# The Fokas method to the Sasa-Satsuma equation on the half-line

Jian Xu  School of Mathematical Sciences
Fudan University
Shanghai 200433
People’s Republic of China<EMAIL_ADDRESS>and  Engui Fan  School of
Mathematical Sciences, Institute of Mathematics and Key Laboratory of
Mathematics for Nonlinear Science
Fudan University
Shanghai 200433
People’s Republic of China  correspondence<EMAIL_ADDRESS>
###### Abstract.

We present a Riemann-Hilbert problem formalism for the initial-boundary value
problem for the Sasa-Satsuma(SS) equation: on the half-line. And we also
analysis the global relation in this paper.

###### Key words and phrases:

Riemann-Hilbert problem, Sasa-Satsuma equation, Initial-boundary value problem

##  1\. Introduction

Several of the most important PDEs in mathematics and physics are integrable.
Integrable PDEs can be analyzed by means of the Inverse Scattering Transform
(IST) formalism. Until the 1990s the IST methodology was pursued almost
entirely for pure initial value problems. However, in many laboratory and
field situations, the wave motion is initiated by what corresponds to the
imposition of boundary conditions rather than initial conditions. This
naturally leads to the formulation of an initial-boundary value (IBV) problem
instead of a pure initial value problem.

In 1997, Fokas announced a new unified approach for the analysis of IBV
problems for linear and nonlinear integrable PDEs [1, 2](see also [3]). The
Fokas method provides a generalization of the IST formalism from initial value
to IBV problems, and over the last fifteen years, this method has been used to
analyze boundary value problems for several of the most important integrable
equations with $2\times 2$ Lax pairs, such as the Korteweg de Vries, the
nonlinear Schrö dinger, the sine-Gordon, and the stationary axisymmetric
Einstein equations, see e.g. [4, 9]. Just like the IST on the line, the
unified method yields an expression for the solution of an IBV problem in
terms of the solution of a Riemann-Hilbert problem. In particular, the
asymptotic behavior of the solution can be analyzed in an effective way by
using this Riemann-Hilbert problem and by employing the nonlinear version of
the steepest descent method introduced by Deift and Zhou [15].

It is well known that the nonlinear Schrödinger(NLS) equation

$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q=0$ (1.1)

describes slowly varying wave envelopes in dispersive media and arises in
various physical systems such as water waves, plasma physics, solid-state
physics and nonlinear optics. One of the most successful among them is the
description of optical solitons in fibers. But, by the advancement of
experomenal accuracy, several phenomena which can not be explained by equation
(1.1) have been observed. In order to understand such phenomena, Kodama and
Hasegawa proposed a higer-order nonlinear Schrödinger equation

$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q+i\varepsilon\\{\beta_{1}q_{xxx}+\beta_{2}|q|^{2}q_{X}+\beta_{3}q(|q|^{2})_{X}\\}=0.$
(1.2)

In general, equation (1.2) may not be completely integrable. However, if some
restrictions are imposed on the real parameters $\beta_{1},\beta_{2}$ and
$\beta_{3}$, then we can apply the IST to solve its initial value problems.
Until now, the following four cases besides the NLS equation itself are konwn
to be solvable:

  * •

the derivative NLS equation-type i@($\beta_{1}:\beta_{2}:\beta_{3}$=0:1:1),

  * •

the derivative NLS equation-type ii@($\beta_{1}:\beta_{2}:\beta_{3}$=0:1:0),

  * •

the Hirota equation($\beta_{1}:\beta_{2}:\beta_{3}$=1:6:0),

  * •

the Sasa-Satsuma equation($\beta_{1}:\beta_{2}:\beta_{3}$=1:6:3).

$iq_{T}+\frac{1}{2}q_{XX}+|q|^{2}q+i\varepsilon(q_{XXX}+6|q|^{2}q_{X}+3q(|q|^{2})_{X})=0$
(1.3)

Recently, Lenells develop a methodology for analyzing IBV problems for
integrable evolution equations with Lax pairs involving $3\times 3$ matrices
[12]. He also used this method to analyze the Degasperis-Procesi equation in
[13]. In this paper we analyze the initial-boundary value problem of the Sasa-
Satsuma equation on the half-line by using this method. The IST formalism for
the initial value problem of the Sasa-Satsuma equation has been obtained in
[10].

According to [10] we introduce variable transformations,

$u(x,t)=q(X,T)\exp\\{\frac{-i}{6\varepsilon}(X-\frac{T}{18\varepsilon})\\},$
(1.4a) $t=T,$ (1.4b) $x=X-\frac{T}{12\varepsilon}.$ (1.4c)

Then equation (1.2) is reduce to a complex modified KdV-type equation

$u_{t}+\varepsilon\\{u_{xxx}+6|u|^{2}u_{x}+3u(|u|^{2})_{x}\\}=0.$ (1.5)

Organization of the paper.In section 2 we perform the spectral analysis of the
associated Lax pair. And we formulate the main Riemann-Hilbert problem in
section 3. We also analysis the global relation in section 4.

##  2\. Spectral analysis

The Lax pair of equation (1.5) is [10],

$\Psi_{x}=U\Psi,\quad\Psi=\left(\begin{array}[]{c}\Psi_{1}\\\ \Psi_{2}\\\
\Psi_{3}\end{array}\right).$ (2.1a) $\Psi_{t}=V\Psi.$ (2.1b)

where

$U=-ik\Lambda+V_{1}.$ (2.2)

and

$V=-4i\varepsilon k^{3}\Lambda+V_{2}$ (2.3)

here

$\Lambda=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&0&-1\end{array}\right),V_{1}=\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\
-\bar{u}&-u&0\end{array}\right),V_{2}=k^{2}V_{2}^{(2)}+kV_{2}^{(1)}+V_{2}^{(0)}.$
(2.4)

where

$\begin{array}[]{l}V_{2}^{(2)}=4\varepsilon\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\ -\bar{u}&-u&0\end{array}\right),\\\
V_{2}^{(1)}=2i\varepsilon\left(\begin{array}[]{ccc}|u|^{2}&u^{2}&u_{x}\\\
\bar{u}^{2}&|u|^{2}&\bar{u}_{x}\\\
\bar{u}_{x}&u_{x}&-2|u|^{2}\end{array}\right),\\\
V_{2}^{(0)}=-4|u|^{2}\varepsilon\left(\begin{array}[]{ccc}0&0&u\\\
0&0&\bar{u}\\\
-\bar{u}&-u&0\end{array}\right)-\varepsilon\left(\begin{array}[]{ccc}0&0&u_{xx}\\\
0&0&\bar{u}_{xx}\\\
-\bar{u}_{xx}&-u_{xx}&0\end{array}\right)+\varepsilon(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\end{array}\right)\end{array}$ (2.5)

In the following, we let $\varepsilon=1$ for the convenient of the analysis.

###  2.1. The closed one-form

Suppose that $u(x,t)$ is sufficiently smooth function of $(x,t)$ in the half-
line domain $\Omega=\\{0<x<\infty,0<t<T\\}$ which decay as
$x\rightarrow\infty$. Introducing a new eigenfunction $\mu(x,t,k)$ by

$\Psi=\mu e^{-i\Lambda kx-4i\Lambda k^{3}t}$ (2.6)

then we find the Lax pair equations

$\left\\{\begin{array}[]{l}\mu_{x}+[ik\Lambda,\mu]=V_{1}\mu,\\\
\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.\end{array}\right.$ (2.7)

the equations in (A.2) can be written in differential form as

$d(e^{(ikx+4ik^{3}t)\hat{\Lambda}}\mu)=W,$ (2.8)

where $W(x,t,k)$ is the closed one-form defined by

$W=e^{(ikx+4ik^{3}t)\hat{\Lambda}}(V_{1}dx+V_{2}dt)\mu.$ (2.9)

###  2.2. The $\mu_{j}$’s

We define three eigenfunctions $\\{\mu_{j}\\}_{1}^{3}$ of (A.2) by the
Volterra integral equations

$\mu_{j}(x,t,k)=\mathbb{I}+\int_{\gamma_{j}}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}W_{j}(x^{\prime},t^{\prime},k).\qquad
j=1,2,3.$ (2.10)

where $W_{j}$ is given by (2.9) with $\mu$ replaced with $\mu_{j}$, and the
contours $\\{\gamma_{j}\\}_{1}^{3}$ are showed in Figure 1.

Figure 1. The three contours $\gamma_{1},\gamma_{2}$ and $\gamma_{3}$ in the
$(x,t)-$domain.

The first, second and third column of the matrix equation (2.10) involves the
exponentials

$\begin{array}[]{ll}\mbox{$[\mu_{j}]_{1}$:}&e^{2ik(x-x^{\prime})+8ik^{3}(t-t^{\prime})},\\\
\mbox{$[\mu_{j}]_{2}$:}&e^{2ik(x-x^{\prime})+8ik^{3}(t-t^{\prime})},\\\
\mbox{$[\mu_{j}]_{3}$:}&e^{-2ik(x-x^{\prime})-8ik^{3}(t-t^{\prime})},e^{-2ik(x-x^{\prime})-8ik^{3}(t-t^{\prime})}.\end{array}$
(2.11)

And we have the following inequalities on the contours:

$\begin{array}[]{ll}\gamma_{1}:&x-x^{\prime}\geq 0,t-t^{\prime}\leq 0,\\\
\gamma_{2}:&x-x^{\prime}\geq 0,t-t^{\prime}\geq 0,\\\
\gamma_{3}:&x-x^{\prime}\leq 0.\end{array}$ (2.12)

So, these inequalities imply that the functions $\\{\mu_{j}\\}_{1}^{3}$ are
bounded and analytic for $k\in{\mathbb{C}}$ such that $k$ belongs to

$\begin{array}[]{ll}\mu_{1}:&(D_{2},D_{2},D_{3}),\\\
\mu_{2}:&(D_{1},D_{1},D_{4}),\\\ \mu_{3}:&(D_{3}\cup D_{4},D_{3}\cup
D_{4},D_{1}\cup D_{2}).\end{array}$ (2.13)

where $\\{D_{n}\\}_{1}^{4}$ denote four open, pairwisely disjoint subsets of
the Riemann $k-$sphere showed in Figure 2.

Figure 2. The sets $D_{n}$, $n=1,\ldots,4$, which decompose the complex
$k-$plane.

And the sets $\\{D_{n}\\}_{1}^{4}$ has the following properties:

$\begin{array}[]{l}D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}>\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}>\mathrm{Re}{z_{3}}\\},\\\
D_{2}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}>\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}<\mathrm{Re}{z_{3}}\\},\\\
D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}<\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}>\mathrm{Re}{z_{3}}\\},\\\
D_{1}=\\{k\in{\mathbb{C}}|\mathrm{Re}{l_{1}}=\mathrm{Re}{l_{2}}<\mathrm{Re}{l_{3}},\mathrm{Re}{z_{1}}=\mathrm{Re}{z_{2}}<\mathrm{Re}{z_{3}}\\},\\\
\end{array}$

where $l_{i}(k)$ and $z_{i}(k)$ are the diagonal entries of matrices
$-ik\Lambda$ and $-4ik^{3}\Lambda$, respectively.

In fact, for $x=0$, $\mu_{1}(0,t,k)$ has enlarged domain of boundedness:
$(D_{2}\cup D_{4},D_{2}\cup D_{4},D_{1}\cup D_{3})$, and $\mu_{2}(0,t,k)$ has
enlarged domain of boundedness: $(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup
D_{4})$.

###  2.3. The $M_{n}$’s

For each $n=1,\ldots,4$, define a solution $M_{n}(x,t,k)$ of (A.2) by the
following system of integral equations:

$(M_{n})_{ij}(x,t,k)=\delta_{ij}+\int_{\gamma_{ij}^{n}}(e^{(-ikx-4ik^{3}t)\hat{\Lambda}}W_{n}(x^{\prime},t^{\prime},k))_{ij},\quad
k\in D_{n},\quad i,j=1,2,3.$ (2.14)

where $W_{n}$ is given by (2.9) with $\mu$ replaced with $M_{n}$, and the
contours $\gamma_{ij}^{n}$, $n=1,\ldots,4$, $i,j=1,2,3$ are defined by

$\gamma_{ij}^{n}=\left\\{\begin{array}[]{lclcl}\gamma_{1}&if&\mathrm{Re}l_{i}(k)<\mathrm{Re}l_{j}(k)&and&\mathrm{Re}z_{i}(k)\geq\mathrm{Re}z_{j}(k),\\\
\gamma_{2}&if&\mathrm{Re}l_{i}(k)<\mathrm{Re}l_{j}(k)&and&\mathrm{Re}z_{i}(k)<\mathrm{Re}z_{j}(k),\\\
\gamma_{3}&if&\mathrm{Re}l_{i}(k)\geq\mathrm{Re}l_{j}(k)&&.\\\
\end{array}\right.\quad\mbox{for }\quad k\in D_{n}.$ (2.15)

The following proposition ascertains that the $M_{n}$’s defined in this way
have the properties required for the formulation of a Riemann-Hilbert problem.

######  Proposition 2.1.

For each $n=1,\ldots,4$, the function $M_{n}(x,t,k)$ is well-defined by
equation (2.14) for $k\in\bar{D}_{n}$ and $(x,t)\in\Omega$. For any fixed
point $(x,t)$, $M_{n}$ is bounded and analytic as a function of $k\in D_{n}$
away from a possible discrete set of singularities $\\{k_{j}\\}$ at which the
Fredholm determinant vanishes. Moreover, $M_{n}$ admits a bounded and contious
extension to $\bar{D}_{n}$ and

$M_{n}(x,t,k)=\mathbb{I}+O(\frac{1}{k}),\qquad k\rightarrow\infty,\quad k\in
D_{n}.$ (2.16)

###### Proof.

The bounedness and analyticity properties are established in appendix B in
[12]. And substituting the expansion

$M=M_{0}+\frac{M^{(1)}}{k}+\frac{M^{(2)}}{k^{2}}+\cdots,\qquad
k\rightarrow\infty.$

into the Lax pair (A.2) and comparing the terms of the same order of $k$ yield
the equation (2.16). ∎

###  2.4. The jump matrices

We define spectral functions $S_{n}(k)$, $n=1,\ldots,4$, and

$S_{n}(k)=M_{n}(0,0,k),\qquad k\in D_{n},\quad n=1,\ldots,4.$ (2.17)

Let $M$ denote the sectionally analytic function on the Riemann $k-$sphere
which equals $M_{n}$ for $k\in D_{n}$. Then $M$ satisfies the jump conditions

$M_{n}=M_{m}J_{m,n},\qquad k\in\bar{D}_{n}\cap\bar{D}_{m},\qquad
n,m=1,\ldots,4,\quad n\neq m,$ (2.18)

where the jump matrices $J_{m,n}(x,t,k)$ are defined by

$J_{m,n}=e^{(-ikx-4ik^{3}t)\hat{\Lambda}}(S_{m}^{-1}S_{n}).$ (2.19)

According to the definition of the $\gamma^{n}$, we find that

$\begin{array}[]{ll}\gamma^{1}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{2}&\gamma_{2}&\gamma_{3}\end{array}\right)&\gamma^{2}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\\\
\gamma_{1}&\gamma_{1}&\gamma_{3}\end{array}\right)\\\
\gamma^{3}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{1}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\end{array}\right)&\gamma^{4}=\left(\begin{array}[]{lll}\gamma_{3}&\gamma_{3}&\gamma_{2}\\\
\gamma_{3}&\gamma_{3}&\gamma_{2}\\\
\gamma_{3}&\gamma_{3}&\gamma_{3}\end{array}\right).\end{array}$ (2.20)

###  2.5. The adjugated eigenfunctions

We will also need the analyticity and boundedness properties of the minors of
the matrices $\\{\mu_{j}(x,t,k)\\}_{1}^{3}$. We recall that the adjugate
matrix $X^{A}$ of a $3\times 3$ matrix $X$ is defined by

$X^{A}=\left(\begin{array}[]{ccc}m_{11}(X)&-m_{12}(X)&m_{13}(X)\\\
-m_{21}(X)&m_{22}(X)&-m_{23}(X)\\\
m_{31}(X)&-m_{32}(X)&m_{33}(X)\end{array}\right),$

where $m_{ij}(X)$ denote the $(ij)$th minor of $X$.

It follows from (A.2) that the adjugated eigenfunction $\mu^{A}$ satisfies the
Lax pair

$\left\\{\begin{array}[]{l}\mu_{x}^{A}-[ik\Lambda,\mu^{A}]=-V_{1}^{T}\mu^{A},\\\
\mu_{t}^{A}-[4ik^{3}\Lambda,\mu^{A}]=-V_{2}^{T}\mu^{A}.\end{array}\right.$
(2.21)

where $V^{T}$ denote the transform of a matrix $V$. Thus, the eigenfunctions
$\\{\mu_{j}^{A}\\}_{1}^{3}$ are solutions of the integral equations

$\mu_{j}^{A}(x,t,k)=\mathbb{I}-\int_{\gamma_{j}}e^{ik(x-x^{\prime})+4ik^{3}(t-t^{\prime})\hat{\Lambda}}(V_{1}^{T}dx+V_{2}^{T})\mu^{A},\quad
j=1,2,3.$ (2.22)

Then we can get the following analyticity and boundedness properties:

$\begin{array}[]{ll}\mu_{1}^{A}:&(D_{3},D_{3},D_{2}),\\\
\mu_{2}^{A}:&(D_{4},D_{4},D_{1}),\\\ \mu_{3}^{A}:&(D_{1}\cup D_{2},D_{1}\cup
D_{2},D_{3}\cup D_{4}).\end{array}$ (2.23)

In fact, for $x=0$, $\mu_{1}^{A}(0,t,k)$ has enlarged domain of boundedness:
$(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup D_{4})$, and $\mu_{2}^{A}(0,t,k)$
has enlarged domain of boundedness: $(D_{2}\cup D_{4},D_{2}\cup
D_{4},D_{1}\cup D_{3})$.

###  2.6. The $J_{m,n}$’s computation

Let us define the $3\times 3-$matrix value spectral functions $s(k)$ and
$S(k)$ by

$\mu_{3}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}s(k),$ (2.24a)
$\mu_{1}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S(k),$ (2.24b)

Thus,

$s(k)=\mu_{3}(0,0,k),\qquad S(k)=\mu_{1}(0,0,k).$ (2.25)

And we deduce from the properties of $\mu_{j}$ and $\mu_{j}^{A}$ that $s(k)$
and $S(k)$ have the following boundedness properties:

$\begin{array}[]{ll}s(k):&(D_{3}\cup D_{4},D_{3}\cup D_{4},D_{1}\cup
D_{2}),\\\ S(k):&(D_{2}\cup D_{4},D_{2}\cup D_{4},D_{1}\cup D_{3}),\\\
s^{A}(k):&(D_{1}\cup D_{2},D_{1}\cup D_{2},D_{3}\cup D_{4}),\\\
S^{A}(k):&(D_{1}\cup D_{3},D_{1}\cup D_{3},D_{2}\cup D_{4}).\end{array}$

Moreover,

$M_{n}(x,t,k)=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{n}(k),\quad
k\in D_{n}.$ (2.26)

######  Proposition 2.2.

The $S_{n}$ can be expressed in terms of the entries of $s(k)$ and $S(k)$ as
follows:

$\begin{array}[]{l}S_{1}=\left(\begin{array}[]{ccc}\frac{m_{22}(s)}{s_{33}}&\frac{m_{21}(s)}{s_{33}}&s_{13}\\\
\frac{m_{12}(s)}{s_{33}}&\frac{m_{11}(s)}{s_{33}}&s_{23}\\\
0&0&s_{33}\end{array}\right),\\\
S_{2}=\left(\begin{array}[]{ccc}\frac{m_{22}(s)m_{33}(S)-m_{32}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{21}(s)m_{33}(S)-m_{31}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}&s_{13}\\\
\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{11}(s)m_{33}(S)-m_{31}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&s_{23}\\\
\frac{m_{12}(s)m_{23}(S)-m_{22}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&\frac{m_{11}(s)m_{23}(S)-m_{21}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}&s_{33}\end{array}\right),\\\
\end{array}$ (2.27a)
$\begin{array}[]{ll}S_{3}=\left(\begin{array}[]{ccc}s_{11}&s_{12}&\frac{S_{13}}{(S^{T}s^{A})_{33}}\\\
s_{21}&s_{22}&\frac{S_{23}}{(S^{T}s^{A})_{33}}\\\
s_{31}&s_{32}&\frac{S_{33}}{(S^{T}s^{A})_{33}}\end{array}\right),&S_{4}=\left(\begin{array}[]{ccc}s_{11}&s_{12}&0\\\
s_{21}&s_{22}&0\\\
s_{31}&s_{32}&\frac{1}{m_{33}(s)}\end{array}\right).\end{array}$ (2.27b)

###### Proof.

Let $\gamma_{3}^{X_{0}}$ denote the contour $(X_{0},0)\rightarrow(x,t)$ in the
$(x,t)-$plane, here $X_{0}>0$ is a constant. We introduce
$\mu_{3}(x,t,k;X_{0})$ as the solution of (2.10) with $j=3$ and with the
contour $\gamma_{3}$ replaced by $\gamma_{3}^{X_{0}}$. Similarly, we define
$M_{n}(x,t,k;X_{0})$ as the solution of (2.14) with $\gamma_{3}$ replaced by
$\gamma_{3}^{X_{0}}$. We will first derive expression for
$S_{n}(k;X_{0})=M_{n}(0,0,k;X_{0})$ in terms of $S(k)$ and
$s(k;X_{0})=\mu_{3}(0,0,k;X_{0})$. Then (2.27) will follow by taking the limit
$X_{0}\rightarrow\infty$.

First, We have the following relations:

$\left\\{\begin{array}[]{l}M_{n}(x,t,k;X_{0})=\mu_{1}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}R_{n}(k;X_{0}),\\\
M_{n}(x,t,k;X_{0})=\mu_{2}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{n}(k;X_{0}),\\\
M_{n}(x,t,k;X_{0})=\mu_{3}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}T_{n}(k;X_{0}).\end{array}\right.$
(2.28)

Then we get $R_{n}(k;X_{0})$ and $T_{n}(k;X_{0})$ are fedined as follows:

$R_{n}(k;X_{0})=e^{4ik^{3}T\hat{\Lambda}}M_{n}(0,T,k;X_{0}),$ (2.29a)
$T_{n}(k;X_{0})=e^{ikx\hat{\Lambda}}M_{n}(X_{0},0,k;X_{0}).$ (2.29b)

The relations (2.28) imply that

$s(k;X_{0})=S_{n}(k;X_{0})T^{-1}_{n}(k;X_{0}),\qquad
S(k)=S_{n}(k;X_{0})R^{-1}_{n}(k;X_{0}).$ (2.30)

These equations constitute a matrix factorization problem which, given
$\\{s,S\\}$ can be solved for the $\\{R_{n},S_{n},T_{n}\\}$. Indeed, the
integral equations (2.14) together with the definitions of
$\\{R_{n},S_{n},T_{n}\\}$ imply that

$\left\\{\begin{array}[]{lll}(R_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{1},\\\
(S_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{2},\\\
(T_{n}(k;X_{0}))_{ij}=0&if&\gamma_{ij}^{n}=\gamma_{3}.\end{array}\right.$
(2.31)

It follows that (2.30) are 18 scalar equations for 18 unknowns. By computing
the explicit solution of this algebraic system, we find that
$\\{S_{n}(k;X_{0})\\}_{1}^{4}$ are given by the equation obtained from (2.27)
by replacing $\\{S_{n}(k),s(k)\\}$ with $\\{S_{n}(k;X_{0}),s(k;X_{0})\\}$.
taking $X_{0}\rightarrow\infty$ in this equation, we arrive at (2.27). ∎

###  2.7. The global relation

The spectral functions $S(k)$ and $s(k)$ are not independent but satisfy an
important relation. Indeed, it follows from (2.24) that

$\mu_{1}(x,t,k)e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S^{-1}(k)s(k)=\mu_{3}(x,t,k),\quad
k\in(D_{3}\cup D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (2.32)

Since $\mu_{1}(0,T,k)=\mathbb{I}$, evaluation at $(0,T)$ yields the following
global relation:

$S^{-1}(k)s(k)=e^{4ik^{3}T\hat{\Lambda}}c(T,k),\quad k\in(D_{3}\cup
D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (2.33)

where $c(T,k)=\mu_{3}(0,T,k)$.

###  2.8. The residue conditions

Since $\mu_{2}$ is an entire function, it follows from (2.26) that M can only
have sigularities at the points where the $S_{n}^{\prime}s$ have
singularities. We infer from the explicit formulas (2.27) that the possible
singularities of $M$ are as follows:

  * •

$[M]_{1}$ could have poles in $D_{1}\cup D_{2}$ at the zeros of $s_{33}(k)$;

  * •

$[M]_{1}$ could have poles in $D_{2}$ at the zeros of $(s^{T}S^{A})_{33}(k)$;

  * •

$[M]_{2}$ could have poles in $D_{1}\cup D_{2}$ at the zeros of $s_{33}(k)$;

  * •

$[M]_{2}$ could have poles in $D_{2}$ at the zeros of $(s^{T}S^{A})_{33}(k)$;

  * •

$[M]_{3}$ could have poles in $D_{3}$ at the zeros of $(S^{T}s^{A})_{33}(k)$;

  * •

$[M]_{3}$ could have poles in $D_{3}\cup D_{4}$ at the zeros of
$m_{33}(s)(k)$;

We denote the above possible zeros by $\\{k_{j}\\}_{1}^{N}$ and assume they
satisfy the following assumption.

######  Assumption 2.3.

We assume that

  * •

$s_{33}(k)$ has $n_{0}$ possible simple zeros in $D_{1}$ denoted by
$\\{k_{j}\\}_{1}^{n_{0}}$;

  * •

$s_{33}(k)$ has $n_{1}-n_{0}$ possible simple zeros in $D_{2}$ denoted by
$\\{k_{j}\\}_{n_{0}+1}^{n_{1}}$;

  * •

$(s^{T}S^{A})_{33}(k)$ has $n_{2}-n_{1}$ possible simple zeros in $D_{2}$
denoted by $\\{k_{j}\\}_{n_{1}+1}^{n_{2}}$;

  * •

$(S^{T}s^{A})_{33}(k)$ has $n_{3}-n_{2}$ possible simple zeros in $D_{3}$
denoted by $\\{k_{j}\\}_{n_{2}+1}^{n_{3}}$;

  * •

$m_{33}(s)(k)$ has $n_{4}-n_{3}$ possible simple zeros in $D_{3}$ denoted by
$\\{k_{j}\\}_{n_{3}+1}^{n_{4}}$;

  * •

$m_{33}(s)(k)$ has $n_{5}-n_{4}$ possible simple zeros in $D_{3}$ denoted by
$\\{k_{j}\\}_{n_{4}+1}^{n_{5}}$;

  * •

$m_{33}(s)(k)$ has $N-n_{5}$ possible simple zeros in $D_{4}$ denoted by
$\\{k_{j}\\}_{n_{5}+1}^{N}$;

and that none of these zeros coincide. Moreover, we assume that none of these
functions have zeros on the boundaries of the $D_{n}$’s.

We determine the residue conditions at these zeros in the following:

######  Proposition 2.4.

Let $\\{M_{n}\\}_{1}^{4}$ be the eigenfunctions defined by (2.14) and assume
that the set $\\{k_{j}\\}_{1}^{N}$ of singularitues are as the above
assumption. Then the following residue conditions hold:

${Res}_{k=k_{j}}[M]_{1}=\frac{m_{12}(s)(k_{j})}{\dot{s}_{33}(k_{j})s_{23}(k_{j})}e^{\theta_{31}(k_{j})}[M(k_{j})]_{3},\quad
1\leq j\leq n_{0},k_{j}\in D_{1}$ (2.34a)
${Res}_{k=k_{j}}[M]_{2}=\frac{m_{12}(s)(k_{j})}{\dot{s}_{33}(k_{j})s_{13}(k_{j})}e^{\theta_{32}(k_{j})}[M(k_{j})]_{3},\quad
1\leq j\leq n_{0},k_{j}\in D_{1}$ (2.34b)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{1}=\frac{m_{12}(s)(k_{j})m_{33}(S)(k_{j})-m_{32}(s)(k_{j})m_{13}(S)(k_{j})}{\dot{(s^{T}S^{A})_{33}(k_{j})}s_{23}(k_{j})}e^{\theta_{31}(k_{j})}[M(k_{j})]_{3}\\\
\quad n_{1}+1\leq j\leq n_{2},k_{j}\in D_{2},\end{array}$ (2.34c)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{2}=\frac{m_{21}(s)(k_{j})m_{33}(S)(k_{j})-m_{31}(s)(k_{j})m_{23}(S)(k_{j})}{\dot{(s^{T}S^{A})_{33}(k_{j})}s_{13}(k_{j})}e^{\theta_{32}(k_{j})}[M(k_{j})]_{3}\\\
\quad n_{1}+1\leq j\leq n_{2},k_{j}\in D_{2},\end{array}$ (2.34d)
$\begin{array}[]{rl}Res_{k=k_{j}}[M]_{3}=&\frac{S_{13}(k_{j})s_{32}(k_{j})-S_{33}(k_{j})s_{12}(k_{j})}{\dot{(S^{T}s^{A})_{33}(k_{j})}m_{23}(s)(k_{j})}e^{\theta_{13}(k_{j})}[M(k_{j})]_{1}\\\
&+\frac{S_{33}(k_{j})s_{11}(k_{j})-S_{13}(k_{j})s_{31}(k_{j})}{\dot{(S^{T}s^{A})_{33}(k_{j})}m_{23}(s)(k_{j})}e^{\theta_{23}(k_{j})}[M(k_{j})]_{2},n_{2}+1\leq
j\leq n_{3},k_{j}\in D_{3},\end{array}$ (2.34e)
$\begin{array}[]{r}Res_{k=k_{j}}[M]_{3}=\frac{s_{12}(k_{j})}{\dot{m}_{33}(s)(k_{j})m_{23}(s)(k_{j})}e^{\theta_{13}(k_{j})}[M(k_{j})]_{1}-\frac{s_{11}(k_{j})}{\dot{m}_{33}(s)(k_{j})m_{23}(s)(k_{j})}e^{\theta_{23}(k_{j})}[M(k_{j})]_{2}\\\
\quad n_{4}+1\leq j\leq N,k_{j}\in D_{4}.\end{array}$ (2.34f)

where $\dot{f}=\frac{df}{dk}$, and $\theta_{ij}$ is defined by

$\theta_{ij}(x,t,k)=(l_{i}-l_{j})x+(z_{i}-z_{j})t,\quad i,j=1,2,3.$ (2.35)

that implies that

$\theta_{ij}=0,i,j=1,2;\quad\theta_{13}=\theta_{23}=-\theta_{32}=-\theta_{31}=-2ikx-8ik^{3}t.$

###### Proof.

We will prove (2.34a), (2.34c), (2.34e), (2.34f), the other conditions follow
by similar arguments. Equation (2.26) implies the relation

$M_{1}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{1},$ (2.36a)
$M_{2}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{2}.$ (2.36b)
$M_{3}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{3},$ (2.36c)
$M_{4}=\mu_{2}e^{(-ikx-4ik^{3}t)\hat{\Lambda}}S_{4},$ (2.36d)

In view of the expressions for $S_{1}$ and $S_{2}$ given in (2.27), the three
columns of (2.36a) read:

$[M_{1}]_{1}=[\mu_{2}]_{1}\frac{m_{22}(s)}{s_{33}}+[\mu_{2}]_{2}e^{\theta_{21}}\frac{m_{12}(s)}{s_{33}},$
(2.37a)
$[M_{1}]_{2}=[\mu_{2}]_{1}e^{\theta_{12}}\frac{m_{21}(s)}{s_{33}}+[\mu_{2}]_{2}\frac{m_{11}(s)}{s_{33}},$
(2.37b)
$[M_{1}]_{3}=[\mu_{2}]_{1}e^{\theta_{13}}s_{13}+[\mu_{2}]_{2}e^{\theta_{23}}s_{23}+[\mu_{2}]_{3}s_{33}.$
(2.37c)

while the three columns of (2.36b) read:

$\begin{array}[]{rl}[M_{2}]_{1}&=[\mu_{2}]_{1}\frac{m_{22}(s)m_{33}(S)-m_{32}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}\\\
&+[\mu_{2}]_{2}\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{21}}\\\
&+[\mu_{2}]_{3}\frac{m_{12}(s)m_{23}(S)-m_{22}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{31}}\end{array}$
(2.38a)
$\begin{array}[]{rl}[M_{2}]_{2}&=[\mu_{2}]_{1}\frac{m_{21}(s)m_{33}(S)-m_{31}(s)m_{23}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{12}}\\\
&+[\mu_{2}]_{2}\frac{m_{11}(s)m_{33}(S)-m_{31}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}\\\
&+[\mu_{2}]_{3}\frac{m_{11}(s)m_{23}(S)-m_{21}(s)m_{13}(S)}{(s^{T}S^{A})_{33}}e^{\theta_{32}}\end{array}$
(2.38b)
$[M_{2}]_{3}=[\mu_{2}]_{1}s_{13}e^{\theta_{13}}+[\mu_{2}]_{2}s_{23}e^{\theta_{23}}+[\mu_{2}]_{3}s_{33}.$
(2.38c)

and the three columns of (2.36c) read:

$[M_{3}]_{1}=[\mu_{2}]_{1}s_{11}+[\mu_{2}]_{2}s_{21}e^{\theta_{21}}+[\mu_{2}]_{3}s_{31}e^{\theta_{31}},$
(2.39a)
$[M_{3}]_{2}=[\mu_{2}]_{1}s_{12}e^{\theta_{12}}+[\mu_{2}]_{2}s_{22}+[\mu_{2}]_{3}s_{32}e^{\theta_{32}},$
(2.39b)
$[M_{3}]_{3}=[\mu_{2}]_{1}\frac{S_{13}}{(S^{T}s^{A})_{33}}e^{\theta_{13}}+[\mu_{2}]_{2}\frac{S_{23}}{(S^{T}s^{A})_{33}}e^{\theta_{23}}+[\mu_{2}]_{3}\frac{S_{33}}{(S^{T}s^{A})_{33}}.$
(2.39c)

the three columns of (2.36d) read:

$[M_{4}]_{1}=[\mu_{2}]_{1}s_{11}+[\mu_{2}]_{2}s_{21}e^{\theta_{21}}+[\mu_{2}]_{3}s_{31}e^{\theta_{31}},$
(2.40a)
$[M_{4}]_{2}=[\mu_{2}]_{1}s_{12}e^{\theta_{12}}+[\mu_{2}]_{2}s_{22}+[\mu_{2}]_{3}s_{32}e^{\theta_{32}},$
(2.40b) $[M_{4}]_{3}=[\mu_{2}]_{3}\frac{1}{m_{33}(s)}.$ (2.40c)

We first suppose that $k_{j}\in D_{1}$ is a simple zero of $s_{33}(k)$.
Solving (2.37c) for $[\mu_{2}]_{2}$ and substituting the result in to (2.37a),
we find

$[M_{1}]_{1}=\frac{m_{12}(s)}{s_{33}s_{23}}e^{\theta_{31}}[M_{1}]_{3}+\frac{m_{32}(s)}{s_{23}}[\mu_{2}]_{2}-\frac{m_{12}(s)}{s_{23}}e^{\theta_{31}}[\mu_{2}]_{3}.$

Taking the residue of this equation at $k_{j}$, we find the condition (2.34a)
in the case when $k_{j}\in D_{1}$. Similarly, Solving (2.38c) for
$[\mu_{2}]_{2}$ and substituting the result in to (2.38a), we find

$[M_{2}]_{1}=\frac{m_{12}(s)m_{33}(S)-m_{32}(s)m_{13}(S)}{(s^{T}S^{A})_{33}s_{23}}e^{\theta_{31}}[M_{1}]_{3}-\frac{m_{32}(s)}{s_{23}}[\mu_{2}]_{1}-\frac{m_{12}(s)}{s_{23}}e^{\theta_{31}}[\mu_{2}]_{3}.$

Taking the residue of this equation at $k_{j}$, we find the condition (2.34c)
in the case when $k_{j}\in D_{2}$.

In order to prove (2.34e), we solve (2.39a) and (2.39b) for $[\mu_{2}]_{1}$
and $[\mu_{2}]_{3}$, then substituting the result into (2.39c), we find

$[M_{3}]_{3}=\frac{S_{13}s_{32}-S_{33}s_{12}}{(S^{T}s^{A})_{33}m_{23}(s)}e^{\theta_{13}}[M_{3}]_{1}+\frac{S_{33}s_{11}-S_{13}s_{31}}{(S^{T}s^{A})_{33}(k_{j})m_{23}(s)}e^{\theta_{23}}[M_{3}]_{2}+\frac{1}{m_{23}(s)}[\mu_{2}]_{3}.$

Taking the residue of this equation at $k_{j}$, we find the condition (2.34e)
in the case when $k_{j}\in D_{3}$. Similarly, solving (2.40a) and (2.40b) for
$[\mu_{2}]_{1}$ and $[\mu_{2}]_{3}$, then substituting the result into
(2.40c), we find

$[M_{4}]_{3}=\frac{s_{12}}{m_{33}(s)m_{23}(s)}e^{\theta_{13}}[M_{4}]_{1}-\frac{s_{11}}{m_{33}(s)m_{23}(s)}e^{\theta_{13}}[M_{4}]_{2}-\frac{1}{m_{23}(s)}e^{\theta_{23}}[\mu_{2}]_{2}.$

Taking the residue of this equation at $k_{j}$, we find the condition (2.34f)
in the case when $k_{j}\in D_{4}$. ∎

##  3\. The Riemann-Hilbert problem

The sectionally analytic function $M(x,t,k)$ defined in section 2 satisfies a
Riemann-Hilbert problem which can be formulated in terms of the initial and
boundary values of $u(x,t)$. By solving this Riemann-Hilbert problem, the
solution of (1.5)(then (1.3)) can be recovered for all values of $x,t$.

######  Theorem 3.1.

Suppose that $u(x,t)$ is a solution of (1.5) in the half-line domain $\Omega$
with sufficient smoothness and decays as $x\rightarrow\infty$. Then $u(x,t)$
can be reconstructed from the initial value $\\{u_{0}(x)\\}$ and boundary
values $\\{g_{0}(t),g_{1}(t),g_{2}(t)\\}$ defined as follows,

$u_{0}(x)=u(x,0),\quad g_{0}(t)=u(0,t),\quad g_{1}(t)=u_{x}(0,t),\quad
g_{2}(t)=u_{xx}(0,t).$ (3.1)

Use the initial and boundary data to define the jump matrices $J_{m,n}(x,t,k)$
as well as the spectral $s(k)$ and $S(k)$ by equation (2.24). Assume that the
possible zeros $\\{k_{j}\\}_{1}^{N}$ of the functions
$s_{33}(k),(s^{T}S^{A})_{33}(k),(S^{T}s^{A})_{33}(k)$ and $m_{33}(s)(k)$ are
as in assumption 2.3.

Then the solution $\\{u(x,t)\\}$ is given by

$u(x,t)=2i\lim_{k\rightarrow\infty}(kM(x,t,k))_{13}.$ (3.2)

where $M(x,t,k)$ satisfies the following $3\times 3$ matrix Riemann-Hilbert
problem:

  * •

$M$ is sectionally meromorphic on the Riemann $k-$sphere with jumps across the
contours $\bar{D}_{n}\cap\bar{D}_{m},n,m=1,\cdots,4$, see Figure 2.

  * •

Across the contours $\bar{D}_{n}\cap\bar{D}_{m}$, $M$ satisfies the jump
condition

$M_{n}(x,t,k)=M_{m}(x,t,k)J_{m,n}(x,t,k),\quad
k\in\bar{D}_{n}\cap\bar{D}_{m},n,m=1,2,3,4.$ (3.3)

  * •

$M(x,t,k)=\mathbb{I}+O(\frac{1}{k}),\qquad k\rightarrow\infty$.

  * •

The residue condition of $M$ is showed in Proposition 2.4.

###### Proof.

It only remains to prove (3.2) and this equation follows from the large $k$
asymptotics of the eigenfunctions, see the appendix A. ∎

##  4\. Non-linearizable Boundary Conditions

A major difficulty of initial-boundary value problems is that some of the
boundary values are unkown for a well-posed problem. All boundary values are
needed for the definition of $S(k)$, and hence for the formulation of the
Riemann-Hilbert problem. Our main result expresses the spectral function
$S(k)$ in terms of the prescribed boundary data and the initial data via the
solution of a system of nonlinear integral equations.

###  4.1. Asymptotics

An analysis of (A.2) shows that the eigenfunctions $\\{\mu_{j}\\}_{1}^{3}$
have the following asymptotics as $k\rightarrow\infty$ (see the appendix A):

$\begin{array}[]{l}\mu_{j}(x,t,k)=\mathbb{I}+\frac{1}{k}\left(\begin{array}[]{lll}\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta&\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}&\frac{1}{2i}u\\\
\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}&\frac{i}{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta&\frac{1}{2i}\bar{u}\\\
\frac{1}{2i}\bar{u}&\frac{1}{2i}u&-i\int_{(x_{j},t_{j})}^{(x,t)}\Delta\end{array}\right)\\\
+\frac{1}{k^{2}}\left(\begin{array}[]{lll}-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}(\eta+\nu_{1})&-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}\eta_{1}&\mu^{(2)}_{13}\\\
-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}\eta_{2}&-\frac{1}{4}\int_{(x_{j},t_{j})}^{(x,t)}(\eta+\nu_{2})&\mu^{(2)}_{23}\\\
\mu^{(2)}_{31}&\mu^{(2)}_{32}&\int_{(x_{j},t_{j})}^{(x,t)}\eta_{3}\\\
\end{array}\right)\\\
+\frac{1}{k^{3}}\left(\begin{array}[]{lll}\mu^{(3)}_{11}&\mu^{(3)}_{12}&\mu^{(3)}_{13}\\\
\mu^{(3)}_{21}&\mu^{(3)}_{22}&\mu^{(3)}_{23}\\\
\mu^{(3)}_{31}&\mu^{(3)}_{32}&\mu^{(3)}_{33}\end{array}\right)+O(\frac{1}{k^{4}})\end{array}$
(4.1a)

where

$\begin{array}[]{l}\Delta=-|u|^{2}dx+(u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4})dt\\\
\Delta_{1}=-u^{2}dx+(uu_{xx}+u_{xx}u-(u_{x})^{2}+6|u|^{2}u^{2})dt\\\
\Delta_{2}=-\bar{u}^{2}dx+(\bar{u}\bar{u}_{xx}+\bar{u}_{xx}\bar{u}-(\bar{u}_{x})^{2}+6|u|^{2}\bar{u}^{2})dt\end{array}$
(4.2a)
$\begin{array}[]{l}\mu^{(2)}_{13}=-\frac{1}{2}u\int_{(x_{j},t_{j})}^{(x,t)}\Delta+\frac{1}{4}u_{x}\\\
\mu^{(2)}_{23}=-\frac{1}{2}\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta+\frac{1}{4}\bar{u}_{x}\\\
\mu^{(2)}_{31}=\frac{1}{4}(\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta+u\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2})-\frac{1}{4}\bar{u}_{x}\\\
\mu^{(2)}_{32}=\frac{1}{4}(\bar{u}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+u\int_{(x_{j},t_{j})}^{(x,t)}\Delta)-\frac{1}{4}u_{x}.\end{array}$
(4.2b)
$\begin{array}[]{l}\eta=d[\frac{1}{2}(\int_{(x_{j},t_{j})}^{(x,t)}\Delta)^{2}]\\\
\nu_{1}=\Delta_{1}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}+(u\bar{u}_{x})dx+(u\bar{u}_{t}-2|u|^{2}(u\bar{u}_{x}-u_{x}\bar{u})-(u_{xx}\bar{u}_{x}-u_{x}\bar{u}_{xx}))dt\\\
\eta_{1}=\int_{(x_{j},t_{j})}^{(x,t)}\Delta\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+u^{2}\\\
\eta_{2}=\int_{(x_{j},t_{j})}^{(x,t)}\Delta\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{2}+\bar{u}^{2}\\\
\nu_{2}=\Delta_{2}\int_{(x_{j},t_{j})}^{(x,t)}\Delta_{1}+(\bar{u}u_{x})dx+(\bar{u}u_{t}-2|u|^{2}(\bar{u}u_{x}-\bar{u}_{x}u)-(\bar{u}_{xx}u_{x}-\bar{u}_{x}u_{xx}))dt,\\\
\eta_{3}=d[-\frac{1}{2}(\int_{(x_{j},t_{j})}^{(x,t)}\Delta)^{2}-\frac{1}{4}|u|^{2}].\end{array}$
(4.2c) and in the following we just use
$\mu^{(3)}_{13},\mu^{(3)}_{23},\mu^{(3)}_{31}$ and $\mu^{(3)}_{32}$, so we
only compute these functions
$\begin{array}[]{l}\mu^{(3)}_{13}=\frac{1}{2i}u\mu^{(2)}_{33}+\frac{1}{4}u_{x}\mu^{(1)}_{33}+\frac{i}{4}|u|^{2}u+\frac{i}{8}u_{xx}\\\
\mu^{(3)}_{23}=\frac{1}{2i}\bar{u}\mu^{(2)}_{33}+\frac{1}{4}\bar{u}_{x}\mu^{(1)}_{33}+\frac{i}{4}|u|^{2}\bar{u}+\frac{i}{8}\bar{u}_{xx}\\\
\mu^{(3)}_{31}=\frac{1}{2i}(\bar{u}\mu^{(2)_{11}}+u\mu^{(2)}_{21})-\frac{1}{4}(\bar{u}_{x}\mu^{(1)_{11}}+u_{x}\mu^{(1)}_{21})+\frac{i}{4}|u|^{2}\bar{u}+\frac{i}{8}\bar{u}_{xx}\\\
\mu^{(3)}_{32}=\frac{1}{2i}(\bar{u}\mu^{(2)_{12}}+u\mu^{(2)}_{22})-\frac{1}{4}(\bar{u}_{x}\mu^{(1)_{12}}+u_{x}\mu^{(1)}_{22})+\frac{i}{4}|u|^{2}u+\frac{i}{8}u_{xx}\end{array}$
(4.2d)

From the global relation (2.33)and replacing $T$ by $t$, we find

$\mu_{2}(0,t,k)e^{-4ik^{3}t\hat{\Lambda}}s(k)=c(t,k),\quad k\in(D_{3}\cup
D_{4},D_{3}\cup D_{4},D_{1}\cup D_{2}).$ (4.3)

We define functions $\\{\Phi_{13}(t,k),\Phi_{23}(t,k),\Phi_{33}(t,k)\\}$ and
$\\{c_{j}(t,k)\\}_{1}^{3}$ by:

$\mu_{2}(0,t,k)=\left(\begin{array}[]{lll}\Phi_{11}(t,k)&\Phi_{12}(t,k)&\Phi_{13}(t,k)\\\
\Phi_{21}(t,k)&\Phi_{22}(t,k)&\Phi_{23}(t,k)\\\
\Phi_{31}(t,k)&\Phi_{32}(t,k)&\Phi_{33}(t,k)\end{array}\right),\quad\frac{[c(t,k)]_{3}}{s_{33}(k)}=\left(\begin{array}[]{l}c_{1}(t,k)\\\
c_{2}(t,k)\\\ c_{3}(t,k)\end{array}\right).$ (4.4)

we can write the $(13)$ and $(23)$ entries of the global relation as

$\Phi_{11}(t,k)e^{-8ik^{3}t}\frac{s_{13}}{s_{33}}+\Phi_{12}(t,k)e^{-8ik^{3}t}\frac{s_{23}}{s_{33}}+\Phi_{13}(t,k)=c_{1}(t,k),\quad
k\in D_{1}\cup D_{2},$ (4.5a)
$\Phi_{21}(t,k)e^{-8ik^{3}t}\frac{s_{13}}{s_{33}}+\Phi_{22}(t,k)e^{-8ik^{3}t}\frac{s_{23}}{s_{33}}+\Phi_{23}(t,k)=c_{2}(t,k),\quad
k\in D_{1}\cup D_{2},$ (4.5b)

The functions $\\{c_{j}(t,k)\\}_{1}^{3}$ are analytic and bounded in
$D_{1}\cup D_{2}$ away from the possible zeros of $s_{33}(k)$ and of order
$O(\frac{1}{k})$ as $k\rightarrow\infty$.

From the asymptotic of $\mu_{j}(x,t,k)$ in (4.1a) we have

$\left(\begin{array}[]{l}s_{13}(k)\\\ s_{23}(k)\\\
s_{33}(k)\end{array}\right)=\left(\begin{array}[]{l}0\\\ 0\\\
1\end{array}\right)+\frac{1}{2ik}\left(\begin{array}[]{l}u(0,0)\\\
\bar{u}(0,0)\\\
2\int_{(\infty,0)}^{(0,0)}\Delta\end{array}\right)+O(\frac{1}{k^{2}}).$ (4.6)

and

$\Phi_{j3}(t,k)=\frac{\Phi_{j3}^{(1)}(t)}{k}+\frac{\Phi_{j3}^{(2)}(t)}{k^{2}}+\frac{\Phi_{j3}^{(3)}(t)}{k^{3}}+O(\frac{1}{k^{4}}),$
(4.7a)
$\Phi_{33}(t,k)=1+\frac{\Phi_{33}^{(1)}(t)}{k}+\frac{\Phi_{33}^{(2)}(t)}{k^{2}}+O(\frac{1}{k^{3}}),\quad
k\rightarrow\infty,k\in D_{1}\cup D_{2}.$ (4.7b) where
$\begin{array}[]{ll}\Phi_{j3}^{(1)}(t)=\frac{1}{2i}g_{0}(t)^{T},&\Phi_{j3}^{(2)}(t)=\frac{1}{4}g_{1}(t)^{T}-\frac{1}{2}g_{0}^{T}\int_{(0,0)}^{(x,t)}\Delta\\\
\Phi_{j3}^{(3)}(t)=\frac{1}{2i}g_{0}^{T}\Phi^{(2)}_{33}+\frac{1}{4}g_{1}^{T}\Phi^{(1)}_{33}+\frac{i}{4}|u|^{2}g_{0}^{T}+\frac{i}{8}g_{2}^{T},&\\\
\Phi_{33}^{(1)}(t)=-i\int_{(0,0)}^{(x,t)}\Delta,&\Phi_{33}^{(2)}(t)=\int_{(x_{j},t_{j})}^{(x,t)}\eta_{3}.\end{array}$
Here the definition of $\Phi_{j3}(t,k)$ can be found in the appendix A.

In particular, we find the following expressions for the boudary values:

$g_{0}^{T}=2i\Phi_{j3}^{(1)}(t),$ (4.8a)
$g_{1}^{T}=2ig_{0}^{T}\Phi_{33}^{(1)}(t)+4\Phi_{j3}^{(2)}(t),$ (4.8b)
$g_{2}^{T}=-2|g_{0}|^{2}g_{0}^{T}+2ig_{1}^{T}\Phi_{33}^{(1)}(t)+4g_{0}^{T}\Phi_{33}^{(2)}(t)-8i\Phi_{j3}^{(3)}(t).$
(4.8c)

We will also need the asymptotic of $c_{j}(t,k)$,

######  Lemma 4.1.

The global relation (4.5) implies that the large $k$ behavior of $c_{j}(t,k)$
satisfies

$c_{j}(t,k)=\frac{\Phi_{j3}^{(1)}(t)}{k}+\frac{\Phi_{j3}^{(2)}(t)}{k}+\frac{\Phi_{j3}^{(3)}(t)}{k}+O(\frac{1}{k^{4}}),\quad
k\rightarrow\infty,k\in D_{1}.$ (4.9)

###### Proof.

See the appendix B. ∎

###  4.2. The Dirichlet and Neumann problems

We can now derive effective characterizations of spectral function $S(k)$ for
the Dirichlet ($g_{0}$ prescribed), the first Neumann ($g_{1}$ prescribed),
and the second Neumann ($g_{2}$ prescribed) problems.

Define $\alpha$ by $\alpha=e^{\frac{2\pi i}{3}}$ and let
$\\{\Pi_{j}(t,k),\hat{\Pi}_{j}(t,k),\tilde{\Pi}_{j}(t,k)\\}_{1}^{3}$ denote
the following combinations formed from $\\{\Phi_{j3}(t,k)\\}_{1}^{3}$:

$\begin{array}[]{l}\Pi_{j}(t,k)=\Phi_{j3}(t,k)+\alpha\Phi_{j3}(t,\alpha
k)+\alpha^{2}\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3,\\\
\hat{\Pi}_{j}(t,k)=\Phi_{j3}(t,k)+\alpha^{2}\Phi_{j3}(t,\alpha
k)+\alpha\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3,\\\
\tilde{\Pi}_{j}(t,k)=\Phi_{j3}(t,k)+\Phi_{j3}(t,\alpha
k)+\Phi_{j3}(t,\alpha^{2}k),\quad j=1,2,3.\end{array}$ (4.10)

And let $R(k)=\Phi_{11}\frac{s_{13}}{s_{33}}+\Phi_{12}\frac{s_{23}}{s_{33}}$.

Let $D_{1}=D_{1}^{{}^{\prime}}\cup D_{1}^{{}^{\prime\prime}}$ where
$D_{1}^{{}^{\prime}}=D_{1}\cap\\{\mathrm{Re}k>0\\}$ and
$D_{1}^{{}^{\prime\prime}}=D_{1}\cap\\{\mathrm{Re}k<0\\}$. Similarly, let
$D_{4}=D_{4}^{{}^{\prime}}\cup D_{4}^{{}^{\prime\prime}}$ where
$D_{4}^{{}^{\prime}}=D_{4}\cap\\{\mathrm{Re}k>0\\}$ and
$D_{4}^{{}^{\prime\prime}}=D_{1}\cap\\{\mathrm{Re}k<0\\}$.

######  Theorem 4.2.

Let $T<\infty$. Let $u_{0}(x),u\geq 0$, be a function of Schwartz class.

For the Dirichlet problem it is assumed that the function $g_{0}(t),0\leq
t<T$, has sufficient smoothness and is compatible with $u_{0}(x)$ at $x=t=0$.

For the first Neumann problem it is assumed that the function $g_{1}(t),0\leq
t<T$, has sufficient smoothness and is compatible with $u_{0}(x)$ at $x=t=0$.

Similarly, for the second Neumann problem it is assumed that the function
$g_{2}(t),0\leq t<T$, has sufficient smoothness and is compatible with
$u_{0}(x)$ at $x=t=0$.

Suppose that $s_{33}(k)$ has a finite number of simple zeros in $D_{1}$.

Then the spectral function $S(k)$ is given by

$S(k)=\left(\begin{array}[]{ccc}A(k)&B(k)&e^{8ik^{3}T}C(k)\\\
D(k)&E(k)&e^{8ik^{3}T}F(k)\\\
e^{-8ik^{3}T}G(k)&e^{-8ik^{3}T}H(k)&I(k)\end{array}\right)$ (4.11)

where

$\begin{array}[]{ll}A(k)=\Phi_{22}(k)\Phi_{33}(k)-\Phi_{23}(k)\Phi_{32}(k)&B(k)=\Phi_{13}(k)\Phi_{22}(k)-\Phi_{12}(k)\Phi_{33}(k)\\\
C(k)=\Phi_{12}(k)\Phi_{23}(k)-\Phi_{13}(k)\Phi_{22}(k)&D(k)=\Phi_{23}(k)\Phi_{31}(k)-\Phi_{21}(k)\Phi_{33}(k)\\\
E(k)=\Phi_{11}(k)\Phi_{33}(k)-\Phi_{13}(k)\Phi_{31}(k)&F(k)=\Phi_{21}(k)\Phi_{13}(k)-\Phi_{11}(k)\Phi_{23}(k)\\\
G(k)=\Phi_{21}(k)\Phi_{32}(k)-\Phi_{22}(k)\Phi_{31}(k)&H(k)=\Phi_{12}(k)\Phi_{31}(k)-\Phi_{11}(k)\Phi_{32}(k)\\\
I(k)=\Phi_{11}(k)\Phi_{22}(k)-\Phi_{12}(k)\Phi_{21}(k)&\end{array}$

and the complex-value functions $\\{\Phi_{l3}(t,k)\\}_{l=1}^{3}$ satisfy the
following system of integral equations:

$\begin{array}[]{rl}\Phi_{13}(t,k)&=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{13}\right.\\\
&\left.+g_{0}^{2}\Phi_{23}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12a)
$\begin{array}[]{rl}\Phi_{23}(t,k)&=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{13}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{23}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12b)
$\begin{array}[]{rl}\Phi_{33}(t,k)&=1+\int_{0}^{t}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{13}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{23}+-4ik|g_{0}|^{2}\Phi_{33}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.12c)

and $\\{\Phi_{l1}(t,k)\\}_{l=1}^{3},\\{\Phi_{l2}(t,k)\\}_{l=1}^{3}$ satisfy
the following system of integral equations:

$\begin{array}[]{rl}\Phi_{11}(t,k)&=1+\int_{0}^{t}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{11}\right.\\\
&\left.+g_{0}^{2}\Phi_{21}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13a)
$\begin{array}[]{rl}\Phi_{21}(t,k)&=\int_{0}^{t}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{11}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{21}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13b)
$\begin{array}[]{rl}\Phi_{33}(t,k)&=\int_{0}^{t}e^{8ik^{3}(t-t^{\prime})}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{11}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{21}+-4ik|g_{0}|^{2}\Phi_{31}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.13c)
$\begin{array}[]{rl}\Phi_{12}(t,k)&=\int_{0}^{t}\left[(2ik|g_{0}|^{2}+(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{12}\right.\\\
&\left.+g_{0}^{2}\Phi_{22}+(4k^{2}g_{0}+2ikg_{1}-4|g_{0}|^{2}g_{0}-g_{2})\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14a)
$\begin{array}[]{rl}\Phi_{22}(t,k)&=1+\int_{0}^{t}\left[(2ik|g_{0}|^{2}-(g_{0}\bar{g}_{1}-g_{1}\bar{g}_{0}))\Phi_{12}\right.\\\
&\left.+\bar{g}_{0}^{2}\Phi_{22}+(4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}-4|g_{0}|^{2}\bar{g}_{0}-\bar{g}_{2})\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14b)
$\begin{array}[]{rl}\Phi_{32}(t,k)&=\int_{0}^{t}e^{8ik^{3}(t-t^{\prime})}\left[(-4k^{2}\bar{g}_{0}+2ik\bar{g}_{1}+4|g_{0}|^{2}+\bar{g}_{2})\Phi_{12}\right.\\\
&\left.+(-4k^{2}g_{0}+2ikg_{1}+4|g_{0}|^{2}+g_{2})\Phi_{22}+-4ik|g_{0}|^{2}\Phi_{32}\right](t^{\prime},k)dt^{\prime}\end{array}$
(4.14c)

  1. (i)

For the Dirichlet problem, the unknown Neumann boundary values $g_{1}(t)$ and
$g_{2}(t)$ are given by

$\begin{array}[]{rl}g_{1}(t)=&\frac{2g_{0}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk+\frac{2}{\pi i}\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk\\\ &-\frac{2}{\pi
i}\int_{\partial D_{3}}ke^{-8ik^{3}t}[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)]dk\\\
&+4\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.15a) and
$\begin{array}[]{rl}g_{2}(t)=&g_{0}(t)^{3}-\frac{4}{\pi}\int_{\partial
D_{3}}\left[k^{2}\Pi_{1}(t,k)-\frac{3kg_{0}(t)}{2i}\right]dk\\\
&+\frac{4}{\pi}\int_{\partial D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)R(\alpha
k)+(1-\alpha^{2})R(\alpha^{2}k)\right]dk\\\
&-8i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k)\\\
&+\frac{4g_{0}(t)}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{3}(t,k)dk+\frac{2g_{1}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.15b)

  2. (ii)

For the first Neumann problem, the unknown boundary values $g_{0}(t)$ and
$g_{2}(t)$ are given by

$\begin{array}[]{rl}g_{0}(t)=&\frac{1}{\pi}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk-\frac{1}{\pi}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
&+2i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k),\end{array}$
(4.16a) and
$\begin{array}[]{rl}g_{2}(t)=&g_{0}^{3}(t)-\frac{4}{\pi}\int_{\partial
D_{3}}\left(k^{2}\hat{\Pi}_{1}(t,k)-\frac{3}{\pi i}\int_{\partial
D_{3}}l\hat{\Pi}_{1}(t,l)dl\right)dk\\\ &+\frac{4}{\pi}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha^{2})R(\alpha
k)+(1-\alpha)R(\alpha^{2}k)\right]dk\\\ &-8i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k)\\\
&+\frac{4g_{0}(t)}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{3}(t,k)dk+\frac{2g_{1}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.16b)

  3. (iii)

For the second Neumann problem, the unknown boundary values $g_{0}(t)$ and
$g_{1}(t)$ are given by

$\begin{array}[]{rl}g_{0}(t)=&\frac{1}{\pi}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk-\frac{1}{\pi}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
&+2i\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k),\end{array}$
(4.17a) and $\begin{array}[]{rl}g_{1}(t)=&\frac{2g_{0}(t)}{\pi}\int_{\partial
D_{3}}\Pi_{3}(t,k)dk+\frac{2}{\pi i}\int_{\partial
D_{3}}k\tilde{\Pi}_{1}(t,k)dk\\\ &-\frac{2}{\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-1)R(\alpha
k)+(\alpha-1)R(\alpha^{2}k)\right]dk\\\ &+4\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.17b)

###### Proof.

The representations (4.11) follow from the relation
$S(k)=e^{8ik^{3}T}\mu_{2}^{A}(0,T,k)^{T}$. And the system (4.12) is the direct
result of the Volteral integral equations of $\mu_{2}(0,t,k)$.

  1. (i)

In order to derive (4.15a) we note that equation (4.8b) expresses $g_{1}$ in
terms of $\Phi_{33}^{(1)}$ and $\Phi_{13}^{(2)}$. Furthermore, equation (4.7)
and Cauchy theorem imply

$-\frac{2\pi i}{3}\Phi_{33}^{(1)}(t)=2\int_{\partial
D_{2}}[\Phi_{33}(t,k)-1]dk=\int_{\partial D_{4}}[\Phi_{33}(t,k)-1]dk$

and

$-\frac{2\pi i}{3}\Phi_{13}^{(2)}(t)=2\int_{\partial
D_{2}}\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk=\int_{\partial
D_{4}}\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk.$

Thus,

$\begin{array}[]{l}i\pi\Phi_{33}^{(1)}(t)=-\left(\int_{\partial
D_{2}}+\int_{\partial D_{4}}\right)[\Phi_{33}(t,k)-1]dk=\left(\int_{\partial
D_{1}}+\int_{\partial D_{3}}\right)[\Phi_{33}(t,k)-1]dk\\\ =\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk+\alpha\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk+\alpha^{2}\int_{\partial
D_{3}}[\Phi_{33}(t,k)-1]dk\\\ =\int_{\partial
D_{3}}\Pi_{3}(t,k)dk.\end{array}$ (4.18)

Similarly,

$\begin{array}[]{l}i\pi\Phi_{13}^{(2)}(t)=\left(\int_{\partial
D_{3}}+\int_{\partial
D_{1}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk\\\
=\left(\int_{\partial D_{3}}+\alpha^{2}\int_{\partial
D_{1}^{{}^{\prime}}}+\alpha\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk+I(t)\\\
=\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk+I(t).\end{array}$
(4.19)

where $I(t)$ is defined by

$I(t)=\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[k\Phi_{13}(t)-\frac{g_{0}(t)}{2i}\right]dk$

The last step involves using the global relation to compute $I(t)$

$\begin{array}[]{r}I(t)=\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)\left[kc_{1}(t,k)-\frac{g_{0}(t)}{2i}\right]dk\\\
-\left((1-\alpha^{2})\int_{\partial
D_{1}^{{}^{\prime}}}+(1-\alpha)\int_{\partial
D_{1}^{{}^{\prime\prime}}}\right)ke^{-8ik^{3}t}R(k)dk\end{array}$ (4.20)

Using the asymptotic (4.9) and Cauchy theorem to compute the first term on the
right-hand side of equation (4.20) and using the transformation
$k\rightarrow\alpha k$ and $k\rightarrow\alpha^{2}k$ in the second term on the
right-hand side of (4.20), we find

$\begin{array}[]{r}I(t)=-i\pi\Phi_{13}^{(2)}(t)-\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)\right]dk\\\ +2\pi
i\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}ke^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.21)

Equations (4.19) and (4.21) imply

$\begin{array}[]{l}\Phi_{13}^{(2)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k\Pi_{1}(t,k)-\frac{3g_{0}(t)}{2i}\right]dk\\\ -\frac{1}{2\pi
i}\int_{\partial D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-\alpha)R(\alpha
k)+(\alpha-\alpha^{2})R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$

This equation together with (4.8b) and (4.18) yields (4.15a).

In order to derive (4.15b), we note that (4.8c) expresses $g_{2}$ in terms of
$\Phi_{13}^{(3)}$, $\Phi_{33}^{(2)}$ and $\Phi_{33}^{(1)}$. Equation (4.15b)
follows from the expression (4.18) for $\Phi_{33}^{(1)}$ and the following
formulas:

$\Phi_{33}^{(2)}(t)=\frac{1}{\pi i}\int_{\partial D_{3}}k\hat{\Pi}_{3}dk,$
(4.22a) $\begin{array}[]{l}\Phi_{13}^{(3)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k^{2}\Pi_{1}(t,k)-\frac{3kg_{0}(t)}{2i}\right]dk\\\ -\frac{1}{2\pi
i}\int_{\partial D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)R(\alpha
k)+(1-\alpha^{2})R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.22b)

  2. (ii)

In order to derive the representations (4.16) relevant for the first Neumann
problem, we use (4.8) together with (4.18), (4.22a) and the following
formulas:

$\begin{array}[]{l}\Phi_{13}^{(1)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\hat{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-\alpha^{2})R(\alpha
k)+(\alpha^{2}-\alpha)R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.23a) $\Phi_{13}^{(2)}(t)=\frac{1}{\pi i}\int_{\partial
D_{3}}k\hat{\Pi}_{1}dk,$ (4.23b)
$\begin{array}[]{l}\Phi_{13}^{(3)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\left[k^{2}\hat{\Pi}_{1}(t,k)-3\Phi_{13}^{(2)}\right]dk\\\
-\frac{1}{2\pi i}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha^{2})R(\alpha
k)+(1-\alpha)R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}k_{j}^{2}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.23c)

  3. (iii)

In order to derive the representations (4.17) relevant for the second Neumann
problem, we use (4.8) together with (4.18) and the following formulas:

$\begin{array}[]{l}\Phi_{13}^{(1)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}\tilde{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}e^{-8ik^{3}t}\left[(\alpha-1)R(\alpha
k)+(\alpha^{2}-1)R(\alpha^{2}k)\right]dk\\\
\left\\{(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.24a) $\begin{array}[]{l}\Phi_{13}^{(2)}(t)=\frac{1}{2\pi i}\int_{\partial
D_{3}}k\tilde{\Pi}_{1}(t,k)dk\\\ -\frac{1}{2\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}\left[(\alpha^{2}-1)R(\alpha
k)+(\alpha-1)R(\alpha^{2}k)\right]dk\\\ \left\\{(1-\alpha)\sum_{k_{j}\in
D_{1}^{{}^{\prime}}}+(1-\alpha^{2})\sum_{k_{j}\in
D_{1}^{{}^{\prime\prime}}}\right\\}e^{-8ik_{j}^{3}t}Res_{k_{j}}R(k).\end{array}$
(4.24b)

∎

###  4.3. Effective characterizations

Substituting into the system (4.12) the expressions

$\Phi_{ij}=\Phi^{(0)}_{ij}+\varepsilon\Phi^{(1)}_{ij}+\varepsilon^{2}\Phi^{(2)}_{ij}+\cdots,\quad
i,j=1,2,3.$ (4.25a) $g_{0}=\varepsilon g_{01}+\varepsilon^{2}g_{02}+\cdots,$
(4.25b) $g_{1}=\varepsilon g_{11}+\varepsilon^{2}g_{12}+\cdots,$ (4.25c)
$g_{2}=\varepsilon g_{21}+\varepsilon^{2}g_{22}+\cdots,$ (4.25d)

where $\varepsilon>0$ is a small parameter, we find that the terms of $O(1)$
give $\Phi^{(0)}_{13}=\Phi^{(0)}_{23}=0$ and $\Phi^{(0)}_{33}=1$. Moreover,
the terms of $O(\varepsilon)$ give $\Phi^{(1)}_{33}=0$ and

$O(\varepsilon):\quad\Phi^{(1)}_{13}(t,k)=\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}(4k^{2}g_{01}+2ikg_{11}-g_{21})(t^{\prime},k)dt^{\prime},$
(4.26)

From the above equation (4.26) we can get

$\Pi^{(1)}_{1}(t,k)=12k^{2}\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{01}(t^{\prime})dt^{\prime},$
(4.27a)
$\hat{\Pi}^{(1)}_{1}(t,k)=6ik\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{11}(t^{\prime})dt^{\prime},$
(4.27b)
$\tilde{\Pi}^{(1)}_{1}(t,k)=-3\int_{0}^{t}e^{-8ik^{3}(t-t^{\prime})}g_{11}(t^{\prime})dt^{\prime},$
(4.27c)

The Dirichlet problem can now be solved perturbatively as follows: assuming
for simplicity that $s_{33}(k)$ has no zeros and expanding (4.15a) and
(4.15b), we find

$\begin{array}[]{rl}g_{11}=&\frac{2}{\pi i}\int_{\partial
D_{3}}\left[k\Pi^{(1)}_{1}(t,k)-\frac{3g_{01}(t)}{2i}\right]dk\\\
&-\frac{2}{\pi i}\int_{\partial
D_{3}}ke^{-8ik^{3}t}[(\alpha^{2}-\alpha)s_{131}(\alpha
k)+(\alpha-\alpha^{2})s_{131}(\alpha^{2}k)]dk\end{array}$ (4.28a)
$\begin{array}[]{rl}g_{21}=&-\frac{4}{\pi}\int_{\partial
D_{3}}\left[k^{2}\Pi^{(1)}_{1}(t,k)-\frac{3kg_{01}(t)}{2i}\right]dk\\\
&+\frac{4}{\pi}\int_{\partial
D_{3}}k^{2}e^{-8ik^{3}t}\left[(1-\alpha)s_{131}(\alpha
k)+(1-\alpha^{2})s_{131}(\alpha^{2}k)\right]dk\end{array}$ (4.28b)

Using equation (4.27a) to determine $\Pi^{(1)}_{1}$, we can determine
$g_{11},g_{21}$ from (4.28), then $\Phi^{(1)}_{13}$ can be found from (4.26),
And these arguments can be extended to higher orders and also can be extended
to the systems (4.13a) and (4.14a), thus yields a constructive scheme for
computing $S(k)$ to all orders.

Similarly, these arguments also can be used to the first Neumann problem and
the second Neumann problem. That is to say, in all cases, the system can be
solved perturbatively to all orders.

##  Appendix A The asymptotic behavior of the functions
$\\{\mu_{j}(x,t,k)\\}_{1}^{3}$

We denote some symbols as follows:

$\Lambda=\left(\begin{array}[]{ll}\mathbb{I}_{2\times 2}&0\\\
0&-1\end{array}\right),$ (A.1a)
$\begin{array}[]{l}V_{1}=\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right),\\\
V_{2}^{(2)}=4\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right),\\\
V_{2}^{(1)}=2i\left(\begin{array}[]{ll}U^{T}\bar{U}&U_{x}^{T}\\\
\bar{U}_{x}&-2|u|^{2}\end{array}\right),\\\
V_{2}^{(0)}=-4|u|^{2}\left(\begin{array}[]{ll}0&U^{T}\\\
-\bar{U}&0\end{array}\right)-\left(\begin{array}[]{ll}0&U_{xx}^{T}\\\
-\bar{U}_{xx}&0\end{array}\right)+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}&0\\\
0&0\end{array}\right).\end{array}$ (A.1b) where $\mathbb{I}_{2\times
2}=\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$ and $U=(u,\bar{u})$.

From the Lax pair of $\mu$

$\left\\{\begin{array}[]{l}\mu_{x}+[ik\Lambda,\mu]=V_{1}\mu,\\\
\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.\end{array}\right.$ (A.2)

Suppose that

$\mu(x,t,k)=D_{0}+\frac{D_{1}}{k}+\frac{D_{2}}{k^{2}}+\frac{D_{3}}{k^{3}}+\cdots.$
(A.3)

We substitute the equation (A.3) into the Lax pair (A.2), and compare the
order of $k$, we find that:

$\begin{array}[]{ll}O(k):&[i\Lambda,D_{0}]=0,\\\
O(1):&D_{0x}+[i\Lambda,D_{1}]=V_{1}D_{0},\\\
O(k^{-1}):&D_{1x}+[i\Lambda,D_{2}]=V_{1}D_{1},\\\
O(k^{-2}):&D_{2x}+[i\Lambda,D_{3}]=V_{1}D_{2},\\\ \end{array}$ (A.4a)
$\begin{array}[]{ll}O(k^{3}):&[4i\Lambda,D_{0}]=0,\\\
O(k^{2}):&[4i\Lambda,D_{1}]=V_{2}^{(2)}D_{0},\\\
O(k^{1}):&[4i\Lambda,D_{2}]=V_{2}^{(2)}D_{1}+V_{2}^{(1)}D_{0},\\\
O(1):&D_{0t}+[4i\Lambda,D_{3}]=V_{2}^{(2)}D_{2}+V_{2}^{(1)}D_{1}+V_{2}^{(0)}D_{0},\\\
O(k^{-1}):&D_{1t}+[4i\Lambda,D_{4}]=V_{2}^{(2)}D_{3}+V_{2}^{(1)}D_{4}+V_{2}^{(0)}D_{1},\\\
O(k^{-2}):&D_{2t}+[4i\Lambda,D_{5}]=V_{2}^{(2)}D_{4}+V_{2}^{(1)}D_{3}+V_{2}^{(0)}D_{2},\\\
\end{array}$ (A.4b)

And we denote the $D_{l}$ by $D_{l}=\left(\begin{array}[]{ll}D_{2\times
2}^{(l)}&D_{j3}^{(l)}\\\ D_{3j}^{(l)}&D_{33}^{(l)}\end{array}\right),\quad
j=1,2$.

Then, from $O(k^{3})$,we have

$D_{j3}^{(0)}=0,\quad D_{3j}^{(0)}=0.$ (A.5)

$O(k^{2})$, we get

$4i\left(\begin{array}[]{ll}0&2D_{j3}^{(1)}\\\
-2D_{3j}^{(1)}&0\end{array}\right)=4\left(\begin{array}[]{ll}0&U^{T}D_{33}^{(0)}\\\
-\bar{U}D_{2\times 2}^{(0)}&0\end{array}\right),$ (A.6a) this implies that
$\left\\{\begin{array}[]{l}D_{j3}^{(1)}=-\frac{i}{2}U^{T}D_{33}^{(0)}\\\
D_{3j}^{(1)}=-\frac{i}{2}\bar{U}D_{2\times 2}^{(0)}.\end{array}\right.$ (A.6b)

$O(k)$, we find

$\begin{array}[]{l}4i\left(\begin{array}[]{ll}0&2D_{j3}^{(2)}\\\
-2D_{3j}^{(2)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(1)}&U^{T}D_{33}^{(1)}\\\
-\bar{U}D_{2\times
2}^{(1)}&-\bar{U}D_{j3}^{(1)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(0)}&U_{x}^{T}D_{33}^{(0)}\\\ -\bar{U}_{x}D_{2\times
2}^{(0)}&-2|u|^{2}D_{33}^{(0)}\end{array}\right),\end{array}$ (A.7a) this
implies that
$\left\\{\begin{array}[]{l}D_{j3}^{(2)}=-\frac{i}{2}U^{T}D_{33}^{(1)}+\frac{1}{4}U_{x}^{T}D_{33}^{0}\\\
D_{3j}^{(2)}=-\frac{i}{2}\bar{U}D_{2\times
2}^{(1)}-\frac{1}{4}\bar{U}_{x}D_{2\times 2}^{(0)}.\end{array}\right.$ (A.7b)

$O(1)$, we have

$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times 2t}^{(0)}&0\\\
0&D_{33t}^{(0)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(3)}\\\
-2D_{3j}^{(3)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(2)}&U^{T}D_{33}^{(2)}\\\
-\bar{U}D_{2\times
2}^{(2)}&-\bar{U}D_{j3}^{(2)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(1)}+U_{x}^{T}D_{3j}^{(1)}&U^{T}\bar{U}D_{j3}^{(1)}+U_{x}^{T}D_{33}^{(1)}\\\
-\bar{U}_{x}D_{2\times
2}^{(1)}-2|u|^{2}D_{3j}^{(1)}&\bar{U}_{x}D_{j3}^{(1)}-2|u|^{2}D_{33}^{(1)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}0&U^{T}D_{33}^{(0)}\\\ -\bar{U}D_{2\times
2}^{(0)}&0\end{array}\right)-\left(\begin{array}[]{ll}0&U_{xx}^{T}D_{33}^{(0)}\\\
-\bar{U}_{xx}D_{2\times 2}^{(0)}&0\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(0)}&0\\\ 0&0\end{array}\right).\end{array}$ (A.8a) this implies that
$\begin{array}[]{l}D_{2\times 2t}^{(0)}=0\quad D_{33t}^{(0)}=0\\\
\left\\{\begin{array}[]{l}D_{j3}^{(3)}=-\frac{i}{2}U^{T}D_{33}^{(2)}+\frac{1}{4}U_{x}^{T}D_{33}^{(1)}+\frac{i}{4}|u|^{2}U^{T}D_{33}^{(0)}+\frac{i}{8}U_{xx}^{T}D_{33}^{(0)}\\\
D_{3j}^{(3)}=-\frac{i}{2}\bar{U}D_{2\times
2}^{(2)}-\frac{1}{4}\bar{U}_{x}D_{2\times
2}^{(1)}+\frac{i}{4}|u|^{2}\bar{U}D_{2\times
2}^{(0)}+\frac{i}{8}\bar{U}_{xx}D_{2\times
2}^{(0)}.\end{array}\right.\end{array}$ (A.8b)

$O(k^{-1})$, we get

$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times
2t}^{(1)}&D_{j3t}^{(1)}\\\
D_{3jt}^{(1)}&D_{33t}^{(1)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(4)}\\\
-2D_{3j}^{(4)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(3)}&U^{T}D_{33}^{(3)}\\\
-\bar{U}D_{2\times
2}^{(3)}&-\bar{U}D_{j3}^{(3)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(2)}+U_{x}^{T}D_{3j}^{(2)}&U^{T}\bar{U}D_{j3}^{(2)}+U_{x}^{T}D_{33}^{(2)}\\\
-\bar{U}_{x}D_{2\times
2}^{(2)}-2|u|^{2}D_{3j}^{(2)}&\bar{U}_{x}D_{j3}^{(2)}-2|u|^{2}D_{33}^{(2)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}U^{T}D_{3j}^{(1)}&U^{T}D_{33}^{(1)}\\\
-\bar{U}D_{2\times
2}^{(1)}&-\bar{U}D_{j3}^{(1)}\end{array}\right)-\left(\begin{array}[]{ll}U_{xx}^{T}D_{3j}^{(1)}&U_{xx}^{T}D_{33}^{(1)}\\\
-\bar{U}_{xx}D_{2\times
2}^{(1)}&-\bar{U}_{xx}D_{j3}^{(1)}\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(1)}&\sigma_{3}D_{j3}^{(1)}\\\ 0&0\end{array}\right).\end{array}$ (A.9a)
this implies that $\begin{array}[]{l}\left\\{\begin{array}[]{l}D_{2\times
2t}^{(1)}=\frac{i}{2}\\{U^{T}\bar{U}_{xx}+U_{xx}\bar{U}-U_{x}^{T}\bar{U}_{x}+6|u|^{2}U^{T}\bar{U}\\}D_{2\times
2}^{(0)}\\\
D_{33t}^{(1)}=-i\\{u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4}\\}D_{33}^{(0)}.\end{array}\right.\\\
\left\\{\begin{array}[]{l}D_{j3}^{(4)}=\frac{1}{16}U_{t}^{T}D_{33}^{(0)}-\frac{i}{2}U^{T}D_{33}^{(3)}+\frac{1}{4}U_{x}^{T}D_{33}^{(2)}+\frac{i}{4}|u|^{2}U^{T}D_{33}^{(1)}+\frac{i}{8}U_{xx}^{T}D_{33}^{(1)}+\frac{1}{8}|u|^{2}U_{x}^{T}D_{33}^{(0)}\\\
D_{3j}^{(3)}=-\frac{1}{16}\bar{U}_{t}D_{2\times
2}^{(0)}-\frac{i}{2}\bar{U}D_{2\times
2}^{(3)}-\frac{1}{4}\bar{U}_{x}D_{2\times
2}^{(2)}+\frac{i}{4}|u|^{2}\bar{U}D_{2\times
2}^{(1)}+\frac{i}{8}\bar{U}_{xx}D_{2\times
2}^{(1)}-\frac{1}{8}|u|^{2}\bar{U}_{x}D_{2\times
2}^{(0)}.\end{array}\right.\end{array}$ (A.9b)

$O(k^{-2})$, we get

$\begin{array}[]{l}\left(\begin{array}[]{ll}D_{2\times
2t}^{(2)}&D_{j3t}^{(2)}\\\
D_{3jt}^{(2)}&D_{33t}^{(2)}\end{array}\right)+4i\left(\begin{array}[]{ll}0&2D_{j3}^{(5)}\\\
-2D_{3j}^{(5)}&0\end{array}\right)=\\\
4\left(\begin{array}[]{ll}U^{T}D_{3j}^{(4)}&U^{T}D_{33}^{(4)}\\\
-\bar{U}D_{2\times
2}^{(4)}&-\bar{U}D_{j3}^{(4)}\end{array}\right)+2i\left(\begin{array}[]{ll}U^{T}\bar{U}D_{2\times
2}^{(3)}+U_{x}^{T}D_{3j}^{(3)}&U^{T}\bar{U}D_{j3}^{(3)}+U_{x}^{T}D_{33}^{(3)}\\\
-\bar{U}_{x}D_{2\times
2}^{(3)}-2|u|^{2}D_{3j}^{(3)}&\bar{U}_{x}D_{j3}^{(3)}-2|u|^{2}D_{33}^{(3)}\end{array}\right)\\\
-4|u|^{2}\left(\begin{array}[]{ll}U^{T}D_{3j}^{(2)}&U^{T}D_{33}^{(2)}\\\
-\bar{U}D_{2\times
2}^{(2)}&-\bar{U}D_{j3}^{(2)}\end{array}\right)-\left(\begin{array}[]{ll}U_{xx}^{T}D_{3j}^{(2)}&U_{xx}^{T}D_{33}^{(2)}\\\
-\bar{U}_{xx}D_{2\times
2}^{(2)}&-\bar{U}_{xx}D_{j3}^{(2)}\end{array}\right)\\\
+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}D_{2\times
2}^{(2)}&\sigma_{3}D_{j3}^{(2)}\\\ 0&0\end{array}\right).\end{array}$ (A.10a)
this implies that $\left\\{\begin{array}[]{l}\begin{array}[]{rl}D_{2\times
2t}^{(2)}=&\frac{i}{2}\\{U^{T}\bar{U}_{xx}+U_{xx}\bar{U}-U_{x}^{T}\bar{U}_{x}+6|u|^{2}U^{T}\bar{U}\\}D_{2\times
2}^{(1)}\\\
&+\\{-\frac{1}{4}U^{T}\bar{U}_{t}+\frac{1}{2}|u|^{2}(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}+\frac{1}{4}(u_{xx}\bar{u}_{x}-u_{x}\bar{u}_{xx})\sigma_{3}\\}\end{array}\\\
D_{33t}^{(2)}=-i\\{u\bar{u}_{xx}+u_{xx}\bar{u}-u_{x}\bar{u}_{x}+6|u|^{4}\\}D_{33}^{(1)}-\frac{1}{4}(|u|^{2})_{t}D_{33}^{(0)}.\end{array}\right.$
(A.10b)

Also, from the $x-$part of the Lax pair, we have the following equations

$D_{2\times 2x}^{(0)}=0,\quad D_{33x}^{(0)}=0.$ (A.11a)
$\left\\{\begin{array}[]{l}D_{2\times
2x}^{(1)}=-\frac{i}{2}U^{T}\bar{U}D_{2\times 2}^{(0)}\\\
D_{33x}^{(1)}=i|u|^{2}D_{33}^{(0)}.\end{array}\right.$ (A.11b)
$\left\\{\begin{array}[]{l}D_{2\times
2x}^{(2)}=-\frac{i}{2}U^{T}\bar{U}D_{2\times
2}^{(1)}-\frac{1}{4}U^{T}\bar{U}_{x}D_{2\times 2}^{(0)}\\\
D_{33x}^{(2)}=i|u|^{2}D_{33}^{(1)}-\frac{1}{4}(|u|^{2})_{x}D_{33}^{(0)}.\end{array}\right.$
(A.11c)

Then from the integral contours $\gamma_{j}$, we can get

$D_{2\times 2}^{(0)}=\mathbb{I}_{2\times 2},\quad D_{33}^{(0)}=1.$ (A.12)

##  Appendix B The asymptotic behavior of $c_{j}(t,k)$

Let

$\mu_{2}(0,t,k)=\left(\begin{array}[]{ll}\Phi_{2\times 2}&\Phi_{j3}\\\
\Phi_{3j}&\Phi_{33}\end{array}\right).$

The global relation shows that

$\Phi_{2\times 2}\frac{s_{j3}}{s_{33}}e^{-8ik^{3}t}+\Phi_{j3}=c_{j}.$ (B.1)

And from equation

$\mu_{t}+[4ik^{3}\Lambda,\mu]=V_{2}\mu.$

we get

$\begin{array}[]{l}\left(\begin{array}[]{ll}\Phi_{2\times 2}&\Phi_{j3}\\\
\Phi_{3j}&\Phi_{33}\end{array}\right)_{t}+4ik^{3}\left(\begin{array}[]{ll}0&2\Phi_{j3}\\\
-2\Phi_{3j}&0\end{array}\right)=4k^{2}\left(\begin{array}[]{ll}U^{T}\Phi_{3j}&U^{T}\Phi_{33}\\\
-\bar{U}\Phi_{2\times 2}&-\bar{U}\Phi_{j3}\end{array}\right)\\\
+2ik\left(\begin{array}[]{ll}U^{T}\bar{U}\Phi_{2\times
2}+U_{x}^{T}\Phi_{3j}&U^{T}\bar{U}\Phi_{j3}+U_{x}^{T}\Phi_{33}\\\
\bar{U}_{x}\Phi_{2\times
2}-2|u|^{2}\Phi_{3j}&-\bar{U}_{x}\Phi_{j3}-2|u|^{2}\Phi_{33}\end{array}\right)-4|u|^{2}\left(\begin{array}[]{ll}U^{T}\Phi_{3j}&U^{T}\Phi_{33}\\\
-\bar{U}\Phi_{2\times 2}&-\bar{U}\Phi_{j3}\end{array}\right)\\\
-\left(\begin{array}[]{ll}U_{xx}^{T}\Phi_{3j}&U_{xx}^{T}\Phi_{33}\\\
-\bar{U}_{xx}\Phi_{2\times
2}&-\bar{U}_{xx}\Phi_{j3}\end{array}\right)+(u\bar{u}_{x}-u_{x}\bar{u})\left(\begin{array}[]{ll}\sigma_{3}\Phi_{2\times
2}&\sigma_{3}\Phi_{j3}\\\ 0&0\end{array}\right).\end{array}$ (B.2)

From the second column of the equation (B.2) we get

$\left\\{\begin{array}[]{l}\begin{array}[]{rl}\Phi_{j3t}+8ik^{3}\Phi_{j3}=&4k^{2}U^{T}\Phi_{33}+2ik(U^{T}\bar{U}\Phi_{j3}+U^{T}_{x}\Phi_{33})\\\
&-4|u|^{2}U^{T}\Phi_{33}-U_{xx}^{T}\Phi_{33}+(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}\Phi_{j3}\end{array}\\\
\Phi_{33t}=-4k^{2}\bar{U}\Phi_{j3}+2ik(\bar{U}_{x}\Phi_{j3}-2|u|^{2}\Phi_{33})+4|u|^{2}\bar{U}\Phi_{j3}+\bar{U}_{xx}\Phi_{j3}.\end{array}\right.$
(B.3)

Suppose

$\left(\begin{array}[]{l}\Phi_{j3}\\\
\Phi_{33}\end{array}\right)=(\alpha_{0}(t)+\frac{\alpha_{1}(t)}{k}+\frac{\alpha_{2}(t)}{k^{2}}+\cdots)+(\beta_{0}(t)+\frac{\beta_{1}(t)}{k}+\frac{\beta_{2}(t)}{k^{2}}+\cdots)e^{-8ik^{3}t}$
(B.4)

where the coefficients $\alpha_{l}(t)$ and $\beta_{l}(t)$, $l\geq 0$, are
independent of $k$. To determine these coefficients,we substitute the above
equation into equation (B.3) and use the initial conditions

$\alpha_{0}(0)+\beta_{0}(0)=(0_{1\times
2},1)^{T},\quad\alpha_{1}(0)+\beta_{1}(0)=(0_{1\times 2},0)^{T}.$

Then we get

$\begin{array}[]{rl}\left(\begin{array}[]{l}\Phi_{j3}\\\
\Phi_{33}\end{array}\right)=&\left(\begin{array}[]{l}0_{1\times 2}\\\
1\end{array}\right)+\frac{1}{k}\left(\begin{array}[]{l}\Phi_{j3}^{(1)}\\\
\Phi_{33}^{(1)}\end{array}\right)+\frac{1}{k^{2}}\left(\begin{array}[]{l}\Phi_{j3}^{(2)}\\\
\Phi_{33}^{(2)}\end{array}\right)+\cdots\\\
&+\left[-\frac{1}{k}\left(\begin{array}[]{l}\Phi_{j3}^{(1)}(0)\\\
0\end{array}\right)+\cdots\right]e^{-8ik^{3}t}\end{array}$ (B.5)

From the first column of the equation (B.2) we get

$\left\\{\begin{array}[]{l}\begin{array}[]{rl}\Phi_{2\times
2t}=&4k^{2}U^{T}\Phi_{3j}+2ik(U^{T}\bar{U}\Phi_{2\times
2}+U_{x}^{T}\Phi_{3j})\\\
&-4|u|^{2}U^{T}\Phi_{3j}-U_{xx}^{T}\Phi_{3j}+(u\bar{u}_{x}-u_{x}\bar{u})\sigma_{3}\Phi_{2\times
2}\end{array}\\\ \Phi_{3jt}-8ik^{3}\Phi_{3j}=-4k^{2}\bar{U}\Phi_{2\times
2}+2ik(\bar{U}_{x}\Phi_{2\times
2}-2|u|^{2}\Phi_{3j})+4|u|^{2}\bar{U}\Phi_{2\times
2}+\bar{U}_{xx}\Phi_{2\times 2}.\end{array}\right.$ (B.6)

Suppose

$\left(\begin{array}[]{l}\Phi_{2\times 2}\\\
\Phi_{3j}\end{array}\right)=(\xi_{0}(t)+\frac{\xi_{1}(t)}{k}+\frac{\xi_{2}(t)}{k^{2}}+\cdots)+(\nu_{0}(t)+\frac{\nu_{1}(t)}{k}+\frac{\nu_{2}(t)}{k^{2}}+\cdots)e^{8ik^{3}t}$
(B.7)

where the coefficients $\xi_{l}(t)$ and $\nu_{l}(t)$, $l\geq 0$, are
independent of $k$. To determine these coefficients,we substitute the above
equation into equation (B.6) and use the initial conditions

$\xi_{0}(0)+\nu_{0}(0)=(\mathbb{I}_{2\times 2},0_{2\times 1})^{T},$

Then we get

$\begin{array}[]{rl}\left(\begin{array}[]{l}\Phi_{2\times 2}\\\
\Phi_{3j}\end{array}\right)=&\left(\begin{array}[]{l}\mathbb{I}_{2\times 2}\\\
0_{2\times
1}\end{array}\right)+\frac{1}{k}\left(\begin{array}[]{l}\Phi_{2\times
2}^{(1)}\\\ \Phi_{3j}^{(1)}\end{array}\right)+\cdots\\\
&+\left[\frac{1}{k^{2}}\left(\begin{array}[]{l}0\\\
\nu_{2}^{(2)}\end{array}\right)+\cdots\right]e^{8ik^{3}t}\end{array}$ (B.8)

So, from the equation (B.1) and the asymptotic of $s_{j3}(k)$ and $s_{33}(k)$,
we get the asymptotic behavior of $c_{j}(t,k)$ as $k\rightarrow\infty$,

$c_{j}(t,k)=\frac{\Phi_{j3}^{(1)}}{k}+\frac{\Phi_{j3}^{(2)}}{k^{2}}+\frac{\Phi_{j3}^{(3)}}{k^{3}}+\cdots.$
(B.9)

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