pi_{2}(z_{n})&0\\\
0&0&0\\\ 0&0&0\end{array}\right),&\text{ for
}n\in\mathcal{N}_{1}\setminus\Delta_{1}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&0&0\\\
c_{n}^{-1}e^{-iz_{n}t\theta_{12}}(1/\Pi_{1})^{\prime}(z_{n})^{-2}\Pi_{2}(z_{n})^{-1}&0&0\\\
0&0&0\end{array}\right),&\text{ for }n\in\Delta_{1}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&0&0\\\
0&0&c_{n}e^{iz_{n}t\theta_{23}}\Pi_{1}(z_{n})\\\
0&0&0\end{array}\right),&\text{ for
}n\in\mathcal{N}_{2}\setminus\Delta_{2}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&c_{n}^{-1}e^{-iz_{n}t\theta_{23}}(1/\Pi_{2})^{\prime}(z_{n})^{-2}\Pi_{1}(z_{n})^{-1}&0\end{array}\right),&\text{
for }n\in\Delta_{2}^{-}(I);\end{array}\right.,$ (6.24)

and

$\displaystyle\tilde{\Gamma}_{n}^{\Delta(I)}=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{ccc}0&0&0\\\
\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{12}}\Pi_{2}(\bar{z}_{n})^{-1}&0&0\\\
0&0&0\end{array}\right),&\text{ for
}n\in\mathcal{N}_{1}\setminus\Delta_{1}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&\tilde{c}_{n}^{-1}e^{i\bar{z}_{n}t\theta_{12}}\Pi_{1}^{\prime}(\bar{z}_{n})^{-2}\Pi_{2}(\bar{z}_{n})&0\\\
0&0&0\\\ 0&0&0\end{array}\right),&\text{ for }n\in\Delta_{1}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\
0&\tilde{c}_{n}e^{-i\bar{z}_{n}t\theta_{23}}\Pi_{1}^{-1}(\bar{z}_{n})&0\end{array}\right),&\text{
for }n\in\mathcal{N}_{2}\setminus\Delta_{2}^{-}(I);\\\
\left(\begin{array}[]{ccc}0&0&0\\\
0&0&\tilde{c}_{n}^{-1}e^{i\bar{z}_{n}t\theta_{23}}\Pi_{2}^{\prime}(\bar{z}_{n})^{-2}\Pi_{1}(\bar{z}_{n})\\\
0&0&0\end{array}\right),&\text{ for }n\in\Delta_{2}^{-}(I)\end{array}\right.,$
(6.41)

For $x=x_{0}+v_{0}t$ with $x_{1}\leq x_{0}\leq x_{2}$ and $v_{1}\leq v_{0}\leq
v_{2}$, consider the exponent in nilpotent matrix corresponding to pole
$z_{n}$ in residue condition (2.57) and (2.65), when $n=1,...,N_{1}$,

$\displaystyle|e^{iz_{n}[(a_{1}-a_{2})x+(b_{1}-b_{2})t]}|=|e^{-\text{Im}z_{n}(a_{1}-a_{2})x_{0}}||e^{-\text{Im}z_{n}t(a_{1}-a_{2})(v_{0}-v_{z_{n}})}|,$
(6.42)

and when $n=N_{1},...,N_{1}+N_{2}$,

$\displaystyle|e^{iz_{n}[(a_{2}-a_{3})x+(b_{2}-b_{3})t]}|=|e^{-\text{Im}z_{n}(a_{2}-a_{3})x_{0}}||e^{-\text{Im}z_{n}t(a_{2}-a_{3})(v_{0}-v_{z_{n}})}|.$
(6.43)

Above equations imply that for $z_{n}\notin\mathcal{Z}(I)$, their residue
exponentially small as

$\displaystyle\parallel\Gamma_{n}^{\Delta(I)}\parallel=\mathcal{O}(e^{-a\mu(I)t}).$
(6.44)

So for the poles $z_{n}$ not in $\mathcal{Z}(I)$, we want to trap them for
jumps along small closed loops enclosing themselves as
$\mathbb{D}(z_{n},\varrho)$ respectively and prove that this jump is uniformly
exponentially near identity. By the definition of $\varrho$ in (5.22), for
every $n\in\mathcal{N}$, $\mathbb{D}(z_{n},\varrho)$ are pairwise disjoint and
are disjoint with $\mathbb{R}$. Further, from the symmetry of poles , this
definition guarantee $\mathbb{D}(\bar{z}_{n},\varrho)$ have same property. To
achieve this purpose, we denote a piecewise matrix function

$G(z)=\left\\{\begin{array}[]{ll}I-\frac{\Gamma_{n}^{\Delta(I)}(\xi)}{z-z_{n}},&\text{as
}z\in\mathbb{D}(z_{n},\varrho),z_{n}\in\mathcal{Z}\setminus\mathcal{Z}(I);\\\\[12.0pt]
I-\frac{\tilde{\Gamma}_{n}^{\Delta(I)}(\xi)}{z-\bar{z}_{n}},&\text{as
}z\in\mathbb{D}(\bar{z}_{n},\varrho),z_{n}\in\mathcal{Z}\setminus\mathcal{Z}(I);\\\
I&\text{as }z\text{ in elsewhere};\end{array}\right..$ (6.45)

Introduce a new transformation as

$\displaystyle\widetilde{m}(z;\tilde{\mathcal{D}})=N^{\Delta(I)}(z;\tilde{\mathcal{D}})G(z).$
(6.46)

Comparing with $N^{\Delta(I)}(z;\tilde{\mathcal{D}})$, the new matrix function
$\widetilde{m}(z;\tilde{\mathcal{D}})$ has new jump in each
$\partial\mathbb{D}(\bar{z}_{n},\varrho)$ which denote by $\widetilde{V}(z)$.
Then by (6.44), direct calculation shows that as $t\to\infty$ with $(x,t)\in
C(x_{1},x_{2},v_{1},v_{2})$,

$\parallel\widetilde{V}(z)-I\parallel_{L^{\infty}(\widetilde{\Sigma})}=\mathcal{O}(e^{-a\mu(I)t}),\hskip
14.22636pt\widetilde{\Sigma}=\cup_{z_{k}\in\mathcal{Z}\setminus\mathcal{Z}(I)}\left(\partial
D_{k}\cup\partial\bar{D}_{k}\right).$ (6.47)

This property inspire us to consider following solution of RHP 5.

######  Proposition 8.

Denote $M^{sol}_{\mathcal{Z}(I)}(z;\mathcal{D}(I))$ as the solution of the
Riemann-Hilbert problem 5 with scattering data
$\mathcal{D}(I)=\left\\{\vec{r}(z)=(r_{1}(z),...,r_{4}(z)),\left\\{z_{n},c_{n}(I)\right\\}_{z_{n}\in\mathcal{Z}(I)}\right\\}$,
where

$\displaystyle c_{k}(I)=\left\\{\begin{array}[]{ll}c_{k}\Pi_{2}(z_{n}),&\text{
for }n\in\mathcal{N}_{1}\cap\mathcal{Z}(I);\\\ c_{k}\Pi_{1}(z_{n}),&\text{ for
}n\in\mathcal{N}_{2}\cap\mathcal{Z}(I);\\\ \end{array}\right.$ (6.50)

Then there is a reflectionless solution of the original RHP 1 with modified
scattering data $\tilde{\mathcal{D}}(I)$ defined by

$\displaystyle
N(z;\tilde{\mathcal{D}}(I))=M^{sol}_{\mathcal{Z}(I)}(z;\mathcal{D}(I))\left(\prod_{n\in\Delta(\xi),z_{n}\in\mathcal{Z}(I)}\dfrac{z-z_{n}}{z-\bar{z}_{n}}\right)^{-\sigma_{3}},$
(6.51)

with

$\displaystyle\tilde{\mathcal{D}}(I)=\left\\{0,\left\\{z_{n},\widetilde{c}_{n}(I)\right\\}_{z_{n}\in\mathcal{Z}(I)}\right\\},\
\mathring{c}_{n}(I)=c_{n}(I)\exp\left(2i\int_{I(\xi)}\dfrac{\nu(s)ds}{s-z}\right).$
(6.52)

Analogously, From (2.70), denote $p_{ij}^{sol}(x,t;\tilde{\mathcal{D}}(I))$ is
the soliton solution of a reflectionless scattering data
$\tilde{\mathcal{D}}(I)$ to the original RHP 1 with

$p_{ij}^{sol}(x,t;\tilde{\mathcal{D}}(I))=-i(a_{i}-a_{j})\lim_{z\to\infty}[zN(z;\tilde{\mathcal{D}}(I))]_{ij}.$

Then by (6.2), it also has

$\displaystyle
p_{ij}^{sol}(x,t;\tilde{\mathcal{D}}(I))=-i(a_{i}-a_{j})\lim_{z\to\infty}[zM_{\mathcal{Z}(I)}^{sol}(z;\mathcal{D}(I))]_{ij}.$
(6.53)

######  Proposition 9.

For $M^{sol}(z;\mathcal{D})$ and
$M^{sol}_{\mathcal{Z}(I)}(z;\tilde{\mathcal{D}}(I))$ defined above, as
$t\to\infty$ with $(x,t)\in C(x_{1},x_{2},v_{1},v_{2})$, there exists a
positive constant $a=\min\left\\{a_{1}-a_{2},a_{2}-a_{3}\right\\}$

$\displaystyle
M^{sol}(z;\mathcal{D})=\left(I+\mathcal{O}(e^{-a\mu(I)t})\right)M^{sol}_{\mathcal{Z}(I)}(z;\tilde{\mathcal{D}}(I))\left(\prod_{n\in\Delta(\xi),z_{n}\notin\mathcal{Z}(I)}\dfrac{z-z_{n}}{z-\bar{z}_{n}}\right)^{-\sigma_{3}},$
(6.54)

where

$\mu(I)=\underset{z_{k}\in\mathcal{Z}\setminus\mathcal{Z}(I)}{\min}\left\\{\text{\rm
Im}(z_{k})\text{dist}(v_{z_{k}},\mathcal{I})\right\\}=\underset{z_{k}\in\mathcal{Z}\setminus\mathcal{Z}(I),i=1,2}{\min}\left\\{\text{\rm
Im}(z_{k})|v_{i}-v_{z_{k}}|\right\\}>0.$

###### Proof.

To arrive at (6.54), we consider the asymptotic error between
$N(z;\tilde{\mathcal{D}})$ and $N(z;\tilde{\mathcal{D}}(I))$. Since
$\widetilde{m}(z;\tilde{\mathcal{D}})$ has same poles and residue conditions
with $N(z;\tilde{\mathcal{D}}(I))$, then

$m_{0}(z)=\widetilde{m}(z;\tilde{\mathcal{D}})N(z;\tilde{\mathcal{D}}(I))^{-1}$

has no poles, but it has jump matrix for $z\in\widetilde{\Sigma}$,

$m_{0}^{+}(z)=m_{0}^{-}(z)V_{m_{0}}(z),$ (6.55)

where the jump matrix $V_{m_{0}}(z)$ given by

$V_{m_{0}}(z)=N(z;\tilde{\mathcal{D}}(I))\widetilde{V}(z)N(z;\tilde{\mathcal{D}}(I))^{-1},$
(6.56)

which, by using (6.47), also admits the same decaying estimate

$\parallel
V_{m_{0}}(z)-I\parallel_{L^{\infty}(\widetilde{\Sigma})}=\parallel\widetilde{V}(z)-I\parallel_{L^{\infty}(\widetilde{\Sigma})}=\mathcal{O}(e^{-a\mu(I)t}),\
\ t\rightarrow+\infty.$

Then by using the theory of small norm RH problem, we find that $m_{0}(z)$
exists and

$m_{0}(z)=I+\mathcal{O}(e^{-a\mu(I)t}),\ \ t\to\infty,$

which together with (6.45) gives the formula (6.54). ∎

######  Corollary 2.

For the soliton solution of the reflectionless scattering data
$\tilde{\mathcal{D}}$ and $\tilde{\mathcal{D}}(I)$ to the original RHP 1
$p_{ij}^{sol}(x,t;\tilde{\mathcal{D}})$ and
$p_{ij}^{sol}(x,t;\tilde{\mathcal{D}}(I))$ defined in (6.3) and (6.53)
respectively, as $t\to\infty$ with $(x,t)\in C(x_{1},x_{2},v_{1},v_{2})$,
their error is exponentially small with

$\displaystyle
p_{ij}^{sol}(x,t;\tilde{\mathcal{D}})=p_{ij}^{sol}(x,t;\tilde{\mathcal{D}}(I))+\mathcal{O}(e^{-a\mu(I)t}).$
(6.57)

##  7 The small norm RH problem for error function

In this section, we consider the error matrix-function $E(z)$ and show that
the error function $E(z)$ solves a small norm Riemann-Hilbert problem which
can be expanded asymptotically for large times. From the definition (5.65), we
can obtain a RH problem for the matrix function $E(z)$.

######  RHP 6.

Find a matrix-valued function $E(z)$ with following identities:

$\blacktriangleright$ Analyticity: $E(z)$ is analytical in
$\mathbb{C}\setminus\Sigma^{(2)}$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle E(z)\sim I+\mathcal{O}(z^{-1}),\hskip
14.22636pt|z|\rightarrow\infty;$ (7.1)

$\blacktriangleright$ Jump condition: $E$ has continuous boundary values
$E_{\pm}$ on $\Sigma^{(2)}$ satisfying

$E_{+}(z)=E_{-}(z)V^{E},$

where the jump matrix $V^{E}$ is given by

$V^{E}(z)=M^{sol}(z)V^{(2)}(z)M^{sol}(z)^{-1}.$ (7.2)

Proposition 7 implies that $M^{sol}(z)$ is bound on $\Sigma^{(2)}$. By using
proposition 6 and Corollary 1, we have the following evaluation

$\parallel V^{E}-I\parallel_{p}\lesssim\parallel
V^{(2)}-I\parallel_{p}=\mathcal{O}(e^{-\varrho\rho_{0}t}),\hskip
8.5359pt\text{for $1\leq p\leq+\infty$.}$ (7.3)

This uniformly vanishing bound $\parallel V^{E}-I\parallel$ establishes RHP 6
as a small-norm Riemann-Hilbert problem. Therefore, the existence and
uniqueness of the RHP 6 is shown by using a small-norm RH problem [31, 32]
with

$E(z)=I+\frac{1}{2\pi
i}\int_{\Sigma^{(2)}}\dfrac{\left(I+\eta(s)\right)(V^{E}-I)}{s-z}ds,$ (7.4)

where the $\eta\in L^{2}(\Sigma^{(2)})$ is the unique solution of following
equation:

$(1-C_{E})\eta=C_{E}\left(I\right).$ (7.5)

Here $C_{E}$:$L^{2}(\Sigma^{(2)})\to L^{2}(\Sigma^{(2)})$ is a integral
operator defined by

$C_{E}(f)(z)=C_{-}\left(f(V^{E}-I)\right),$ (7.6)

with the Cauchy projection operator $C_{-}$ on $\Sigma^{(2)}$ :

$C_{-}(f)(s)=\lim_{z\to\Sigma^{(2)}_{-}}\frac{1}{2\pi
i}\int_{\Sigma^{(2)}}\dfrac{f(s)}{s-z}ds.$ (7.7)

Then by (7.2) we have

$\parallel C_{E}\parallel\leq\parallel C_{-}\parallel\parallel
V^{E}-I\parallel_{\infty}\lesssim\mathcal{O}(e^{-\varrho\rho_{0}t}),$ (7.8)

which means $\parallel C_{E}\parallel<1$ for sufficiently large t, therefore
$1-C_{E}$ is invertible, and $\eta$ exists and is unique. Moreover,

$\parallel\eta\parallel_{L^{2}(\Sigma^{(2)})}\lesssim\dfrac{\parallel
C_{E}\parallel}{1-\parallel
C_{E}\parallel}\lesssim\mathcal{O}(e^{-\varrho\rho_{0}t}).$ (7.9)

Then we have the existence and boundedness of $E(z)$. In order to reconstruct
the solution $p_{ij}(x,t)$ of (1.1), we need the asymptotic behavior of $E(z)$
as $z\to\infty$ .

######  Proposition 10.

For $E(z)$ defined in (7.4), as $z\to\infty$, $E(z)$ has Laurent expansion as

$\displaystyle E(z)=I+\frac{E_{1}}{z}+\mathcal{O}(z^{-2}),$ (7.10)

where

$E_{1}=-\frac{1}{2\pi i}\int_{\Sigma^{(2)}}\left(I+\eta(s)\right)(V^{E}-I)ds.$
(7.11)

Moreover, $E_{1}$ satisfies following long time asymptotic behavior condition:

$|E_{1}|\lesssim\mathcal{O}(e^{-\varrho\rho_{0}t}),\hskip
14.22636ptE_{1}\lesssim\mathcal{O}(e^{-2\rho_{0}t}).$ (7.12)

###### Proof.

By combining (7.9) and (7.3), we obtain the result promptly. ∎

##  8  Asymptotic analysis on the pure $\bar{\partial}$-Problem

Now we consider the long time asymptotics behavior of $M^{(3)}$. The
$\bar{\partial}$-problem 4 of $M^{(3)}$ is equivalent to the integral equation

$M^{(3)}(z)=I-\frac{1}{\pi}\iint_{\mathbb{C}}\dfrac{M^{(3)}(s)W^{(3)}(s)}{s-z}dm(s),$
(8.1)

where $m(s)$ is the Lebesgue measure on the $\mathbb{C}$. Denote $C_{z}$ as
the left Cauchy-Green integral operator,

$fC_{z}(z)=\frac{1}{\pi}\iint_{C}\dfrac{f(s)W^{(3)}(s)}{z-s}dm(s).$

Then above equation can be rewritten as

$M^{(3)}(z)=I\cdot\left(I-C_{z}\right)^{-1}.$ (8.2)

To prove the existence of operator $\left(I-C_{z}\right)^{-1}$, we have
following Lemma.

######  Lemma 6.

The norm of the integral operator $C_{z}$ decay to zero as $t\to\infty$:

$\parallel C_{z}\parallel_{L^{\infty}\to L^{\infty}}\lesssim|t|^{-1/2},$ (8.3)

which implies that $\left(I-C_{z}\right)^{-1}$ exists.

###### Proof.

For any $f\in L^{\infty}$,

$\displaystyle\parallel fC_{z}\parallel_{L^{\infty}}$
$\displaystyle\leq\parallel
f\parallel_{L^{\infty}}\frac{1}{\pi}\iint_{C}\dfrac{|W^{(3)}(s)|}{|z-s|}dm(s).$

Consequently, we only need to evaluate the integral
$\frac{1}{\pi}\iint_{C}\dfrac{|W^{(3)}(s)|}{|z-s|}dm(s)$. As $W^{(3)}(s)$ is a
sectorial function, we only need to consider it on ever sector. Recall the
definition of $W^{(3)}(s)=M^{r}(z)\bar{\partial}R^{(2)}(z)M^{r}(z)^{-1}$.
$W^{(3)}(s)\equiv 0$ out of $\bar{\Omega}$. We only detail the case for matrix
functions having support in the sector $\Omega_{1}$ as $-n_{13}<\xi<-n_{12}$,
because the case $\xi<-n_{12}$ is more trivial. Then in this case,
$\theta_{12}<0$. Proposition 10 and 7 implies the boundedness of $M^{r}(z)$
and $M^{r}(z)^{-1}$ for $z\in\bar{\Omega}$, so

$\frac{1}{\pi}\iint_{\Omega_{1}}\dfrac{|W^{(3)}(s)|}{|z-s|}dm(s)\lesssim\iint_{\Omega_{1}}\dfrac{|\bar{\partial}R_{11}(s)e^{-izt\theta_{12}}|+|\bar{\partial}R_{12}(s)e^{izt\theta_{23}}|+|\bar{\partial}R_{13}(s)e^{izt\theta_{13}}|}{|z-s|}dm(s).$
(8.4)

Give the detail of estimation to first integral, and the others are similarly.
Referring to (5.37) in proposition 5, the integral
$\iint_{\Omega_{1}}\dfrac{|\bar{\partial}R_{1}(s)e^{-izt\theta_{12}}|}{|z-s|}dm(s)$
can be divided to five part:

$\displaystyle\iint_{\Omega_{1}}\dfrac{|\bar{\partial}R_{1}(s)e^{-izt\theta_{12}}|}{|z-s|}dm(s)\lesssim
I_{11}+I_{12}+I_{21}+I_{22}+I_{3},$ (8.5)

with

$\displaystyle
I_{11}=\iint_{\Omega_{10}}\dfrac{|q_{11}^{\prime}(\text{Re}s)e^{-izt\theta_{12}}|}{|z-s|}dm(s),\hskip
14.22636ptI_{12}=\iint_{\Omega_{10}}\dfrac{|q_{11}(\text{Re}s)e^{-izt\theta_{12}}|}{|z-s|}dm(s)$
(8.6) $\displaystyle
I_{21}=\iint_{\Omega_{11}}\dfrac{|q_{11}^{\prime}(|s-i\varrho|)e^{-izt\theta_{12}}|}{|z-s|}dm(s),\hskip
14.22636ptI_{22}=\iint_{\Omega_{11}}\dfrac{|\mathcal{X}_{2}^{\prime}(|s-i\varrho|)e^{-izt\theta_{12}}|}{|z-s|}dm(s)$
(8.7) $\displaystyle
I_{3}=\iint_{\Omega_{11}}\dfrac{|s-i\varrho|^{-1/2}|e^{-izt\theta_{12}}|}{|z-s|}dm(s).$
(8.8)

For $I_{11}$ and $I_{12}$, let $s=u+vi$, $z=x+yi$. In the following
computation, we will use the inequality

$\displaystyle\parallel|s-z|^{-1}\parallel_{L^{q}(0,+\infty)}$
$\displaystyle=\left\\{\int_{0}^{+\infty}\left[\left(\frac{u-x}{v-y}\right)^{2}+1\right]^{-q/2}d\left(\frac{u-x}{|v-y|}\right)\right\\}^{1/q}|v-y|^{1/q-1}$
$\displaystyle\lesssim|v-y|^{1/q-1},$ (8.9)

with $1\leq q<+\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. Moreover, by
$-n_{13}<\xi<-n_{12}$ we have the constant $\theta_{12}<0$. Therefore,

$\displaystyle I_{11}$
$\displaystyle\leq\int_{0}^{\varrho}\int_{0}^{+\infty}\dfrac{|q_{11}^{\prime}(u)|}{|z-s|}e^{tv\theta_{12}}dudv\leq\int_{0}^{\varrho}\parallel|s-z|^{-1}\parallel_{L^{2}(\mathbb{R}^{+})}\parallel
q_{11}^{\prime}\parallel_{L^{2}(\mathbb{R}^{+})}e^{\theta_{12}tv}dv$
$\displaystyle\lesssim\int_{0}^{+\infty}|v-y|^{-1/2}e^{tv\theta_{12}}dv\lesssim
t^{-1/2}.$ (8.10)

It also deduce to that $I_{12}\lesssim t^{-1/2}$ in the same way. For the
$\Omega_{10}$, we take another change of variable as $s=u+(v+\varrho)i$,
$z=x+(y+\varrho)i$. Recall $\varphi$ which is the angle of $\Sigma_{1}$, then

$\displaystyle I_{21}$
$\displaystyle\leq\int_{0}^{+\infty}\int_{\frac{v}{\tan\varphi}}^{+\infty}\dfrac{|q_{11}^{\prime}(\sqrt{u^{2}+v^{2}})|}{|z-s|}e^{tv\theta_{12}}dudve^{t\varrho\theta_{12}}$
$\displaystyle\leq\int_{0}^{+\infty}\parallel|s-z|^{-1}\parallel_{L^{2}(\mathbb{R}^{+})}\parallel
q_{11}^{\prime}\parallel_{L^{2}(\mathbb{R}^{+})}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}$
$\displaystyle\lesssim\int_{0}^{+\infty}|v-y|^{-1/2}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}\lesssim
t^{-1/2}e^{t\varrho\theta_{12}}.$ (8.11)

In the second inequality we use that
d$u=\sqrt{1+\left(\frac{v}{u}\right)^{2}}$d$\sqrt{u^{2}+v^{2}}\lesssim$d$\sqrt{u^{2}+v^{2}}$.
This evaluation is also practicable for $I_{22}$ because
$\mathcal{X}_{2}^{\prime}\in L^{2}(\mathbb{R})$. Before we estimating the last
item, we consider for $p>2$,

$\displaystyle\left(\int_{v}^{+\infty}|\sqrt{u^{2}+v^{2}}|^{-\frac{p}{2}}du\right)^{\frac{1}{p}}$
$\displaystyle=\left(\int_{v}^{+\infty}|l|^{-\frac{p}{2}+1}u^{-1}dl\right)^{\frac{1}{p}}\lesssim
v^{-\frac{1}{2}+\frac{1}{p}}.$ (8.12)

By Cauchy-Schwarz inequality,

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{+\infty}\parallel|s-z|^{-1}\parallel_{L^{q}(\mathbb{R}^{+})}\parallel|z-i\varrho|^{-1/2}\parallel_{L^{p}(\mathbb{R}^{+})}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}$
$\displaystyle\lesssim\int_{0}^{+\infty}|v-y|^{1/q-1}v^{-\frac{1}{2}+\frac{1}{p}}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}$
$\displaystyle\lesssim\int_{0}^{+\infty}v^{-\frac{1}{2}}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}\lesssim
t^{-1/2}e^{t\varrho\theta_{12}}.$ (8.13)

So the proof is completed. ∎

To reconstruct the solution of (1.1), we need following proposition.

######  Proposition 11.

As $z\to\infty$, The solution $M^{(3)}(z)$ of $\bar{\partial}$-problem admits
Laurent expansion:

$M^{(3)}(z)=I+\frac{1}{z}M^{(3)}_{1}(x,t)+\mathcal{O}(z^{-2}),$ (8.14)

where $M^{(3)}_{1}$ is a $z$-independent coefficient with

$M^{(3)}_{1}(x,t)=-\frac{1}{\pi}\iint_{C}M^{(3)}(s)W^{(3)}(s)dm(s).$ (8.15)

There exist constants $T_{1}$, such that for all $t>T_{1}$, $M^{(3)}_{1}(x,t)$
satisfies

$|M^{(3)}_{1}(x,t)|\lesssim t^{-1}.$ (8.16)

###### Proof.

The proof proceeds along the same lines as it of above Proposition. Lemma 6
and (8.2) implies that for large $t$, $\parallel
M^{(3)}\parallel_{\infty}\lesssim 1$. And for same reason, we only estimate
the integral on sector $\Omega_{1}$ as $-n_{13}<\xi<-n_{12}$. Referring to
(5.37) in proposition 5, the region $\Omega_{1}$ as a domain of integration is
divided into three parts shown in Figure 4. In this Figure,
$\Omega_{12}=\Omega_{10}\cap\mathbb{D}(i\varrho,2)$, and
$\Omega_{13}=\Omega_{10}\setminus\Omega_{12}$ with $\Omega_{10}$ shown in
Figure 2.

$\Sigma_{1}$Re$z$Im$z$$\Omega_{11}$$0$$\Omega_{13}$$\Omega_{12}$$\Omega_{2}$$i\varrho$
Figure 4: The cyan region is the integral domain $\Omega_{1}$, which is
divided into three parts as $\Omega_{1i}$, $i=1,2,3$. The red arc is
$\Omega_{10}\cap\partial\mathbb{D}(i\varrho,2)$, which divides $\Omega_{10}$
in figure 2 into two parts:
$\Omega_{12}=\Omega_{10}\cap\mathbb{D}(i\varrho,2)$, and
$\Omega_{13}=\Omega_{10}\setminus\Omega_{12}$.

We also divide $M^{(3)}_{1}$ to six parts, but this time we use another
estimation (5.44):

$|\frac{1}{\pi}\iint_{C}M^{(3)}(s)W^{(3)}(s)dm(s)|\lesssim
I_{41}+I_{42}+I_{51}+I_{52}+I_{61}+I_{62},$ (8.17)

with

$\displaystyle
I_{41}=\iint_{\Omega_{11}}|q_{11}^{\prime}(\text{Re}s)|e^{t\text{Im}s\theta_{12}}dm(s),\hskip
14.22636ptI_{42}=\iint_{\Omega_{11}}|q_{11}(\text{Re}s)|e^{t\text{Im}s\theta_{12}}dm(s),$
$\displaystyle
I_{51}=\iint_{\Omega_{10}}|q_{11}^{\prime}(|s-i\varrho|)|e^{t\text{Im}s\theta_{12}}dm(s),\hskip
14.22636ptI_{52}=\iint_{\Omega_{10}}|\mathcal{X}_{2}^{\prime}(|s-i\varrho|)|e^{t\text{Im}s\theta_{12}}dm(s)$
(8.18) $\displaystyle
I_{61}=\iint_{\Omega_{12}}\frac{|\mathcal{X}_{2}(|z-i\varrho|)||q_{11}(|z-i\varrho|)-q_{11}(0)|}{|s-i\varrho|}e^{t\text{Im}s\theta_{12}}dm(s),$
(8.19) $\displaystyle
I_{62}=\iint_{\Omega_{13}}\frac{|\mathcal{X}_{2}(|z-i\varrho|)||q_{11}(|z-i\varrho|)-q_{11}(0)|}{|s-i\varrho|}e^{t\text{Im}s\theta_{12}}dm(s).$
(8.20)

Because $r\in H^{1,1}(\mathbb{R})$, $r^{\prime}$ and $r$ are in
$L^{1}(\mathbb{R})$, which also implies $q_{11}^{\prime}$ and $q_{11}$ in
$L^{1}(\mathbb{R})$. For $I_{41}$ and $I_{42}$, let $s=u+vi$

$\displaystyle I_{41}\leq$
$\displaystyle\int_{0}^{+\infty}\int_{0}^{\varrho}|q_{11}^{\prime}(u)|e^{tv\theta_{12}}dudv$
$\displaystyle\leq\int_{0}^{\varrho}\parallel
q_{11}^{\prime}\parallel_{L^{1}(\mathbb{R}^{+})}e^{tv\theta_{12}}dv\lesssim\int_{0}^{\varrho}e^{tv\theta_{12}}dv\leq
t^{-1}.$

$I_{42}$ has same evaluation. For the rest of integral, we take another change
of variable as $s=u+(v+\varrho)i$ with $|s-i\varrho|=l$. Then Recall
$du\lesssim dl$ in $\Omega_{11}$,

$\displaystyle
I_{51}=\int_{0}^{+\infty}\int_{\frac{v}{\tan\varphi}}^{+\infty}|q_{11}^{\prime}(|s-i\varrho|)|e^{t\text{Im}s\theta_{12}}dudve^{t\varrho\theta_{12}}$
$\displaystyle\lesssim\int_{0}^{+\infty}\parallel
q_{11}^{\prime}\parallel_{L^{2}(\mathbb{R}^{+})}e^{tv\theta_{12}}dve^{t\varrho\theta_{12}}\lesssim
t^{-1}e^{t\varrho\theta_{12}}.$

Similarly, $I_{52}\lesssim t^{-1}e^{t\varrho\theta_{12}}$ from
$\mathcal{X}_{2}^{\prime}\in L^{2}(\mathbb{R})$. Recall the definition of
$\mathcal{X}_{2}$ in (5.27). Then from (5.45), $I_{61}$ has

$\displaystyle I_{61}$
$\displaystyle\leq\int_{0}^{2\tan\varphi}\int_{\frac{v}{\sin\varphi}}^{\frac{2}{\cos\varphi}}|\mathcal{X}_{2}(l)|l^{-1/2}e^{tv\theta_{12}}dldve^{tv\varrho\theta_{12}}$
$\displaystyle\lesssim\parallel\mathcal{X}_{2}(l)l^{-1/2}\parallel_{L^{1}(0,2)}\int_{0}^{2\tan\varphi}v^{-1/2}e^{tv\theta_{12}}dve^{tv\varrho\theta_{12}}\lesssim
t^{-1/2}e^{tv\varrho\theta_{12}}.$ (8.21)

And

$\displaystyle I_{62}$
$\displaystyle=\iint_{\Omega_{13}}\frac{|q_{11}(|z-i\varrho|)-q_{11}(0)|}{|s-i\varrho|^{2}}e^{t\text{Im}s\theta_{12}}dm(s)$
$\displaystyle\leq\int_{0}^{+\infty}\int_{2}^{+\infty}l^{-2}dle^{tv\theta_{12}}dve^{tv\varrho\theta_{12}}\lesssim
t^{-1}e^{tv\varrho\theta_{12}}.$

Combining above inequality we obtain the result. ∎

##  9 Asymptotic stability of N-soliton solutions

Now we begin to construct the long time asymptotics of the tree-wave equation
(1.1). Recalling a serial of transformations (4.28), (5.46), (5.59) and
(5.65), we have

$\displaystyle M(z)=$ $\displaystyle
M^{(3)}(z)E(z)M^{sol}(z)R^{(2)}(z)^{-1}T(z)^{-\sigma_{3}}.$ (9.1)

To reconstruct the solution $q_{ij}(x,t)$ for $i,j=1,2,3$, by using (2.70), we
take $z\to\infty$ out of $\bar{\Omega}$. In this case, $R^{(2)}(z)=I$. Further
using Propositions 4, 10 and 11, we can obtain that as $z\to\infty$ behavior

$\displaystyle
M(z)=\left(I+M^{(3)}_{1}z^{-1}+\cdots\right)\left(I+E_{1}z^{-1}+\cdots\right)(I+M^{sol}_{1}z^{-1}+\cdots)\left(I-iT_{1}^{-\sigma_{3}}z^{-1}+\cdots\right),$

from which we obtain that

$\displaystyle
M(z)=I+(M^{sol}_{1}-iT_{1}^{-\sigma_{3}}+\mathcal{O}(t^{-1}))z^{-1}+\mathcal{O}(z^{-2}).$
(9.2)

Substituting above estimation into (2.70) leads to

$\displaystyle p_{ij}(x,t)$
$\displaystyle=-i(a_{i}-a_{j})\lim_{z\to\infty}[zM]_{ij}=p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}}(I))+\mathcal{O}(t^{-1}),$
(9.3)

where $p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}}(I))$ is shown in Corollary 2.
Therefore, we achieve main result of this paper.

######  Theorem 1.

Let $p_{ij}(x,t)$ be the solution for the initial-value problem (1.1) with
generic data $p_{ij,0}(x)\in H^{1,2}(\mathbb{R})\cap P_{0}$ and scatting data
$\left\\{\vec{r}(z)=(r_{1}(z),...,r_{4}(z)),\left\\{z_{n},c_{n}\right\\}^{4N_{1}+2N_{2}}_{n=1}\right\\}$.
Denote $p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}}(I))$ be the
$\mathcal{N}(I)$-soliton solution corresponding to scattering data
$\widetilde{\mathcal{D}}(I)=\left\\{0,\left\\{z_{n},\widetilde{c}_{n}(I)\right\\}_{n\in\mathcal{Z}(I)}\right\\}$
shown in Corollary 2. There exist a large constant $t_{0}=t_{0}(\xi)$, for all
$t>t_{0}$,

$\displaystyle
p_{ij}(x,t)=p_{ij}^{sol}(x,t;\widetilde{\mathcal{D}}(I))+\mathcal{O}(t^{-1}).$
(9.4)

Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant No. 11671095, 51879045).

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