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

#  On the long-time asymptotics of the Camassa-Holm equation in solitonic
regions

Kai XU1 ,Yiling YANG1 and Engui FAN1 Email address<EMAIL_ADDRESS>Email address<EMAIL_ADDRESS>Corresponding author and email
address<EMAIL_ADDRESS>
###### Abstract

We study the long time asymptotic behavior for the Cauchy problem of the
Camassa-Holm (CH) equation with the appropriate Sobolev initial data. in the
solitonic regions. Our main technical tool is the $\overline{\partial}$
generalization of Deift-Zhou steepest descent method. Through introducing a
new scale $(y,t)$ and constructing the RH problem, we derive different long
time asymptotic expansion of the solution $u(y,t)$ in different space-time
solitonic regions of $\xi=y/t$. We divide the half-plane
$\\{(y,t):-\infty<y<\infty,\ t>0\\}$ into two asymptotic regions: 1\. phase
point absent region: $\xi\in(-\infty,-1/4)\cup(2,+\infty)$, corresponding
asymptotic approximations can be characterized with an $N(\Lambda)$-solitons
with diverse residual error order $\mathcal{O}(t^{-1+2\rho})$; 2. phase points
region: $\xi\in(0,2)$ and $\xi\in(-1/4,0)$. The corresponding asymptotic
approximations can be characterized with an $N(\Lambda)$-soliton and an
interaction term between soliton solutions and the dispersion term with
diverse residual error order $\mathcal{O}(t^{-3/4})$. Our results also confirm
the soliton resolution conjecture and asymptotically stability of the
N-soliton solutions for the CH equation.
Keywords: Camassa-Holm equation, Riemann-Hilbert problem,
$\overline{\partial}$ steepest descent method, long time asymptotics,
asymptotic stability, soliton resolution.
MSC 2020: 35Q51; 35Q15; 37K15; 35C20.

###### Contents

  1. 1 Introduction
  2. 2 Spectral analysis and the RH problem
    1. 2.1 Spectral analysis on the Lax pair
    2. 2.2 A RH formalism with Cauchy value problem
    3. 2.3 Deformation of the RH problem
  3. 3 Long-time asymptotic without phase point
    1. 3.1 Deformation of the RH problem
      1. 3.1.1 Conjugation
      2. 3.1.2 A mixed $\bar{\partial}$-RH problem
    2. 3.2 Analysis on a pure RH problem
      1. 3.2.1 Classify the poles
      2. 3.2.2 Soliton solutions
      3. 3.2.3 Error estimate between $M_{R}$ and $M^{\Lambda}$
    3. 3.3 Analysis on a pure $\bar{\partial}$ problem
    4. 3.4 Long time asymptotic behaviors
  4. 4 Long-time asymptotic with phase points

##  1 Introduction

In this paper, we study the long time asymptotic behavior for the Cauchy
problem of the Camassa-Holm (CH) equation in the solitonic regions

$\displaystyle u_{t}-u_{txx}+2\omega u_{x}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx},$
(1.1) $\displaystyle u(x,0)=u_{0}(x),$ (1.2)

where $\omega$ is a positive constant. The CH equation (1.1) derived as a
model for unidirectional propagation of small amplitude shallow water waves by
Camassa and Holm [1], but already appeared earlier in a list by Fuchssteiner
and Fokas [2]. It also arises in the study of the propagation of axially
symmetric waves in hyperelastic rods [3, 4] and its high-frequency limit
models nematic liquid crystals [5, 6]. It has attracted considerable interest
and been studied extensively due to their rich mathematical structure and
remarkable properties. For example, the CH equation (1.1) is an infinite-
dimensional Hamiltonian system that is completely integrable [7, 8]. The CH
equation admits non-smooth peakon-type solitary or periodic traveling wave
solutions and it was shown that these solutions are orbitally stable [9, 10].
It has been shown that the solitary waves of CH equation are smooth if
$\omega>0$ [11, 12, 13, 14] and peaked weak solutions if $\omega=0$ [1, 15,
16, 17] . The stability of peakons and orbital stability of solitary wave
solution for the CH equation were further shown by Constantin [19, 18]. The
global well-posedness for the Cauchy problem of the CH equation (1.1) was
studied [20, 21, 22]. The algebro-geometric quasiperiodic solutions were
constructed by using algebro-geometric method [23, 24]. With the aid of
reciprocal transformation, Darboux transformation and multi-soliton solutions
for the CH equation were given [25]. The inverse spectral transform for the
conservative Camassa-Holm equation was studied [26, 27, 28].

The inverse scattering transform (IST) procedure, as one of the most powerful
tools to investigate solitons of nonlinear integrable models, was first
discovered by Gardner, Green, Kruskal and Miura [29]. The CH equation admits a
Lax pair which allows using the IST method to study its initial value problem
with $\omega>0$ [8, 30, 31, 32]. The related Riemann-Hilbert (RH) problem are
further studied by Boutet de Monvel [33, 34].

In general, the initial value problems of integrable systems can be solved by
suing inverse scattering transform or RH method only in the case of
refectioness potentials. So it is a natural idea to study the asymptotic
behavior of solutions to integrable systems. The nonlinear steepest descent
method [35] developed by Deift and Zhou a very effective tool to rigorously
obtain the long-time asymptotics behavior of systems [36, 37, 38, 39, 40, 41,
42, 43, 44, 45]. In recent years, Boutet de Monvel et al first developed a RH
and Deift and Zhou steepest descent method to study long-time asymptotics for
CH equation on the line and half-line with Schwartz initial data [46, 47].
They further found Painleve-type asymptotics for Camassa-Holm equation [48].
Minakov obtained long-time for the Camassa-Holm equation with step-like
initial data [49].

In our present paper, we study the long time asymptotic behavior for the
initial value problem of the CH equation (1.1)-(1.2). It is known that if
$m(x,0)+\omega>0$ for all x, where $m:=u-u_{xx}$, then $m(x,t)$ exists for all
t>0; moreover, $m(x,t)+\omega>0$, which justifies the equivalent form of the
CH equation

$\displaystyle(\sqrt{m+\omega})_{t}=-(u\sqrt{m+\omega})_{x},$ (1.3)

Different from the above results [46, 47, 48, 49], in our paper, we remove the
restrictions on the Schwartz initial data by allowing presence of discrete
spectrum and also allowing the weighted Sobolev initial data $u_{0}(x)\in
H^{4,2}(\mathbb{R}),$ where

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

We present different leading order asymptotic approximation for the CH
equation (1.1) in different space-time solitonic regions (see Figure 1 )

$\displaystyle
u(x,t)=u^{r}(x,t;\tilde{\mathcal{D}})+\mathcal{O}(t^{-1+2\rho}),\ \ {\rm for}\
\ \xi\in(-\infty,-1/4)\cup(2,+\infty),$ (1.4)

and

$\displaystyle
u(x,t)=u^{r}(x,t;\tilde{\mathcal{D}})+f_{11}t^{-1/2}+\mathcal{O}(t^{-3/4}),\ \
{\rm for}\ \ \xi\in(-1/4,2).$ (1.5)

Our main technical tool is the representation of the Cauchy problem
(1.1)-(1.2) with an associated matrix RH problem and the consequent asymptotic
analysis of this RH problem via the $\overline{\partial}$ generalization of
Deift-Zhou steepest descent method. This method was a $\bar{\partial}$
generalization to the Deift-Zhou steepest descent method proposed by
McLaughlin and Miller to analyze asymptotic of orthogonal polynomials with
non-analytical weights [51, 52]. In recent years this method has been
successfully used to investigate longtime asymptotics, soliton resolution and
symptotic stability of N-soliton solutions to integrable systems in a weighted
Sobolev space [53, 54, 55, 56, 57, 58, 59].

$0$yt$u^{r}+f_{11}t^{-1/2}+\mathcal{O}(t^{-3/4})$$u^{r}+\mathcal{O}(t^{-1+2\rho})$$u^{r}+\mathcal{O}(t^{-1+2\rho})$$\xi=-1/4$$\xi=2$
Figure 1: Asymptotic approximations of the CH equation in different space-time
solitonic regions, where without stationary phase points in yellow region;
four and eight stationary phase points in green and blue regions respectively.

Our paper is arranged as follows. In section 2, we recall some main results on
the construction process of RH problem [46, 47], which will be used to analyze
long-time asymptotics of the CH equation in our paper. In section 3, we
present long-time asymptotics for the CH equation in the regions in which the
jump contours admit no phase points. In this case, the main contribution to
the asymptotic expansion comes from discrete spectrum and a
$\bar{\partial}$-equation. In section 4, we present long-time asymptotics for
the CH equation in the regions in which there are two and four phase points on
the jump contours. Our results also confirm the soliton resolution conjecture
and asymptotically stability of the N-soliton solutions for the CH equation.

##  2 Spectral analysis and the RH problem

###  2.1 Spectral analysis on the Lax pair

Without loss of generality, we fix $\omega=1$ in the CH equation (1.1) since
the result for the case $\omega=1$ works for any $\omega>0$ if we replace m by
$\frac{m}{\omega}$and $\lambda$ by $\lambda\omega$. The CH equation (1.1) is
completely integrable and admits the Lax pair as follow:

$\Psi_{x}=X\Psi,\hskip 14.22636pt\Psi_{t}=T\Psi,$ (2.1)

where

$X=\left(\begin{array}[]{cc}0&1\\\
\frac{1}{4}+\lambda(m+1)&0\end{array}\right),$
$T=\frac{1}{2}u_{x}\sigma_{3}+\left(\begin{array}[]{cc}0&\frac{1}{2\lambda}-u\\\
(\frac{1}{2\lambda}-u)(\frac{1}{4}+\lambda(m+1))+\frac{1}{2}u_{xx}&0\end{array}\right),$

The $X$ and $T$ in above Lax pair (2.1) are traceless matrixes, which implies
that $\det\Psi(x,t)$ is independent of $x$ and $t$ according to the Abel
theorem.

Usually we use the $x$-part of Lax pair to analyze the initial value problem,
while the $t$-part is used to determine the time evolution of the scattering
data by inverse scattering transform method. Here, different from NLS and
derivative NLS equations, the Lax pair (2.1) for the CH equation has
singularities at $\lambda=0,\lambda=\infty$ , so the asymptotic behavior of
their eigenfunctions should be controlled. The asymptotic behaviors of Lax
pair (2.1) can not be directly obtained as $\lambda\to\infty$. This difficulty
is solved by introducing an appropriate transformation due to Boutet de Monvel
and Shepelsky. Following this idea, we need to use different transformations
to analyze these singularities $\lambda=0$ and $\lambda=\infty$ respectively,
and give a new scale to construct RH problem.

Case I: $\lambda=\infty$, $m+1>0$

Let $u(x,t)$ be a solution of (1.1)with $\omega=1$ such that
$u(x,t)\rightarrow 0$ as $\lvert x\rvert\rightarrow\infty$ for all t.
Introducing

$\displaystyle k^{2}:=-\lambda-\frac{1}{4},\quad
Y=\left(\begin{array}[]{cc}1&1\\\
-ik&ik\end{array}\right),\quad\widetilde{\Phi}:=Y^{-1}\left(\begin{array}[]{cc}(m+1)^{\frac{1}{4}}&0\\\
0&(m+1)^{\frac{-1}{4}}\end{array}\right)\Psi$ (2.6)

we change the Lax pair (2.1) into

$\displaystyle\widetilde{\Phi}_{x}=-ik\sqrt{m+1}\sigma_{3}\widetilde{\Phi}+U\widetilde{\Phi},$
(2.7)
$\displaystyle\widetilde{\Phi}_{t}=-ik(\frac{1}{2\lambda}-u\sqrt{m+1})\sigma_{3}\widetilde{\Phi}+V\widetilde{\Phi},$
(2.8)

where

$\displaystyle
U(x,t,k)=\frac{1}{4}\frac{m_{x}}{m+1}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)-\frac{1}{8ik}\frac{m}{\sqrt{m+1}}\left(\begin{array}[]{cc}-1&-1\\\
1&1\end{array}\right),$ (2.13) $\displaystyle
V(x,t,k)=-uU+\frac{ik}{4\lambda}\sqrt{m+1}\left(\begin{array}[]{cc}-1&1\\\
-1&1\end{array}\right)+(\frac{u}{4ik}+\frac{ik}{4\lambda})\frac{1}{\sqrt{m+1}}\left(\begin{array}[]{cc}-1&-1\\\
1&1\end{array}\right)+\frac{ik}{2\lambda}\sigma_{3}$ (2.18)

Eqs.(2.7),(2.8) suggest us to define a scalar fuction $p(x,t,k)$ such that
$p_{x}=\sqrt{m+1}$ and $p_{t}=\frac{1}{2\lambda}-u\sqrt{m+1}$; due to
(1.3),this is possible, so we define $p$ by

$p(x,t,k)=x-\int_{x}^{\infty}(\sqrt{m(\xi,t)+1}-1)d\xi+\frac{t}{2\lambda(k)},$
(2.19)

By making a transformation

$\Phi(x,t,k):=\widetilde{\Phi}e^{ikp(x,t,k)\sigma_{3}},$ (2.20)

(2.7)-(2.8) becomes

$\displaystyle\Phi_{x}=-ikp_{x}[\sigma_{3},\Phi]+U\Phi,$ (2.21)
$\displaystyle\Phi_{t}=-ikp_{t}[\sigma_{3},\Phi]+V\Phi,$ (2.22)

In addition, the transformation above also implies that

$\displaystyle\Phi(x,t,k)\sim I,\hskip 14.22636ptx\rightarrow\pm\infty.$

The Lax pair (2.21)-(2.22) can be written into a total differential form

$d\left(e^{ikp\hat{\sigma}_{3}}\Phi\right)=e^{ikp\hat{\sigma}_{3}}[\left(Udx+Vdt\right)\Phi],$
(2.23)

which leads to two Volterra type integrals

$\Phi_{\pm}(x,t,k)=I+\int^{x}_{\pm\infty}e^{-ik(p(x)-p(y))\hat{\sigma}_{3}}(U\Phi_{\pm})(y,t,k)dy.$
(2.24)

Denote

$\Phi_{\pm}(x,t,k)=\left(\Phi_{\pm}^{(1)},\Phi_{\pm}^{(2)}\right),$

where $\Phi_{\pm}^{(1)}$ and $\Phi_{\pm}^{(2)}$ are the first and second
columns of $\Phi_{\pm}(x,t,k)$, respectively. Then from the Volterra integral
equation (2.24), we can show that $\Phi_{-}^{(1)}$ and $\Phi_{+}^{(2)}$ are
analytical in $\mathbb{C}^{+}$; $\Phi_{+}^{(1)}$ and $\Phi_{-}^{(2)}$ are
analytical in $\mathbb{C}^{-}$(see yellow and white domains in Figure2).
Moreover, as
$k\rightarrow\pm\infty$,$(\Phi_{-}^{(1)},\Phi_{+}^{(2)})\rightarrow I$ and
$(\Phi_{+}^{(1)},\Phi_{-}^{(2)})\rightarrow I$.

Next,we show the reduction conditions of the Jost functions:

######  Proposition 1.

The Jost functions $\Phi_{\pm}(k)$ admit two kinds of reduction conditions

$\Phi(x,k)=\overline{\Phi(x,-\bar{k})}.$ (2.25)

and

$\Phi^{(1)}(x,-k)=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)\Phi^{(2)}(x,k)=\overline{\Phi^{(1)}(x,\bar{k})},$ (2.26)

Since $\widetilde{\Phi}_{\pm}$ are two fundamental matrix solutions of the Lax
pair (2.7), there exists a linear relation between $\widetilde{\Phi}_{+}$ and
$\widetilde{\Phi}_{-}$, namely

$\widetilde{\Phi}_{+}(x,t,k)=\widetilde{\Phi}_{-}(x,t,k)S(k),$ (2.27)

Combing with the transformation applied to $\widetilde{\Phi}$ , the equation
(2.27) is changed to

$\Phi_{+}(x,t,k)=\Phi_{-}(x,t,k)\mathrm{e}^{-\mathrm{i}kp(x,t,k)\hat{\sigma}_{3}}S(k),\quad
k\in\mathbb{R},k\neq 0$ (2.28)

where S(k) is called scattering matrix

$S(k)=\left(\begin{array}[]{ll}\overline{a(\bar{k})}&b(k)\\\
\overline{b(\bar{k})}&a(k)\end{array}\right)=\left(\begin{array}[]{ll}a(-k)&b(k)\\\
b(-k)&a(k)\end{array}\right)$ (2.29)

We define the reflection coefficients by

$r(z)=\frac{b(z)}{a(z)},$ (2.30)

which admits symmetry reductions

$r(k)=\overline{r(-\bar{k})}.$

######  Proposition 2.

If the initial data $u_{0}(x)\in H^{4,2}(\mathbb{R})$, then corresponding
reflection coefficient $r(z)\in H^{1,1}(\mathbb{R})$.

From (2.28), $a(k)$ and $b(k)$ can be expressed by $\Phi_{\pm}(k)$ as

$\displaystyle
a(k)=\Phi_{-}^{11}\Phi_{+}^{22}-\Phi_{+}^{12}\Phi_{-}^{21},\quad\quad
b(k)=(\Phi_{+}^{12}\Phi_{-}^{22}-\Phi_{-}^{12}\Phi_{+}^{22})e^{2ikp}.$ (2.31)

So $a(k)$ is analytic on $\mathbb{C}^{+}$. In addition, $\Phi_{\pm}(k)$ admit
the asymptotics

$\displaystyle\Phi_{\pm}(k)=I+\dfrac{D_{1}}{k}+\mathcal{O}(k^{-2}),\hskip
14.22636ptk\rightarrow\infty,$ (2.32)

From (2.31) and (2.32), we obtain the asymptotic of $a(k)$

$\displaystyle a(k)=1+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty.$ (2.33)

It has been proved in […]that the zeros of $a(k)$ in the upper half-plane lie
on the imaginary axis and all zeros of $a(k)$ in the upper half-plane are
simple zeros. Suppose that $a(k)$ has $N$ simple zeros $k_{1},...,k_{N}$ on
$\\{k|k\in\mathbb{C}^{+}\\}$.The symmetries (2.29) imply that

$a(k_{n})=0\Leftrightarrow a\left(-k_{n}\right)=0,\hskip 14.22636ptn=1,...,N,$

Therefore, the discrete spectrum is

$\mathcal{Z}=\left\\{k_{n},-k_{n},\right\\}_{n=1}^{N},$ (2.34)

with $k_{n}\in\mathbb{C}^{+}$ and $-k_{n}\in\mathbb{C}^{-}$. And the
distribution of $\mathcal{Z}$ on the $k$-plane is shown in Figure 2.

Re$z$Im$z$$\mathbb{C}^{+}$$\mathbb{C}^{-}$$k_{n}$$-k_{n}$ Figure 2: Analytical
domains and distribution of the discrete spectrum $\mathcal{Z}$.
$\Phi_{-}^{(1)},\Phi_{+}^{(2)},a(z)$ are analytical in $\mathbb{C}^{+}$
(yellow domain); $\Phi_{+}^{(1)},\Phi_{-}^{(2)}$ are analytical in
$\mathbb{C}^{-}$ (white domain);

Case II: $\lambda=0$, $m+1>0$

Define a new transformation

$\Phi^{0}(k)=Y^{-1}\Psi e^{i(kx+\frac{k}{2\lambda}t)\sigma_{3}},$ (2.35)

then

$\Phi^{0}_{\pm}(k)\sim I,\hskip 14.22636ptx\rightarrow\pm\infty,$

and the Lax pair (2.1) change to

$\displaystyle(\Phi^{0})_{x}=-ik[\sigma_{3},\Phi^{0}]+U^{0}\Phi^{0},$ (2.36)
$\displaystyle(\Phi^{0})_{t}=\frac{-ik}{2\lambda}[\sigma_{3},\Phi^{0}]+V^{0}\Phi^{0},$
(2.37)

where

$\displaystyle U^{0}=\frac{-\lambda m}{2ik}\sigma_{3}+\frac{\lambda
m}{2ik}\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right),$ (2.40)
$\displaystyle V^{0}=u(ik\sigma_{3}-\frac{\lambda
m}{2ik}\left(\begin{array}[]{cc}-1&-1\\\
1&1\end{array}\right))+\frac{m}{4ik}\left(\begin{array}[]{cc}-1&-1\\\
1&1\end{array}\right)$ (2.45)
$\displaystyle+\frac{1}{2}u_{x}\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right)-\frac{u_{xx}}{4ik}\left(\begin{array}[]{cc}1&1\\\
-1&-1\end{array}\right).$ (2.50)

Consider asymptotic expansion as $k\rightarrow\frac{i}{2}$,

$\displaystyle\Phi^{0}=I+\Phi_{1}^{0}(k-\frac{i}{2})+\mathcal{O}\left((k-\frac{i}{2})^{2}\right),$
(2.51)

where

$\Phi^{0}_{1}=\left(\begin{array}[]{cc}0&i(u-u_{x})\\\
i(u+u_{x})&0\end{array}\right).$

Since $\Phi(k)$ and $\Phi^{0}(k)$ are derived from the same Lax pair(2.1),
they are linearly dependent. The relations (2.20)and (2.35) lead to

$\Phi_{\pm}(k):=F(x,t)\Phi^{0}_{\pm}(k)e^{ik(\int_{\pm\infty}^{x}\sqrt{m+1}-1d\xi)\sigma_{3}},$
(2.52)

where

$\displaystyle
F(x,t)=\frac{1}{2}\left(\begin{array}[]{cc}(m+1)^{\frac{1}{4}}+(m+1)^{-\frac{1}{4}}&(m+1)^{\frac{1}{4}}-(m+1)^{-\frac{1}{4}}\\\
(m+1)^{\frac{1}{4}}-(m+1)^{-\frac{1}{4}}&(m+1)^{\frac{1}{4}}+(m+1)^{-\frac{1}{4}}\end{array}\right).$
(2.55)

###  2.2 A RH formalism with Cauchy value problem

The residue conditions are

$\displaystyle\operatorname*{Res}_{k=k_{n}}\left[\frac{\Phi_{+}^{(2)}(k)}{a(k)}\right]=c_{n}e^{-2ik_{n}p(k_{n})}\Phi_{-}^{(1)}(k_{n}),$
(2.56)
$\displaystyle\operatorname*{Res}_{k=-k_{n}}\left[\frac{\Phi_{+}^{(1)}(k)}{a(-k)}\right]=c_{n}e^{-2ik_{n}p(-k_{n})}\Phi_{-}^{(2)}(-k_{n}),$
(2.57)

where $c_{n}=b(k_{n})/a^{\prime}(k_{n}),\ n=1,\cdots,N$ norming constants, and
the collection $\sigma_{d}=\left\\{k_{n},-k_{n},c_{n}\right\\}^{N}_{n=1}$ is
called the scattering data. The solution of CH equation (1.1) is difficult to
reconstruct, since $p(x,t,k)$ is still unknown. Boutet de Monvel and Shepelsky
proposed an idea to change the spatial variable [46, 47]. Following this idea,
we introduce a new scale

$y(x,t)=x-\int_{x}^{+\infty}\left(\sqrt{m+1}-1\right)d\xi.$ (2.58)

The price to pay for this is that the solution of the initial problem can be
given only implicitly, or perimetrically: it will be given in terms of
functions in the new scale, whereas the original scale will also be given in
terms of functions in the new scale. By the definition of the new scale
$y(x,t)$, we define

$M(k)=M(k;y,t)\triangleq\left\\{\begin{array}[]{ll}\left(\Phi_{-}^{(1)}(k;y,t),\frac{\Phi_{+}^{(2)}(k;y,t)}{a(k)}\right),&\text{as
}k\in\mathbb{C}^{+},\\\\[12.0pt]
\left(\frac{\Phi_{+}^{(1)}(k;y,t)}{a(-k)},\Phi_{-}^{(2)}(k;y,t)\right),&\text{as
}k\in\mathbb{C}^{-},\\\ \end{array}\right.,$ (2.59)

Denote the phase function

$\theta(k)=k\left[\frac{y}{t}+\frac{1}{2\lambda(k)}\right],$ (2.60)

then we can get the RH problem for the new variable $(y,t)$.

######  RHP 1.

Find a matrix-valued function $M(k)\triangleq M(k;y,t)$ which satisfies:

$\blacktriangleright$ Analyticity: $M(k)$ is meromorphic in
$\mathbb{C}\setminus\mathbb{R}$ and has single poles;

$\blacktriangleright$ Symmetry: $M(k)=\overline{M(-\bar{k})}$;

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

$M_{+}(k)=M_{-}(k)V(k),\hskip 14.22636ptk\in\mathbb{R},$ (2.61)

where

$V(k)=\left(\begin{array}[]{cc}1&r(k)e^{-2it\theta}\\\
-\overline{r(k)}e^{2it\theta}&1-|r(k)|^{2}\end{array}\right);$ (2.62)

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M(k)=I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty,$ (2.63) $\displaystyle
M(k)=F\left[I+\Phi_{1}^{0}(k-\frac{i}{2})\right]e^{c\sigma_{3}}+\mathcal{O}\left((k-\frac{i}{2})^{2}\right);$
(2.64)

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

$\displaystyle\operatorname*{Res}_{k=k_{n}}M(k)=\lim_{k\to
k_{n}}M(k)\left(\begin{array}[]{cc}0&c_{n}e^{-2it\theta(k_{n})}\\\
0&0\end{array}\right),$ (2.67)
$\displaystyle\operatorname*{Res}_{k=-k_{n}}M(k)=\lim_{k\to-
k_{n}}M(k)\left(\begin{array}[]{cc}0&0\\\
c_{n}e^{2it\theta(-k_{n})}&0\end{array}\right).$ (2.70)

To reconstruct the potential $u(x,t)$,we can define two functions:

$\displaystyle\mu_{1}(y,t,k):=\frac{\left(\Phi_{-}^{11}\left(y,t,k\right)+\Phi_{-}^{21}\left(y,t,k\right)\right)}{a\left(k\right)}$
$\displaystyle\mu_{2}(y,t,k):=\Phi_{+}^{12}\left(y,t,k\right)+\Phi_{+}^{22}\left(y,t,k\right)$

Therefore, we get the following formula

$\displaystyle
u(y,t)=\frac{1}{2\mathrm{i}}\lim_{k\rightarrow\frac{\mathrm{i}}{2}}\left(\frac{\mu_{1}(y,t;k)\mu_{2}(y,t;k)}{\mu_{1}(y,t;\frac{\mathrm{i}}{2})\mu_{2}(y,t;\frac{\mathrm{i}}{2})}-1\right)\frac{1}{k-\frac{\mathrm{i}}{2}}.$
(2.71) $\displaystyle
x(y,t)=y+\ln\frac{\mu_{1}\left(y,t;\frac{\mathrm{i}}{2}\right)}{\mu_{2}\left(y,t;\frac{\mathrm{i}}{2}\right)}$
(2.72)

Combining (2.59)and(2.71), the solution $u(x,t)$ of the initial value problem
of the CH equation can be represented by the solution of RHP2 :

$\displaystyle
u(y,t)=\frac{1}{2\mathrm{i}}\lim_{k\rightarrow\frac{\mathrm{i}}{2}}\left(\frac{\left(M_{11}(k)+M_{21}(k)\right)\left(M_{12}(k)+M_{22}(k)\right)}{\left(M_{11}(\frac{i}{2})+M_{21}(\frac{i}{2})\right)\left(M_{12}(\frac{i}{2})+M_{22}(\frac{i}{2})\right)}-1\right)\frac{1}{k-\frac{\mathrm{i}}{2}}.$
(2.73)

###  2.3 Deformation of the RH problem

We note that the jump matrix and residue conditions of RHP 1 contain the
exponential function $e^{\pm 2it\theta}$, which is an oscillatory term in
long-time asymptotics. Therefore we need to control the real part of
$2it\theta$ and decompose the jump matrix by decay region.

$\displaystyle\text{Re}(2it\theta)=-2t\text{Im}\theta$
$\displaystyle=-2t\text{Im}k\left[\xi-\frac{2(\text{Re}^{2}k-\text{Im}^{2}k)+(\frac{1}{2}-4\text{Re}^{2}k)}{(2(\text{Re}^{2}k-\text{Im}^{2}k)+\frac{1}{2})^{2}+(4\text{Re}k\text{Im}k)^{2}}\right],$
(2.74)

where $\xi=\frac{y}{t}$.The signature of $\text{Im}\theta$ are shown in Figure
3, which suggest to divide half-plane $-\infty<y<\infty,\ t>0$ in four space-
time regions:

Case I: $\xi<-1/4$ in Figure 3 (a), Case II: $-1/4<\xi<0$ in Figures 3 (b),

Case III: $0\leq\xi<2$ in Figure 3 (c), Case IV: $\xi>2$ in Figure 3 (d).

Figure 3: The classification of $\text{sign Im}\theta$. In the green regions,
$\text{Im}\theta>0$, which implies that $|e^{2it\theta}|\to 0$ as
$t\to\infty$.While in the white regions, $\text{Im}\theta<0$, which implies
$|e^{-2it\theta}|\to 0$ as $t\to\infty$. The green curves $\text{Im}\theta=0$
are critical lines between decay and growth regions.

For the cases I and IV, there is not any stationary phase point on the real
axis, while for the cases II and III, there exist four and two stationary
phase points on the real axis, denoted as $\xi_{1}>...>\xi_{4}$ and
$\xi_{1}>\xi_{2}$ respectively (see Figure 4 and Figure 5).

$0$yt$0<\xi<2$$-1/4<\xi<0$$\xi>2$$\xi<-1/4$$\xi=-1/4$$\xi=2$ Figure 4: The
half-plane $-\infty<y<\infty,\ t>0$ is divided into four regions: The phase
function $\theta(k)$ has no phase point on the real axis for the yellow
regions;The phase function $\theta(k)$ has two phase points on the real axis
for green region and four phase points on the real axis for the blue region.

Re$z$Im$z$$0$$\xi_{4}$$\xi_{3}$$\xi_{1}$$\xi_{2}$

Re$z$Im$z$$0$$\xi_{1}$$\xi_{2}$

Figure 5: Figure (a) and (b) are corresponding to the $-1/4<\xi<0$ and
$0<\xi<2$ respectively. In (a), there are four stationary phase points
$\xi_{1},...,\xi_{4}$ . And in (b), there are two stationary phase points
$\xi_{1},\xi_{2}$ .

Note that the jump matrix in RHP1 also has an item of $e^{2it\theta}$, so we
have to introduce a classical decomposition to make it decay in the target
area.

$\displaystyle V(z)$ $\displaystyle=\left(\begin{array}[]{cc}1&0\\\
-\bar{r}e^{2it\theta}&1\end{array}\right)\left(\begin{array}[]{cc}1&re^{-2it\theta}\\\
0&1\end{array}\right)$ (2.79)
$\displaystyle=\left(\begin{array}[]{cc}1&\frac{re^{-2it\theta}}{1-|r|^{2}}\\\
0&1\end{array}\right)(1-|r|^{2})^{-\sigma_{3}}\left(\begin{array}[]{cc}1&0\\\
\frac{-\bar{r}e^{2it\theta}}{1-|r|^{2}}&1\end{array}\right).$ (2.84)

We will utilize these factorizations to deform the jump contours so that the
oscillating factor $e^{\pm 2it\theta}$ are decaying in corresponding region
respectively.

##  3 Long-time asymptotic without phase point

###  3.1 Deformation of the RH problem

####  3.1.1 Conjugation

One will observe that the long time asymptotic of RHP1 is influenced by the
growth and decay of the oscillatory term. In this section, our aim is to
introduce a transform of $M(k;x,t)\rightarrow M^{(1)}(k;x,t)$ so that
$M^{(1)}(k;x,t)$ is well behaved as $|t|\rightarrow\infty$ along the
characteristic line.

For $\xi<-\frac{1}{4}$ and $\xi>2$, we choose the decomposition (2.79),(2.84)
respectivly. Through simple calculation, one can get the trace formula.

$a(k)=\prod_{n=1}^{N}\frac{k-k_{n}}{k+k_{n}}\exp\left[{-i\int_{-\infty}^{+\infty}\frac{v(s)}{s-k}ds}\right],$
(3.1)
$a(-k)=\prod_{n=1}^{N}\frac{k+k_{n}}{k-k_{n}}\exp\left[{i\int_{-\infty}^{+\infty}\frac{v(s)}{s-k}ds}\right].$
(3.2)

From (2.31), (2.51) and (2.52):

$a(\frac{i}{2})=e^{-\frac{1}{2}\int_{-\infty}^{+\infty}\sqrt{m+1}-1d\xi},$
(3.3)

Introduce the notation that will be used later.Define

$\displaystyle\Omega_{1}=\\{k|Rek=0,Imk>0\\},$
$\displaystyle\Omega_{1}^{-}=\\{k|k\in\Omega_{1},Im\theta(k)<-\delta\\},\
\Omega_{1}^{+}=\\{k|k\in\Omega_{1},Im\theta(k)>\delta\\},$

and

$\displaystyle\Delta_{1}=\Delta_{1}^{+}\bigcup\Delta_{1}^{-},\
\Lambda_{1}=\\{n\in\\{1,\cdot\cdot\cdot,N\\}||Im\theta(k_{n})|<\delta\\}$
$\displaystyle\Delta_{1}^{+}=\\{n\in\\{1,\cdot\cdot\cdot,N\\}|k_{n}\in\Omega_{1}^{+}\\},\
\Delta_{1}^{-}=\\{n\in\\{1,\cdot\cdot\cdot,N\\}|k_{n}\in\Omega_{1}^{-}\\}.$

The first step in our analysis is to introduce a transformation which
renormalizes the Riemann-Hilbert problem such that it is well conditioned for
$t\rightarrow\infty$ with $\xi$ fixed. In order to arrive at a problem which
is well normalized, Let’s introduce some notation. Denote

$\displaystyle L(\xi)=\left\\{\begin{array}[]{ll}\emptyset,&\text{as
}\xi<-1/4,\\\\[4.0pt] \mathbb{R},&\text{as }\xi>2.\\\ \end{array}\right.$
(3.6)

Here, $\emptyset$ means the integral interval is zero, i.e. for any function
$f(x)$, we make a definition of engagement

$\displaystyle\int_{\emptyset}f(x)dx=0.$

Define the function

$T(k)=\prod_{n\in\Delta_{1}^{+}}\frac{k+k_{n}}{k-k_{n}}\delta(k,\xi).$ (3.7)

where

$\delta(k,\xi)=exp\\{-i\int_{L(\xi)}\frac{v_{1}(s)}{s-k}ds\\}\ \ ;\ \
v_{1}(k)=\frac{1}{2\pi}log(1-|r|^{2}).$

######  Proposition 3.

The function defined by (3.7) has following properties:
(a) $T$ is meromorphic in $\mathbb{C}$, and for each $n\in\Delta_{1}^{+}$,
$T(k)$ has simple poles at $k_{n}$ and simple zeros at $-k_{n}$; $T(k)$ is
analytic and nonzero elsewhere.
(b) $\overline{T(\bar{k})}=\frac{1}{T(k)}$;
(c)$\lim_{k\to\infty}T(k)=1$; $T(k)$ is continuous at $k=0$, and $T(0)=1$;
(d)As k approaching the real axis from above and below, $T$ has boundary
values $T_{\pm}$, which satisfy:

$\displaystyle T_{+}(k)=(1-|r(k)|^{2})T_{-}(k),\hskip 14.22636pt\xi>2;$ (3.8)
$\displaystyle T_{+}(k)=T_{-}(k),\hskip 14.22636pt\xi<-\frac{1}{4}.$ (3.9)

(e)For $k=\frac{i}{2}$

$T(k)=\prod_{n\in\Delta_{1}^{+}}\frac{\frac{i}{2}+k_{n}}{\frac{i}{2}-k_{n}}\delta(\frac{i}{2},\xi).$

(f)As $k\to\frac{i}{2}$, $T(k)$ has asymptotic expansion as

$T(k)=T(\frac{i}{2})J_{0}+\left[T(\frac{i}{2})J_{0}J_{1}+\sum_{j,s=1;j\neq
s}^{\mathcal{N}(\Delta_{1}^{+})}\left(\frac{-2k_{n_{s}}(k_{n_{j}}+\frac{i}{2})}{(\frac{i}{2}-k_{n_{j}})(\frac{i}{2}-k_{n_{s}})^{2}}\right)J_{0}\right](k-\frac{i}{2})+\mathcal{O}((k-\frac{i}{2})^{2}),$
(3.10)

with

$J_{0}=exp\\{-\frac{i}{2\pi}\int_{L(\xi)}\frac{ln(1-|r(s)|^{2})}{s-\frac{i}{2}}ds\\};$
(3.11)
$J_{1}=-\frac{i}{2\pi}\int_{L(\xi)}\frac{ln(1-|r(s)|^{2})}{(s-\frac{i}{2})^{2}}ds\\};$
(3.12)

###### Proof.

Properties (a)-(e) can be obtain by simple calculation from the definition of
$T(k)$ in (3.7). To obtain property (f), it is sufficient to do a Laurent
expansion of $T(k)$ at $\frac{i}{2}$. ∎

We define a new unknown function $M^{(1)}(k)$ using the function we defined
above

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

then $M^{(1)}$ solves the following RH problem:

######  RHP 2.

Find a matrix-valued function $M^{(1)}(k)$ which satisfies:

$\blacktriangleright$ Analyticity: $M^{(1)}(k)$ is analytic in
$\mathbb{C}\setminus(\mathbb{R}\bigcup\mathbb{Z})$ and has single poles;

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

$M_{+}^{(1)}(k)=M_{-}^{(1)}(k)V^{(1)}(k,\xi),\hskip 14.22636ptk\in\mathbb{R},$
(3.13)

where

$V^{(1)}(k,\xi)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1&0\\\
-T^{2}\overline{r}e^{2it\theta}&1\end{array}\right)\left(\begin{array}[]{cc}1&T^{-2}re^{-2it\theta}\\\
0&1\end{array}\right),&\text{as }\xi<-\frac{1}{4},k\in\mathbb{R};\\\\[12.0pt]
\left(\begin{array}[]{cc}1&\frac{r(k)e^{-2it\theta}T_{-}^{-2}(k)}{1-|r(k)|^{2}}\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\
\frac{-\bar{r}(k)e^{2it\theta}T_{+}^{2}(k)}{1-|r(k)|^{2}}&1\end{array}\right),&\text{as
}\xi>2,k\in\mathbb{R};\\\\[12.0pt] \end{array}\right.$ (3.14)

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(1)}(k)=I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty,$ (3.15) $\displaystyle
M^{(1)}(k)=Fe^{c\sigma_{3}}T^{\sigma_{3}}(i)+[Fe^{c\sigma_{3}}I_{0}^{\sigma_{3}}+F\Phi_{1}^{0}e^{c\sigma_{3}}T^{\sigma_{3}}(i)](k-\frac{i}{2})$
$\displaystyle+\mathcal{O}\left((k-\frac{i}{2})^{2}\right),\hskip
39.83368ptk\rightarrow\frac{i}{2}.$ (3.16)

$\blacktriangleright$ Residue conditions: $M^{(1)}(k)$ has simple poles at
each point in $\mathcal{Z}$ with:

$\displaystyle\operatorname*{Res}_{k=k_{n}}M^{(1)}(k)=\left\\{\begin{array}[]{ll}\lim_{k\rightarrow
k_{n}}M^{(1)}(k)\left(\begin{array}[]{cc}0&0\\\
c_{n}^{-1}e^{2it\theta{(k_{n})}}[(\frac{1}{T})^{{}^{\prime}}(k_{n})]^{-2}&0\end{array}\right),&\text{as
}n\in\Delta_{1}^{+};\\\\[12.0pt] \lim_{k\rightarrow
k_{n}}M^{(1)}(k)\left(\begin{array}[]{cc}0&c_{n}T^{-2}e^{-2it\theta{(k_{n})}}\\\
0&0\end{array}\right),&\text{as
}n\in\Delta_{1}^{-}\bigcup\bigwedge;\\\\[12.0pt] \end{array}\right.$ (3.23)
$\displaystyle\operatorname*{Res}_{k=-k_{n}}M^{(1)}(k)=\left\\{\begin{array}[]{ll}\lim_{k\rightarrow-
k_{n}}M^{(1)}(k)\left(\begin{array}[]{cc}0&c_{n}^{-1}e^{-2it\theta{(-k_{n})}}[T^{{}^{\prime}}(-k_{n})]^{-2}\\\
0&0\end{array}\right),&\text{as }n\in\Delta_{1}^{+};\\\\[12.0pt]
\lim_{k\rightarrow-k_{n}}M^{(1)}(k)\left(\begin{array}[]{cc}0&0\\\
c_{n}T^{2}e^{2it\theta{(-k_{n})}}&0\end{array}\right),&\text{as
}n\in\Delta_{1}^{-}\bigcup\bigwedge;\\\\[12.0pt] \end{array}\right.$ (3.30)

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

The next step in our analysis is to introduce factorizations of the jump
matrix whose factors admit continuous-but not necessarily analytic-extensions
off the real axis. The price we pay for this non-analytic transformation is
that the new unknown has nonzero $\bar{\partial}$ derivatives inside the
regions in which the extensions are introduced and satisfies a hybrid
$\bar{\partial}$ Riemann-Hilbert problem.

Define the contour as follow:

$\displaystyle\Upsilon_{2n+1}=\left\\{k\in\mathbb{C}|n\pi\leq\arg k\leq
n\pi+\varphi\right\\},$
$\displaystyle\Upsilon_{2n+2}=\left\\{k\in\mathbb{C}|(n+1)\pi-\varphi\leq\arg
k\leq(n+1)\pi\right\\},$

where $n=0,1$. And

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

which is the boundary of $\Upsilon_{k}$ respectively. Take $\varphi$ be a
sufficiently small fixed angle achieving following conditions:
(i): each $\Upsilon_{i}$ doesn’t intersect $\left\\{k\in\mathbb{C}|\text{Im
}\theta(k)=0\right\\}$ ,
(ii): $\sqrt{2(cos2\varphi+1)}>-1+\sqrt{1-\frac{2}{\xi}}$ for
$\xi<-\frac{1}{4}$;
(iii):$1-\sqrt{1-\frac{2}{\xi}}<\sqrt{2(cos2\varphi+1)}<1+\sqrt{1-\frac{2}{\xi}}$
for $\xi>2$.

A more intuitive expression of the above process can be seen in the figure 6.

$\Sigma_{1}$$\Sigma_{2}$$\Sigma_{3}$$\Sigma_{4}$Re$z$Im$z$$\Upsilon_{1}$$\Upsilon_{3}$$\Upsilon_{2}$$\Upsilon_{4}$$0$$+$$-$$-$$+$$-k_{n}$$k_{n}$
Figure 6: Without stationary phase point corresponds the case $\xi<-1/4$ . The
blue regions $\Upsilon_{1}(\xi)$ and $\Upsilon_{2}(\xi)$ have same decay/grow
properties; The same for yellow regions $\Upsilon_{3}(\xi)$ and
$\Upsilon_{4}(\xi)$.

Define a new unknown function

$R^{(2)}(k,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{cc}1&R_{j}(k,\xi)e^{-2it\theta}\\\
0&1\end{array}\right),&\xi<-\frac{1}{4},k\in\Upsilon_{j},j=1,2;\\\ \\\
\left(\begin{array}[]{cc}1&0\\\
R_{j}(k,\xi)e^{2it\theta}&1\end{array}\right),&\xi<-\frac{1}{4},k\in\Upsilon_{j},j=3,4;\\\
\\\ \left(\begin{array}[]{cc}1&0\\\
R_{j}(k,\xi)e^{2it\theta}&1\end{array}\right),&\xi>2,k\in\Upsilon_{j},j=1,2;\\\
\\\ \left(\begin{array}[]{cc}1&R_{j}(k,\xi)e^{-2it\theta}\\\
0&1\end{array}\right),&\xi>2,k\in\Upsilon_{j},j=3,4;\\\ \\\ I,&elsewhere;\\\
\end{array}\right.$ (3.31)

where the $R_{j}$ are given in following Proposition.

######  Proposition 4.

$R_{j}$: $\bar{\Upsilon}_{j}\to C$, $j=1,2,3,4$ have boundary values as
follow:
For $\xi\in(-\infty,-1/4)$,

$\displaystyle
R_{1}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}-rT^{-2}&k\in\mathbb{R}^{+},\\\
0&k\in\Sigma_{1},\\\ \end{array},\hskip
17.07182ptR_{2}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}0&k\in\Sigma_{2},\\\
-rT^{-2}&k\in\mathbb{R}^{-},\\\ \end{array}$ (3.36) $\displaystyle
R_{3}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}-\bar{r}T^{2}&k\in\mathbb{R}^{-},\\\
0&k\in\Sigma_{3},\\\ \end{array},\hskip
17.07182ptR_{4}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}0&k\in\Sigma_{4},\\\
-\bar{r}T^{2}&k\in\mathbb{R}^{+},\\\ \end{array}$ (3.41)

$R_{j}$ have following properties: for $j=1,2,3,4,$

$\displaystyle|\bar{\partial}R_{j}(k)|\lesssim|r^{\prime}(|k|)|+|k|^{-1/2},\
\text{for all $k\in\Upsilon_{j}$, j=1,2}$ (3.42)
$\displaystyle|\bar{\partial}R_{j}(k)|\lesssim|\bar{r}{{}^{\prime}}(|k|)|+|k|^{-1/2},\
\text{for all $k\in\Upsilon_{j}$, j=3,4}$ (3.43)

moreover,

$\displaystyle|\bar{\partial}R_{j}(k)|\lesssim|r^{\prime}(|k|)|+|k|^{-1},\
\text{for all $k\in\Upsilon_{j}$, j=1,2}$ (3.44)
$\displaystyle|\bar{\partial}R_{j}(k)|\lesssim|\bar{r}{{}^{\prime}}(|k|)|+|k|^{-1},\
\text{for all $k\in\Upsilon_{j}$, j=3,4}$ (3.45)
$\displaystyle\bar{\partial}R_{j}(k)=0,\hskip 14.22636pt\text{if }k\in
elsewhere.$

For $\xi\in(2,+\infty)$,

$\displaystyle
R_{1}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}\frac{\bar{r}e^{2it\theta}}{1-|r(k)|^{2}}T_{+}^{2}&k\in\mathbb{R}^{+},\\\
0&k\in\Sigma_{1},\\\ \end{array},\hskip
17.07182ptR_{2}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}0&k\in\Sigma_{2},\\\
\frac{\bar{r}e^{2it\theta}}{1-|r(k)|^{2}}T_{+}^{2}&k\in\mathbb{R}^{-},\\\
\end{array}$ (3.50) $\displaystyle
R_{3}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}\frac{re^{-2it\theta}}{1-|r(k)|^{2}}T_{-}^{-2}&k\in\mathbb{R}^{-},\\\
0&k\in\Sigma_{3},\\\ \end{array},\hskip
17.07182ptR_{4}(k,\xi)=\Bigg{\\{}\begin{array}[]{ll}0&k\in\Sigma_{4},\\\
\frac{re^{-2it\theta}}{1-|r(k)|^{2}}T_{-}^{-2}&k\in\mathbb{R}^{+},\\\
\end{array}$ (3.55)

In this case, $R_{j}$ have the similar properties. Just replace $r$,$\bar{r}$
with
$\frac{\bar{r}e^{2it\theta}}{1-|r(k)|^{2}}$,$\frac{re^{-2it\theta}}{1-|r(k)|^{2}}$
respectively in(3.42)-(3.45).

###### Proof.

The proof is similar with [58]. ∎

Make the matrix transformation:

$\displaystyle M^{(2)}(k)=M^{(1)}(k)R^{(2)}(k),$

then $M^{(2)}$ solves the mixed $\bar{\partial}$-RH problem:

######  RHP 3.

Find a matrix valued function $M^{(2)}(k)$ with following properties:

$\blacktriangleright$ Analyticity: $M^{(2)}(k)$ is continuous in
$\mathbb{C}\backslash\mathbb{Z}$;

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

$M^{(2)}_{+}(k)=M^{(2)}_{-}(k),\hskip 14.22636ptz\in\Sigma^{(2)},$ (3.56)

where $\Sigma^{(2)}=\bigcup\Sigma_{j},j=1,2,3,4$.

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(2)}(k)=I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty,$ (3.57) $\displaystyle
M^{(2)}(k)=Fe^{c\sigma_{3}}T^{\sigma_{3}}(i)+[Fe^{c\sigma_{3}}I_{0}^{\sigma_{3}}+F\Phi_{1}^{0}e^{c\sigma_{3}}T^{\sigma_{3}}(i)](k-\frac{i}{2})$
$\displaystyle+\mathcal{O}\left((k-\frac{i}{2})^{2}\right),\hskip
39.83368ptk\rightarrow\frac{i}{2}.$ (3.58)

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

$\displaystyle\bar{\partial}M^{(2)}(k)=M^{(2)}(k)\bar{\partial}R^{(2)}(k),$
(3.59)

where

$\bar{\partial}R^{(2)}(k,\xi)=\left\\{\begin{array}[]{lll}\left(\begin{array}[]{cc}0&\bar{\partial}R_{j}(k,\xi)e^{-2it\theta}\\\
0&0\end{array}\right),&\xi<-\frac{1}{4},k\in\Upsilon_{j},j=1,2;\\\ \\\
\left(\begin{array}[]{cc}0&0\\\
\bar{\partial}R_{j}(k,\xi)e^{2it\theta}&0\end{array}\right),&\xi<-\frac{1}{4},k\in\Upsilon_{j},j=3,4;\\\
\\\ \left(\begin{array}[]{cc}0&0\\\
\bar{\partial}R_{j}(k,\xi)e^{2it\theta}&0\end{array}\right),&\xi>2,k\in\Upsilon_{j},j=1,2;\\\
\\\ \left(\begin{array}[]{cc}0&\bar{\partial}R_{j}(k,\xi)e^{-2it\theta}\\\
0&0\end{array}\right),&\xi>2,k\in\Upsilon_{j},j=3,4;\\\ \\\ 0,&elsewhere;\\\
\end{array}\right.$ (3.60)

$\blacktriangleright$ Residue conditions: $M^{(2)}$(k) has the same residue
conditions with the RHP2.

In order to solve the RHP3, we decompose it into a pure RH problem
$M^{(R)}(k)$ with $\bar{\partial}R^{(2)}\equiv 0$ and a pure $\bar{\partial}$
problem $M^{(3)}(k)$ with $\bar{\partial}R^{(2)}\not=0$, that is,

$M^{(2)}(k)=M^{(3)}(k)M^{(R)}(k).$ (3.61)

Since the fact that Case I has no stationary point as well as the jump matrix
in $\Sigma^{(2)}$ equals to I, the result of $M^{(R)}$ is only contributed by
its poles and asymptotic behaviors.

The matrix function $M^{(R)}$ satisfies RHP5.

######  RHP 4.

Find a matrix valued function $M^{(R)}(k)$ with following properties:

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

$\blacktriangleright$ Jump condition:

$M^{(R)}_{+}(k)=M^{(R)}_{-}(k);$

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(R)}(k)=I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty;$ (3.62) $\displaystyle
M^{(R)}(k)=Fe^{c\sigma_{3}}T^{\sigma_{3}}(i)+[Fe^{c\sigma_{3}}I_{0}^{\sigma_{3}}+F\Phi_{1}^{0}e^{c\sigma_{3}}T^{\sigma_{3}}(i)](k-\frac{i}{2})$
$\displaystyle+\mathcal{O}\left((k-\frac{i}{2})^{2}\right),\hskip
39.83368ptk\rightarrow\frac{i}{2}.$ (3.63)

$\blacktriangleright$ Residue conditions: $M^{(R)}$(k) has the same residue
conditions with the RHP2.

###  3.2 Analysis on a pure RH problem

####  3.2.1 Classify the poles

In this section, our aim is to convert the poles away from
$\left\\{k\in\mathbb{C}|\text{Im }\theta(k)=0\right\\}$ into jumps. Take the
transform

$M_{R}(k)=M^{(R)}(k)G(k),$ (3.64)

where

$G(k)=\left\\{\begin{array}[]{ll}\left(\begin{array}[]{cc}1&0\\\
-c_{n}^{-1}T(k)e^{2it\theta(k_{n})}[(\frac{1}{T})^{{}^{\prime}}(k_{n})]^{-1}&1\end{array}\right),&\text{as
}k\in\\{k:|k-k_{n}|=\rho\\},n\in\Delta_{1}^{+};\\\\[12.0pt]
\left(\begin{array}[]{cc}1&c_{n}T^{-2}e^{-2it\theta(k_{n})}(k-k_{n})^{-1}\\\
0&1\end{array}\right),&\text{as
}k\in\\{k:|k-k_{n}|=\rho\\},n\in\Delta_{1}^{-};\\\\[12.0pt]
\left(\begin{array}[]{cc}1&-\bar{c}_{n}^{-1}T^{-1}e^{-2it\theta(-k_{n})}[T^{{}^{\prime}}(-k_{n})]^{-1}\\\
0&1\end{array}\right),&\text{as
}k\in\\{k:|k+k_{n}|=\rho\\},n\in\Delta_{1}^{+};\\\\[12.0pt]
\left(\begin{array}[]{cc}1&0\\\
\frac{c_{n}T^{2}e^{2it\theta(-k_{n})}}{k+k_{n}}&1\end{array}\right),&\text{as
}k\in\\{k:|k+k_{n}|=\rho\\},n\in\Delta_{1}^{-},\\\\[12.0pt] I,&\text{elsewhere
}.\end{array}\right.,$ (3.65)

then $M_{R}$ satisfies the following RH problem:

######  RHP 5.

Find a matrix valued function $M_{R}(k)$ with following properties:

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

$\blacktriangleright$ Jump condition:

$M_{+,R}(k)=M_{-,R}(k)V_{R}(k),\hskip 14.22636ptk\in\Sigma^{(R)};$

where $V_{R}(k)=G(k)$. The $\Sigma^{(R)}$ is the union of small circles in
Figure .

$\blacktriangleright$ $\bar{\partial}$-Derivative:
$\bar{\partial}R^{(2)}(k)=0$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{(R)}(k)=$ $\displaystyle I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty;$

$\blacktriangleright$ Residue conditions:

$\displaystyle\operatorname*{Res}_{k=k_{n}}M_{R}(k)=\lim_{k\rightarrow
k_{n}}M_{R}(k)\left(\begin{array}[]{cc}0&c_{n}T^{-2}e^{-2it\theta{(k_{n})}}\\\
0&0\end{array}\right),$ $\displaystyle\text{as }n\in\bigwedge;$ (3.68)
$\displaystyle\operatorname*{Res}_{k=-k_{n}}M_{R}(k)=\lim_{k\rightarrow-
k_{n}}M_{R}(k)\left(\begin{array}[]{cc}0&0\\\
c_{n}T^{2}e^{2it\theta{(-k_{n})}}&0\end{array}\right),$ $\displaystyle\text{as
}n\in\bigwedge;$ (3.71)

$\Sigma_{1}$$\Sigma_{2}$$\Sigma_{3}$$\Sigma_{4}$Re$z$Im$z$$\Upsilon_{1}$$\Upsilon_{3}$$\Upsilon_{2}$$\Upsilon_{4}$$0$$+$$-$$-$$+$$-k_{n}$$k_{n}$
Figure 7: Jump path of $M_{R}$. The blue circle in the figure is the jump
path.

In order to do the estimation error later, here are some simple parametric
estimates for $V_{R}$.

######  Lemma 1.

The jump matrix $V_{R}(k)$ defined by (3.65) satisfies

$\displaystyle\parallel
V_{R}(k)-I\parallel_{L^{\infty}}=\mathcal{\mathcal{O}}(e^{-2\rho_{0}t}),$
(3.72)

where

$\rho_{0}=\min_{n\in\Delta_{1}\setminus\Lambda_{1}}\left\\{|\text{Im}\theta_{n}|>\delta_{0}\right\\}.$
(3.73)

###### Proof.

Take $k\in\\{k:|k-k_{n}|=\rho,n\in\Delta_{1}^{+}\\}$ as an example.

$\displaystyle\parallel V_{R}(k)-I\parallel_{L^{\infty}}$
$\displaystyle=|c_{n}^{-1}T(k)e^{2it\theta(k_{n})}[(\frac{1}{T})^{{}^{\prime}}(k_{n})]^{-1}|$
$\displaystyle\lesssim e^{-\text{Re}(2it\theta_{n})}\lesssim
e^{2t\text{Im}(\theta_{n})}\leq e^{-2\rho_{0}t}.$

The proofs for the rest of the cases are similar. ∎

######  Corollary 1.

For $1\leq p\leq+\infty$, the jump matrix $V_{R}(k)$ satisfies

$\parallel V_{R}(k)-I\parallel_{L^{p}}\leq C_{p}e^{-2\rho_{0}t},$ (3.74)

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

####  3.2.2 Soliton solutions

In this part, we prove the existence and uniqueness of solution of the above
RHP 5. Meanwhile,we show the solution can be approximated by a soliton
solution $M^{\Lambda}(k)$.

######  Proposition 5.

For a given scattering data
$\mathcal{D}=\\{r(k),\\{k_{n},c_{n}\\}_{n\in\aleph}\\}$, the solution
$M_{R}(k)$ of the above RH problem exists uniquely and is equivalent to the
RHP1 corresponding to the reflection-free N soliton solution of the corrected
scattering data
$\tilde{\mathcal{D}}=\\{\tilde{r}=0,\\{k_{n},\tilde{c}_{n}\\}_{n\in\aleph}\\}$
by a display transformation, where the correlation function
$\tilde{c}_{n}=c_{n}\delta(k_{n})$.

###### Proof.

$M_{R}(k)$ contains a number of jump lines consisting of small circumferences
and poles in $\Lambda$.Take a transformation

$\tilde{M}(k)=M_{R}(k)G^{-1}(k)\left(\prod_{n\in\Delta_{1}^{+}}\frac{k+k_{n}}{k-k_{n}}\right)^{-\sigma_{3}},$
(3.75)

Then the transformation has the following effect:
a. $\tilde{M}(k)$ asymptotic to I when $k\rightarrow\infty$;
b. $\tilde{M}(k)$ has no jump on the small circle circumferences;
c. When $n\not\in\Lambda$, the transformation converts the jump into the
poles; For $n\in\Lambda$,

$\displaystyle\operatorname*{Res}_{k=k_{n}}\tilde{M}(k)=\operatorname*{Res}_{k=k_{n}}M_{R}(k)G^{-1}(k)\left(\prod_{n\in\Delta_{1}^{+}}\frac{k+k_{n}}{k-k_{n}}\right)^{-\sigma_{3}}$
$\displaystyle=$ $\displaystyle\lim_{k\rightarrow
k_{n}}\tilde{M}(k)\left(\prod_{n\in\Delta_{1}^{+}}\frac{k+k_{n}}{k-k_{n}}\right)^{-\sigma_{3}}\left(\begin{array}[]{cc}0&c_{n}T^{-2}e^{-2it\theta{(k_{n})}}\\\
0&0\end{array}\right)G^{-1}(k)\left(\prod_{n\in\Delta_{1}^{+}}\frac{k+k_{n}}{k-k_{n}}\right)^{-\sigma_{3}}$
(3.78) $\displaystyle=$ $\displaystyle\lim_{k\rightarrow
k_{n}}\tilde{M}(k)\left(\begin{array}[]{cc}0&c_{n}\delta(k_{n})e^{-2it\theta{(k_{n})}}\\\
0&0\end{array}\right),$ (3.81)

thus, $\tilde{c}_{n}=c_{n}\delta(k_{n})$;

So far, we have proved that $\tilde{M}(k)$ is the solution of RHP 1 under the
corrected scattering data
$\tilde{\mathcal{D}}=\\{\tilde{r}=0,\\{k_{n},c_{n}\delta(k_{n})\\}_{n\in\aleph}\\}$.
And the solution of this kind of RHP is existentially unique, so $M_{R}(k)$
exists uniquely. ∎

Now we try to match $M_{R}(k)$ to the model RHP $M^{\Lambda}(k)$ and show in
the next subsection that the error between the two is a small parametric RHP.

Suppose $M^{\Lambda}(k)$ is the solution of the RHP corresponding to
$M_{R}(k)$ after removing the jump, that is, $M^{\Lambda}(k)$ satisfies the
RHP below:

######  RHP 6.

Find a matrix valued function $M^{\Lambda}(k)$ with following properties:

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

$\blacktriangleright$ Jump condition:

$M_{+}^{\Lambda}(k)=M_{-}^{\Lambda}(k);$

$\blacktriangleright$ $\bar{\partial}$-Derivative:
$\bar{\partial}R^{(2)}(k)=0$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{\Lambda}(k)=$ $\displaystyle I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty;$

$\blacktriangleright$ Residue conditions:$M^{\Lambda}$(k) has the same residue
conditions with the RHP5.

######  Proposition 6.

For a given corrected scattering data
$\tilde{\mathcal{D}}_{\Lambda}=\\{\tilde{r}=0,\\{k_{n},\tilde{c}_{n}\\},n\in\bigwedge\\}$,
the solution of RHP7 exists uniquely and can be shown to be constructed:
I: if $\Lambda=\varnothing$, then

$M^{\Lambda}(k)=I;$ (3.82)

II: if $\Lambda\neq\varnothing$ with
$\Lambda=\left\\{k_{n}\right\\}_{n=1}^{\Xi}$, where the symbol $\Xi$ denotes
the number of poles in $\bigwedge$, then

$\displaystyle M^{\Lambda}(k)$
$\displaystyle=I+\sum_{n=1}^{\Xi}\left(\begin{array}[]{cc}\frac{\gamma_{n}}{k+k_{n}}&\frac{\beta_{n}}{k-k_{n}}\\\
\frac{\zeta_{n}}{k+k_{n}}&\frac{\alpha_{n}}{k-k_{n}}\end{array}\right),$
(3.85)

where $\alpha_{n},\beta_{n},\gamma_{n},\zeta_{n}$ are determined by linearly
dependant equations:

$\displaystyle
c_{n}^{-1}T^{2}(k_{n})e^{2it\theta{(k_{n})}}\beta_{n}=1+\sum_{h=1}^{\Xi}\frac{\gamma_{h}}{k_{n}+k_{h}},$
(3.86) $\displaystyle
c_{n}^{-1}T^{2}(k_{n})e^{2it\theta{(k_{n})}}\alpha_{n}=\sum_{h=1}^{\Xi}\frac{\zeta_{h}}{k_{n}+k_{h}},$
(3.87) $\displaystyle
c_{n}^{-1}T^{2}(k_{n})e^{2it\theta{(k_{n})}}\gamma_{n}=\sum_{h=1}^{\Xi}\frac{-\beta_{h}}{k_{n}+k_{h}},$
(3.88) $\displaystyle
c_{n}^{-1}T^{2}(k_{n})e^{2it\theta{(k_{n})}}\zeta_{n}=1+\sum_{h=1}^{\Xi}\frac{-\alpha_{h}}{k_{n}+k_{h}}.$
(3.89)

###### Proof.

The uniqueness of solution follows from the Liouville’s theorem. The result of
I can be simple obtain.

As for II, the residue and asymptotic conditions ensure that $M^{\Lambda}(k)$
can have an expansion of the form as mentioned above. And in order to obtain
$\alpha_{n},\beta_{n},\gamma_{n},\zeta_{n}$, we substitute (3.85) into (3.68)
and obtain four linearly dependant equations set above. ∎

For convenience, denote the asymptotic expansion of $M^{\Lambda}(k)$ as
$k\to\frac{i}{2}$:

$\displaystyle
M^{\Lambda}(k)=M^{\Lambda}(\frac{i}{2})+M^{\Lambda}_{1}(k-\frac{i}{2})+\mathcal{O}((k-\frac{i}{2})^{2}).$
(3.90)

####  3.2.3 Error estimate between $M_{R}$ and $M^{\Lambda}$

We define $M^{err}$ as the error between $M_{R}$ and $M^{\Lambda}$. More
specifically,

$M_{R}(k)=M^{err}(k)M^{\Lambda}(k).$ (3.91)

From the definition, we can obtain that $M^{err}(k)$ is a solution of the
following small parametric RHP:

######  RHP 7.

Find a matrix-valued function $M^{err}(k)$ with following identities:

$\blacktriangleright$ Analyticity: $M^{err}(k)$ is analytical in
$\mathbb{C}\setminus\Sigma^{(R)}$;

$\blacktriangleright$ Asymptotic behaviors:

$\displaystyle M^{err}(k)\sim I+\mathcal{O}(k^{-1}),\hskip
14.22636pt|k|\rightarrow\infty;$ (3.92)

$\blacktriangleright$ Jump condition: $M^{err}(k)$ has continuous boundary
values $M^{err}_{\pm}(k)$ on $\Sigma^{(R)}$ satisfying

$M^{err}_{+}(k)=M^{err}_{-}(k)V^{err}(k),$

where the jump matrix $V^{err}(k)$ is given by

$V^{err}(k)=M^{\Lambda}(k)V_{R}(k)M^{\Lambda}(k)^{-1}.$ (3.93)

######  Lemma 2.

The jump matrix $V^{err}(k)$ satisfies

$\displaystyle\parallel
V^{err}(k)-I\parallel_{L^{\infty}}=\mathcal{\mathcal{O}}(e^{-2\rho_{0}t}),$
(3.94)

###### Proof.

Take $k\in\\{k:|k-k_{n}|=\rho,n\in\Delta_{1}^{+}\\}$ as an example.

$\displaystyle|V_{R}(k)-I|=|M^{\Lambda}(k)(V_{R}-I)M^{\Lambda}(k)^{-1}|\leq
c|V_{R}-I|\leq e^{-2\rho_{0}t}.$

The proofs for the rest of the cases are similar. ∎

######  Corollary 2.

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

$\parallel V^{err}(k)-I\parallel_{L^{p}}\leq C_{p}e^{-2\rho_{0}t},$ (3.95)

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

Constructing the solution of the RHP 7 according to Beal-Coifman’s theorem.
Consider the mundane decomposition of the jump matrix $V^{err}(k)$

$\displaystyle V^{err}=(b_{-})^{-1}b_{+},\ \ \ b_{-}=I,\ \ \ b_{+}=V^{err},$

thus

$\displaystyle(\omega_{e})_{-}=I-b_{-},\ \ (\omega_{e})_{+}=b_{+}-I,\ \
\omega_{e}=(\omega_{e})_{+}+(\omega_{e})_{-}$ $\displaystyle
C_{\omega_{e}}f=C_{-}(f(\omega_{e})_{+})+C_{+}(f(\omega_{e})_{-})=C_{-}(f(V^{err}-I)),$
(3.96)

where $C_{-}$ is the Cauchy projection operator, defined as follows

$C_{-}f(k)=\operatorname*{lim}\limits_{k^{\prime}\to
k\in\Sigma^{(R)}}\frac{1}{2\pi
i}\int_{\Sigma^{(2)}}\frac{f(s)}{s-k^{\prime}}ds,$ (3.97)

and $||C^{-}|||_{L^{2}}$ is bounded. Then the solution of the above RH problem
can be expressed as

$\displaystyle M^{err}(k)=I+\frac{1}{2\pi
i}\int_{\Sigma^{(R)}}\frac{\mu_{e}(s)(V^{err}(s)-I)}{s-k}ds,$ (3.98)

where $\mu_{e}(k)\in L^{2}(\Sigma^{(R)})$ satisfies

$\displaystyle(1-C_{\omega_{e}})\mu_{e}(k)=I.$ (3.99)

Using (3.94) and (3.96), we can see that

$||C_{\omega_{e}}||_{L^{2}(\Sigma^{(R)})}\leq||C_{-}||_{L^{2}(\Sigma^{(R)})}||V^{err}(k)-I||_{L^{\infty}(\Sigma^{(R)})}\leq
ce^{-2\rho_{0}^{2}t},$ (3.100)

Therefore, the pre-solution operator $(1-C_{\omega_{e}})^{-1}$ exists such
that $\mu_{e}$ and the solution $M^{err}(k)$ of the RH problem exist uniquely.

In order to reconstruct the solution $u(x,t)$ , we need some estimates and
asymptotic behaviors of $M^{err}(k)$.

######  Proposition 7.

For $M^{err}(k)$ defined in (3.91), it stratifies

$\displaystyle|M^{err}(k)-I|\leq ce^{-2\rho_{0}t}$ (3.101)

When $k=\frac{i}{2}$,

$M^{err}(\frac{i}{2})=I+\frac{1}{2\pi
i}\int_{\Sigma^{(R)}}\dfrac{\mu_{e}(s)(V^{err}(s)-I)}{s-\frac{i}{2}}ds,$
(3.102)

As $k\to\frac{i}{2}$, the Laurent expansion of $M^{err}(k)$ is

$\displaystyle
M^{err}(k)=M^{err}(\frac{i}{2})+M_{1}^{err}(k-\frac{i}{2})+\mathcal{O}((k-\frac{i}{2})^{2}),$
(3.103)

where

$M_{1}^{err}=\frac{1}{2\pi
i}\int_{\Sigma^{(R)}}\frac{\mu_{e}(s)(V^{err}(s)-I)}{(s-\frac{i}{2})^{2}}ds.$
(3.104)

In addition, $M^{err}(\frac{i}{2})$ and $M_{1}^{err}$ satisfy following long
time asymptotic behavior condition:

$|M^{err}(\frac{i}{2})-I|\lesssim\mathcal{O}(e^{-2\rho_{0}t}),\hskip
14.22636ptM_{1}^{err}\lesssim\mathcal{O}(e^{-2\rho_{0}t}).$ (3.105)

###### Proof.

By (3.98),

$\displaystyle M^{err}(k)-I=\frac{1}{2\pi
i}\int_{\Sigma^{(R)}}\frac{V^{err}(s)-I}{s-k}ds+\frac{1}{2\pi
i}\int_{\Sigma^{(R)}}\frac{(\mu_{e}(s)-I)(V^{err}(s)-I)}{s-k}ds,$

Further estimates can be given

$\displaystyle|M^{err}(k)-I|$
$\displaystyle\leq\|V^{err}(s)-I\|_{L^{2}(\Sigma^{(2R)})}\bigg{\|}\frac{1}{s-k}\bigg{\|}_{L^{2}(\Sigma^{(R)})}+$
$\displaystyle\ \ \ \
\|V^{err}(s)-I\|_{L^{\infty}(\Sigma^{(R)})}\|\mu_{e}(s)-I\|_{L^{2}(\Sigma^{(R)})}\bigg{\|}\frac{1}{s-k}\bigg{\|}_{L^{2}(\Sigma^{(R)})}$
$\displaystyle\leq ce^{-2\rho_{0}^{2}t}.$

(3.103)-(3.104) can be obtained by doing Taylor expansion on $\frac{1}{s-k}$.
At last, since $\frac{1}{(s-\frac{i}{2})^{2}}$ is bounded in $\Sigma^{(R)}$,
the proof of (3.105) can be given as follow:

$\displaystyle|M_{1}^{err}|\lesssim||V^{err}-I||_{L^{2}}||\mu_{e}(s)-I||_{L^{2}}+||V^{err}-I||_{L^{1}}\lesssim\mathcal{O}(e^{-2\rho_{0}t}).$

∎

###  3.3 Analysis on a pure $\bar{\partial}$ problem

In this subsection, we mainly consider the pure $\bar{\partial}$-problem which
is defined by (3.61). One can get the RHP of $M^{(3)}$ by simple calculation.

######  RHP 8.

(Pure $\bar{\partial}$-problem) Find $M^{(3)}(k)$ with following identities:

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

$\blacktriangleright$ Asymptotic behavior:

$\displaystyle M^{(3)}(k)\sim I+\mathcal{O}(k^{-1}),\hskip
14.22636ptk\rightarrow\infty;$ (3.106)

$\blacktriangleright$ $\bar{\partial}$-Derivative: We have

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

where

$W^{(3)}(k)=M^{(R)}(k)\bar{\partial}R^{(2)}(k)M^{(R)}(k)^{-1}.$ (3.107)

The solution of the pure $\bar{\partial}$ problem $M^{(3)}(k)$ can be given by
the following integral equation

$M^{(3)}(k)=I-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{M^{(3)}W^{(3)}}{s-k}\mathrm{d}A(s),$
(3.108)

where $\mathrm{d}A(s)$ is the Lebesgue measure in the real plane. The equation
(3.108) can also be expressed as an operator equation

$(I-J)M^{(3)}(k)=I\Longleftrightarrow M^{(3)}(k)=I+JM^{(3)}(k),$ (3.109)

where $J$ is the Cauchy operator

$Jf(k)=-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{f(s)W^{(3)}(s)}{s-k}\mathrm{d}A(s)=\frac{1}{\pi
k}\ast f(k)W^{(3)}(k).$ (3.110)

Next we prove that the above operator $J$ is small parametric when $t$ is
sufficiently large.

######  Lemma 3.

For $\xi\in(-\infty,-\frac{1}{4})$, the imaginary part of phase function
(2.60) $Im\theta(k)$ has following estimation:

$\displaystyle\text{Im
}\theta(k)\leq|k||sin\omega|\left(\xi+\frac{1}{1+cos2\varphi+\sqrt{2(1+cos2\varphi)}}\right),\hskip
14.22636pt\text{as }k\in\Upsilon_{1},\Upsilon_{2},$ (3.111)
$\displaystyle\text{Im
}\theta(k)\geq-|k||sin\omega|\left(\xi+\frac{1}{1+cos2\varphi+\sqrt{2(1+cos2\varphi)}}\right),\hskip
8.5359pt\text{as }k\in\Upsilon_{3},\Upsilon_{4}.$ (3.112)

For $\xi\in(2,+\infty)$, the estimation is shown as follow:

$\displaystyle\text{Im
}\theta(k)\geq|k||sin\omega|\left(\xi+\frac{1}{1+cos2\varphi-\sqrt{2(1+cos2\varphi)}}\right),\hskip
14.22636pt\text{as }k\in\Upsilon_{1},\Upsilon_{2},$ (3.113)
$\displaystyle\text{Im
}\theta(k)\leq-|k||sin\omega|\left(\xi+\frac{1}{1+cos2\varphi-\sqrt{2(1+cos2\varphi)}}\right),\hskip
8.5359pt\text{as }k\in\Upsilon_{3},\Upsilon_{4}.$ (3.114)

###### Proof.

We just take $k\in\Upsilon_{1}$ as an example, and the other regions are
similarly.

Since $\theta(k)=k\xi-\frac{k}{2k^{2}+\frac{1}{2}}$, let
$k=|k|e^{i\omega}=u+iv$, then the imaginary part of $\theta(k)$ can be
expressed as

$Im\theta(k)=ksin\omega\left(\xi+\frac{2|k|^{2}-\frac{1}{2}}{(2|k|^{2}cos2\omega+\frac{1}{2})^{2}+4|k|^{2}sin^{2}2\omega}\right),$

Denote

$\displaystyle
h(x;a)=\frac{x}{(x+\frac{1}{2})^{2}+a(x+\frac{1}{2})+\frac{1}{4}},$

where $x=2|k|^{2}-\frac{1}{2}$, $a=cos2\omega$.
Then

$\displaystyle Im\theta=v\left(\xi+h(x;a)\right).$

with $x\in(-\frac{1}{2},+\infty)$, $a\in[cos2\varphi,1]$.

By doing some simple simplification of $h(x,a)$, the image of the function
$\frac{1}{h(x,a)}$ can be drawn, so the value domain of $\frac{1}{h(x,a)}$ is

$\displaystyle\left(-\infty,1+a-\sqrt{2(1+a)}\right)\bigcup\left(1+a+\sqrt{2(1+a)},+\infty\right).$

The conclusion of the lemma can be deduced from the above equation. ∎

######  Corollary 3.

There exist constants $c(\xi)<0$, $\tilde{c}(\xi)>0$ relative to $\xi$ that
the imaginary part of phase function (2.60) $\text{Im }\theta(k)$ have
following evaluation for $k=|k|e^{i\omega}=u+iv$:
When $\xi\in(-\infty,-1/4)$,

$\displaystyle\text{Im }\theta(k)\leq c(\xi)v,\hskip 14.22636pt\text{as
}k\in\Upsilon_{1},\Upsilon_{2},$ (3.115) $\displaystyle\text{Im
}\theta(k)\geq-c(\xi)v,\hskip 14.22636pt\text{as
}k\in\Upsilon_{3},\Upsilon_{4}.$ (3.116)

When $\xi\in(2,+\infty)$,

$\displaystyle\text{Im }\theta(k)\geq\tilde{c}(\xi)v,\hskip 14.22636pt\text{as
}k\in\Upsilon_{1},\Upsilon_{2},$ (3.117) $\displaystyle\text{Im
}\theta(k)\leq-\tilde{c}(\xi)v,\hskip 14.22636pt\text{as
}k\in\Upsilon_{3},\Upsilon_{4}.$ (3.118)

######  Proposition 8.

Cauchy integral operator $J:L^{\infty}(\mathbb{C})\rightarrow
L^{\infty}(\mathbb{C})$. When $\xi\in(-\infty,-1/4)\cup(2,+\infty)$, for
sufficiently large $t$, the operator $J$ is a small parametrization. And

$||J||_{L^{\infty}\rightarrow L^{\infty}}\leqslant ct^{-\frac{1}{2}}.$ (3.119)

Therefore $(1-J)^{-1}$ exists, and thus the operator equation (3.109) has a
solution.

###### Proof.

Here, we just take the case $\xi\in(-\infty,-1/4)$ as an example. In order to
get the esitmate (3.119), It is only necessary to prove that for $f\in
L^{\infty}(\mathbb{C})$,

$||Jf||_{L^{\infty}(\mathbb{C})}\leqslant
ct^{-1/2}||f||_{L^{\infty}(\mathbb{C})}.$

Here, we just prove the case $k\in\Upsilon_{1}$. By (3.110),

$|Jf(k)|\leq
c||f||_{L^{\infty}}\iint_{\Upsilon_{1}}\frac{|W^{(3)}(s)|}{|s-k|}dA(s),$
(3.120)

where

$\displaystyle|W^{(3)}(s)|$
$\displaystyle\leq|M^{(R)}||\bar{\partial}R^{(2)}||(M^{(R)})^{-1}|\leq|M^{(R)}||\bar{\partial}R^{(2)}||\sigma_{2}(M^{(R)})^{T}\sigma_{2}|$
$\displaystyle\leq|M^{(R)}|^{2}|\bar{\partial}R^{(2)}|\leq
c|\bar{\partial}R_{1}(k)||e^{-2it\theta}|,$ (3.121)

thus

$\displaystyle\iint_{\Upsilon_{1}}\frac{|W^{(3)}(s)|}{|s-k|}dA(s)\lesssim\iint_{\Upsilon_{1}}\frac{|\bar{\partial}R_{1}|e^{2tIm\theta}}{|s-k|}dA(s)$
(3.122)

Referring into (3.42) in proposition 4, the integral
$\iint_{\Upsilon_{1}}\frac{|\bar{\partial}R_{1}|e^{2tIm\theta}}{|s-k|}dA(s)$
can be divided to two part:

$\displaystyle\iint_{\Upsilon_{1}}\dfrac{|\bar{\partial}R_{1}(s)|e^{2t\text{Im}\theta}}{|s-k|}dA(s)\lesssim
I_{1}+I_{2},$ (3.123)

with

$\displaystyle
I_{1}=\iint_{\Upsilon_{1}}\dfrac{|r^{\prime}(|s|)|e^{2t\text{Im}\theta}}{|s-k|}dA(s),\hskip
14.22636ptI_{2}=\iint_{\Upsilon_{1}}\dfrac{|s|^{-1/2}e^{2t\text{Im}\theta}}{|s-k|}dA(s).$
(3.124)

Write $s=u+iv$, $k=k_{R}+ik_{I}$, by (3.116),

$\displaystyle\text{Im }\theta(s)\leq c(\xi)2tv\leq c^{\prime}tv.$ (3.125)

Therefore,

$\displaystyle
I_{1}\doteq\int_{0}^{\infty}e^{c^{\prime}tv}dv\int_{v}^{\infty}\frac{|r^{\prime}(|s|)|}{|s-k|}du\leq\int_{0}^{\infty}e^{c^{\prime}tv}\|r^{\prime}(|s|)\|_{L^{2}(v,\infty)}\bigg{\|}\frac{1}{s-z}\bigg{|}\|_{L^{2}(v,\infty)}dv.$
(3.126)

and

$\displaystyle\bigg{\|}\frac{1}{s-k}\bigg{\|}^{2}_{L^{2}(v,\infty)}=\int_{v}^{\infty}\frac{1}{|s-k|^{2}}du\leq\int_{-\infty}^{\infty}\frac{1}{(u-k_{R})^{2}+(v-k_{I})^{2}}du$
$\displaystyle=\frac{1}{|v-k_{I}|}\int_{-\infty}^{\infty}\frac{1}{1+y^{2}}dy=\frac{\pi}{|v-k_{I}|},$
(3.127)

where $y=\frac{u-k_{R}}{v-k_{I}}$.

$\displaystyle\|r^{\prime}(|s|)\|_{L^{2}(v,\infty)}^{2}=\int_{v}^{\infty}\big{|}r^{\prime}(\sqrt{u^{2}+v^{2}})\big{|}^{2}du\leq\int_{\sqrt{2}v}^{\infty}\big{|}r^{\prime}(\tau)\big{|}^{2}\frac{\sqrt{u^{2}+v^{2}}}{u}d\tau$
$\displaystyle\leq\sqrt{2}\int_{\sqrt{2}v}^{\infty}\big{|}r^{\prime}(\tau)\big{|}^{2}d\tau\leq\|r^{\prime}(s)\|_{L^{2}(\mathbb{R})}^{2}.$
(3.128)

Using (3.127)-(3.128), we can directly calculate

$\displaystyle I_{1}\leq
c\|r^{\prime}\|_{L^{2}(\mathbb{R})}\int_{0}^{\infty}e^{c^{\prime}tv}|v-k_{I}|^{-1/2}dv$
$\displaystyle=c\|r^{\prime}\|_{L^{2}(\mathbb{R})}\left[\int_{0}^{k_{I}}\frac{e^{c^{\prime}tv}}{\sqrt{k_{I}-v}}dv+\int_{k_{I}}^{\infty}\frac{e^{c^{\prime}tv}}{\sqrt{v-k_{I}}}dv\right]$
$\displaystyle\lesssim t^{-\frac{1}{2}}.$ (3.129)

To estimate $I_{2}$, consider the following $L^{p}$-estimate $(p>2)$.

$\displaystyle\||s|^{-1/2}\|_{L^{p}(v,\infty)}=\left(\int_{v}^{\infty}\frac{1}{|u+iv|^{p/2}}du\right)^{1/p}=\left(\int_{v}^{\infty}\frac{1}{(u^{2}+v^{2})^{p/4}}du\right)^{1/p}$
$\displaystyle=v^{1/p-1/2}\left(\int_{1}^{\infty}\frac{1}{(1+x^{2})^{p/4}}dx\right)^{1/p}\leq
cv^{1/p-1/2}.$ (3.130)

Similar to the $L^{p}$ estimate above, it can be shown that

$\displaystyle\big{\|}|s-k|^{-1}\big{\|}_{L^{q}(v,\infty)}\leq
c|v-k_{I}|^{1/q-1},\ \ 1/p+1/q=1.$

Using the two estimates above,

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{\infty}e^{c^{\prime}tv}dv\int_{v}^{\infty}\frac{|s|^{-1/2}}{|s-k|}du$
$\displaystyle\leq\int_{0}^{\infty}e^{c^{\prime}tv}\||s|^{-1/2}\|_{L^{p}(v,\infty)}\big{\|}|s-k|^{-1}\big{\|}_{L^{q}(v,\infty)}dv.$
$\displaystyle\leq
c\left[\int_{0}^{k_{I}}e^{c^{\prime}tv}v^{1/p-1/2}|v-k_{I}|^{1/q-1}dv+\int_{k_{I}}^{\infty}e^{c^{\prime}tv}v^{1/p-1/2}|v-k_{I}|^{1/q-1}dv\right].$
(3.131)

Calculating the two integrals by permutation yields,

$\displaystyle I_{2}\leq ct^{-\frac{1}{2}}.$ (3.132)

By (3.120)-(3.123), the result of property 8 is proved. ∎

######  Proposition 9.

There exist a small positive constant $\rho<1/4$, such that the solution
$M^{(3)}(k)$ of $\bar{\partial}$-problem admits the following estimation

$\displaystyle\parallel
M^{(3)}(\frac{i}{2})-I\parallel=\parallel\frac{1}{\pi}\iint_{\mathbb{C}}\dfrac{M^{(3)}(s)W^{(3)}(s)}{s-\frac{i}{2}}dA(s)\parallel\lesssim
t^{-1+2\rho}.$ (3.133)

As $k\to\frac{i}{2}$, $M^{(3)}(k)$ has asymptotic expansion

$M^{(3)}(k)=M^{(3)}(\frac{i}{2})+M^{(3)}_{1}(x,t)(k-\frac{i}{2})+\mathcal{O}((k-\frac{i}{2})^{2}),$
(3.134)

where $M^{(3)}_{1}(x,t)$ is a $k$-independent coefficient with

$M^{(3)}_{1}(x,t)=\frac{1}{\pi}\iint_{C}\dfrac{M^{(3)}(s)W^{(3)}(s)}{(s-\frac{i}{2})^{2}}dA(s),$
(3.135)

and $M^{(3)}_{1}(x,t)$ satisfies

$|M^{(3)}_{1}(x,t)|\lesssim t^{-1+2\rho}.$ (3.136)

###### Proof.

First we estimate (3.133). The proof proceeds along the same steps as the
proof of above Proposition. (3.109) and (3.119) implies that for large $t$,
$\parallel M^{(3)}\parallel_{\infty}\lesssim 1$. And here we only estimate the
integral on sector $\Upsilon_{1}$ as $\xi<-1/4$. Let $s=u+vi$. We also divide
$M^{(3)}(\frac{i}{2})-I$ to two parts, but this time we use another estimation
(3.44):

$\frac{1}{\pi}\iint_{\mathbb{C}}\dfrac{M^{(3)}(s)W^{(3)}(s)}{s-\frac{i}{2}}dA(s)\lesssim
I_{3}+I_{4},$ (3.137)

with

$\displaystyle
I_{3}=\iint_{\Upsilon_{1}}\dfrac{|r^{\prime}(|s|)|e^{2t\text{Im}\theta}}{|\frac{i}{2}-s|}dA(s),\hskip
14.22636ptI_{4}=\iint_{\Upsilon_{1}}\dfrac{|s|^{-1}e^{2t\text{Im}\theta}}{|\frac{i}{2}-s|}dA(s).$
(3.138)

For $s\in\Upsilon_{1}$, $|\frac{i}{2}-s|\leq c$. So

$\displaystyle I_{3}\leq$
$\displaystyle\int_{0}^{+\infty}\int_{v}^{+\infty}\dfrac{|r^{\prime}(|s|)|e^{c^{\prime}tv}}{|\frac{i}{2}-s|}dudv$
$\displaystyle\lesssim\int_{0}^{+\infty}\parallel
r^{\prime}\parallel_{L^{1}(\mathbb{R}^{+})}e^{c^{\prime}tv}dv\lesssim\int_{0}^{+\infty}e^{c^{\prime}tv}dv\lesssim
t^{-1}.$

As for $I_{4}$, we partition it to two parts:

$\displaystyle
I_{4}\leq\int_{0}^{\frac{1}{4}}\int_{v}^{+\infty}\dfrac{|s|^{-1}e^{c^{\prime}tv}}{|\frac{i}{2}-s|}dudv+\int_{\frac{1}{4}}^{+\infty}\int_{v}^{+\infty}\dfrac{|s|^{-1}e^{c^{\prime}tv}}{|\frac{i}{2}-s|}dudv:=I_{41}+I_{42}.$
(3.139)

For $0<v<\frac{1}{4}$,
$|s-\frac{i}{2}|^{2}=u^{2}+(v-\frac{1}{2})^{2}>u^{2}+v^{2}=|s|^{2}$, while as
$v>\frac{1}{4}$, $|s-\frac{i}{2}|^{2}<|s|^{2}$. Then the first integral has:

$\displaystyle
I_{41}\leq\int_{0}^{\frac{1}{4}}\int_{v}^{+\infty}(u^{2}+v^{2})^{-\frac{1}{2}-\rho}(u^{2}+(v-\frac{1}{2})^{2})^{-\frac{1}{2}+\rho}due^{c^{\prime}tv}dv$
$\displaystyle\leq\int_{0}^{\frac{1}{4}}\left[\int_{v}^{+\infty}\left(1+\left(\frac{u}{v}\right)^{2}\right)^{-\frac{1}{2}-\rho}v^{-2\rho}d\frac{u}{v}\right](v^{2}+(v-\frac{1}{2})^{2})^{-\frac{1}{2}+\rho}e^{c^{\prime}tv}dv$
$\displaystyle\lesssim\int_{0}^{\frac{1}{4}}v^{-2\rho}(\frac{1}{8})^{-\frac{1}{2}+\rho}e^{c^{\prime}tv}dv\lesssim
t^{-1+2\rho}.$ (3.140)

The second integral can be bounded in similar way:

$\displaystyle
I_{42}\leq\int_{\frac{1}{4}}^{+\infty}e^{c^{\prime}tv}\parallel|s|^{-1}\parallel_{L^{p}}\parallel|\frac{i}{2}-s|^{-1}\parallel_{L^{q}}dv$
$\displaystyle\leq\int_{\frac{1}{4}}^{+\infty}(v^{2})^{-\frac{1}{2}+\rho}|v-\frac{1}{2}|^{-2\rho}e^{c^{\prime}tv}dv$
$\displaystyle\lesssim\int_{\frac{1}{4}}^{\frac{1}{2}}(\frac{1}{2}-v)^{-2\rho}e^{c^{\prime}tv}dv+\int_{\frac{1}{2}}^{+\infty}(v-\frac{1}{2})^{-2\rho}e^{c^{\prime}tv}dv$
$\displaystyle\leq
e^{\frac{c^{\prime}}{4}t}\int_{\frac{1}{4}}^{\frac{1}{2}}(\frac{1}{2}-v)^{-2\rho}dv+e^{\frac{c^{\prime}}{2}t}\int_{\frac{1}{2}}^{+\infty}(v-\frac{1}{2})^{-2\rho}e^{c^{\prime}t(v-\frac{1}{2})}d(v-\frac{1}{2})\lesssim
e^{\frac{c^{\prime}}{4}t}.$

This estimation is strong enough to obtain the result (3.133). And (3.136) is
obtained by deflating $|\frac{i}{2}-s|^{2}$ to a constant for
$s\in\Upsilon_{1}$. ∎

###  3.4 Long time asymptotic behaviors

Here we begin to construct the long time asymptotics of the CH equation (1.1).
According to the series of RHP transformations we have done before, we have

$\displaystyle M(k)=$ $\displaystyle
M^{(3)}(k)M^{err}(k)M^{\Lambda}(k)G^{-1}(k)R^{(2)}(k)^{-1}T(k)^{-\sigma_{3}}.$
(3.141)

To reconstruct $u(x,t)$ by using (2.73), in above equation , we take
$k\to\frac{i}{2}$ out of $\bar{\Upsilon}$. Further taking the Laurent
expansion of each element into (3.141) , we can obtain that

$\displaystyle M(k)=$
$\displaystyle\left(M^{(3)}(\frac{i}{2})+M^{(3)}_{1}(k-\frac{i}{2})\right)M^{err}(k)M^{\Lambda}(k)G^{-1}(k)$
$\displaystyle\left(T(\frac{i}{2})J_{0}+\left[T(\frac{i}{2})J_{0}J_{1}+\sum_{j,s=1;j\neq
s}^{\mathcal{N}(\Delta_{1}^{+})}\left(\frac{-2k_{n_{s}}(k_{n_{j}}+\frac{i}{2})}{(\frac{i}{2}-k_{n_{j}})(\frac{i}{2}-k_{n_{s}})^{2}}\right)J_{0}\right](k-\frac{i}{2})\right)^{-\sigma_{3}}+\mathcal{O}((k-\frac{i}{2})^{2}).$
(3.142)

Its long time asymptotics can be given by

$\displaystyle M(k)=$ $\displaystyle
M^{\Lambda}(k)\left(T(\frac{i}{2})J_{0}+\left[T(\frac{i}{2})J_{0}J_{1}+\sum_{j,s=1;j\neq
s}^{\mathcal{N}(\Delta_{1}^{+})}\left(\frac{-2k_{n_{s}}(k_{n_{j}}+\frac{i}{2})}{(\frac{i}{2}-k_{n_{j}})(\frac{i}{2}-k_{n_{s}})^{2}}\right)J_{0}\right](k-\frac{i}{2})\right)^{-\sigma_{3}}$
$\displaystyle+\mathcal{O}((k-\frac{i}{2})^{2})+\mathcal{O}(t^{-1+2\rho}),$
(3.143)

and

$\displaystyle M(\frac{i}{2})=$ $\displaystyle
M^{\Lambda}(\frac{i}{2})\left(T(\frac{i}{2})J_{0}\right)^{-\sigma_{3}}+\mathcal{O}(t^{-1+2\rho}).$
(3.144)

Substituting above estimates into (2.73) and (2.72) leads to

$\displaystyle u(x,t)=$ $\displaystyle
u(y(x,t),t)=\frac{1}{2i}\lim_{k\to\frac{i}{2}}\frac{1}{k-\frac{i}{2}}\left(\dfrac{(M_{11}(k)+M_{21}(k))(M_{12}(k)+M_{22}(k))}{(M_{11}(\frac{i}{2})+M_{21}(\frac{i}{2}))(M_{12}(\frac{i}{2})+M_{22}(\frac{i}{2}))}-1\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2i}\lim_{k\to\frac{i}{2}}\frac{1}{k-\frac{i}{2}}\left(\dfrac{([M^{\Lambda}]_{11}(k)+[M^{\Lambda}]_{21}(k))([M^{\Lambda}]_{12}(k)+[M^{\Lambda}]_{22}(k))}{([M^{\Lambda}]_{11}(\frac{i}{2})+[M^{\Lambda}]_{21}(i\frac{i}{2}))([M^{\Lambda}]_{12}(\frac{i}{2})+[M^{\Lambda}]_{22}(\frac{i}{2}))}-1\right)$
$\displaystyle+\mathcal{O}(t^{-1+2\rho})$ $\displaystyle=$ $\displaystyle
u^{r}(x,t;\tilde{\mathcal{D}})+\mathcal{O}(t^{-1+2\rho}),$ (3.145)

and

$\displaystyle x(y,t)$
$\displaystyle=y+\ln\left(\frac{M_{11}(\frac{i}{2})+M_{21}(\frac{i}{2})}{\left[M_{12}(\frac{i}{2})+M_{22}(\frac{i}{2})\right]a^{2}(\frac{i}{2})}\right)$
$\displaystyle=y+\int_{-\infty}^{+\infty}(\sqrt{m+1}-1)d\xi+\ln\left(\frac{M_{11}^{\Lambda}(\frac{i}{2})+M_{21}^{\Lambda}(\frac{i}{2})}{M_{12}^{\Lambda}(\frac{i}{2})+M_{22}^{\Lambda}(\frac{i}{2})}\right)+\mathcal{O}(t^{-1+2\rho}).$
(3.146)

##  4 Long-time asymptotic with phase points

In a similar way to the case without phase points in Section 3, we can obtain
Long-time asymptotic in the regions with phase points as follows

$\displaystyle
u(x,t)=u^{r}(x,t;\tilde{\mathcal{D}})+f_{11}t^{-1/2}+\mathcal{O}(t^{-3/4}),\ \
{\rm for}\ \ \xi\in(-1/4,2).$ (4.1)

The detail analysis will be given in near future.

## References

  * [1] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.
  * [2] B. Fuchssteiner and A. Fokas, Symplectic structures, their Backlund transforms and hereditary symmetries, Phys. D, 4 (1981), 47-66.
  * [3] Constantin A. and Strauss W. Stability of a class of solitary waves in compressible elastic rods, Phys. Lett. A 270 (2000), 140-148.
  * [4] Dai H.-H. Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mechanica 127 (1998), 193-207.
  * [5] J. K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math. 51 (1991), 1498-1521.
  * [6] A. Bressan and A. Constantin Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal. 37 (2005), 996-1026;
  * [7] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Expo. Math. 15 (1997) 53-85.
  * [8] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. Royal Soc. A. 457(2001), 953-970.
  * [9] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603-610
  * [10] J. Lenells, A variational approach to the stability of periodic peakons, J. Nonlinear Math. Phys. 11 (2004) 151-163.
  * [11] R. S. Johnson, On solutions of the Camassa-Holm equation, Proc. Roy. Soc. London A 459 (2003), 1687-1708.
  * [12] A. Parker, On the Camassa-Holm equation and a direct method of solution I. Bilinear form and solitary waves. Proc. Roy. Soc. London A 460 (2004), 2929-2957.
  * [13] A. Parker, On the Camassa-Holm equation and a direct method of solution II. Soliton solutions, Proc. Roy. Soc. London A 461 (2005), 3611-3632.
  * [14] A. Parker, On the Camassa-Holm equation and a direct method of solution III. N-soliton solutions, Proc. Roy. Soc. London A 461 (2005), 3893-3911.
  * [15] R. Beals, Sattinger D. and Szmigielski J. Acoustic scattering and the extended Korteweg-de Vries hierarchy, Adv. Math. 140 (1998), 190-206
  * [16] R. Beals, Sattinger D. and Szmigielski J. Multi-peakons and a theorem of Stieltjes, Inv. Problems 15 (1999), L1-L4.
  * [17] A. Constantin and L. Molinet Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000), 45-61.
  * [18] A Constantin and L Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D 157 (2001), 75-89.
  * [19] A. Constantin, A. Strauss, Stability of peakons, Commun. Pure Appl. Math., 153(2000), 0603-0610.
  * [20] Z. Xin, P. Zhang, On the weak solutions to a shallow water equations, Commun. Pure Appl. Math. 53 (2000) 1411-1433.
  * [21] A Constantin and L Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys. 211 (2000), 45-61.
  * [22] R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Differential Equations, 192 (2003) 429-444.
  * [23] F. Gesztesy, H. Holden, Algebro-geometric solutions of the Camassa-Holm hierarchy, Rev. Mat. Iberoamericana 19(2003), 73-142.
  * [24] Z. J. Qiao, The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold, Commun. Math. Phys. 239(2003), 309-341
  * [25] Y. Li and J. Zhang, The multiple-soliton solution of the Camassa-Holm equation, Proc. R. Soc. London, Ser. A 460 (2004), 2617-2627.
  * [26] J. Eckhardt, G. Teschl, On the isospectral problem of the dispersionless Camassa-Holm equation, Adv. Math. 235(2013), 469-495.
  * [27] J. Eckhardt, The inverse spectral transform for the conservative Camassa-Holm flow with decaying initial data, Arch. Rational Mech. Anal. 224(2017), 21-52.
  * [28] J. Eckhardt, A. Kostenko, The inverse spectral problem for periodic conservative multi-peakon solutions of the Camassa-Holm equation, Int. Meth. Res. Not. 16(2020), 5126-5151.
  * [29] C. S. Gardner, J. M. Green, M. D. Kruskal and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19(1967), 1095-1097.
  * [30] A. Constantin, V. Gerdjikov and R. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inv. Problems 22 (2006), 2197-2207.
  * [31] R.S. Johnson, On solutions of the Camassa-Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459(2035), 1687-1708 (2003)
  * [32] J. Lenells, The scattering approach for the Camassa-Holm equation. J. Nonlinear Math. Phys. 9(4), 389-393 (2003)
  * [33] A. Boutet de Monvel, D. Shepelsky, Riemann-Hilbert approach for the Camassa-Holm equation on the line, C.R. Math. Acad. Sci. Paris 343 (2006) 627-632
  * [34] A. Boutet de Monvel, D. Shepelsky, Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line. In: Probability, Geometry and Integrable Systems. Math. Sci. Res. Inst. Publ., 55(2008), pp. 53-75.
  * [35] P. Deift, X. Zhou,  A steepest descent method for oscillatory Riemann-Hilbert problems. Ann. Math., 137(1993), 295-368.
  * [36] P. Deift, X. Zhou,  Long-time behavior of the non-focusing nonlinear Schr$\ddot{o}$dinger equation–a case study, Lectures in Mathematical Sciences, Graduate School of Mathematical Sciences, University of Tokyo, 1994.
  * [37] P. Deift, X. Zhou,  Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math., 56(2003), 1029-1077.
  * [38] K. Grunert, G. Teschl,  Long-time asymptotics for the Korteweg de Vries equation via noninear steepest descent. Math. Phys. Anal. Geom., 12(2009), 287-324.
  * [39] A. Boutet de Monvel, A. Kostenko, D. Shepelsky, G. Teschl,  Long-time asymptotics for the Camassa-Holm equation, SIAM J. Math. Anal, 41(2009), 1559-1588.
  * [40] J. Xu, E. G. Fan,  Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: Without solitons, J. Differential Equations, 259(2015), 1098-1148.
  * [41] J. Xu,  Long-time asymptotics for the short pulse equation, J. Differential Equations, 265(2018), 3494-3532.
  * [42] H. Liu, X. G. Geng, B. Xue, The Deift-Zhou steepest descent method to long-time asymptotics for the Sasa-Satsuma equation. J. Differential Equations 265(2018), 5984-6008.
  * [43] J. Xu, E. G. Fan, Long-time asymptotic behavior for the complex short pulse equation JianXua, EnguiFanb. J. Differential Equations 269(2020), 10322-10349.
  * [44] J. Xu, E. G. Fan, Y. Chen,  Long-time asymptotic for the derivative nonlinear Schr$\ddot{o}$dinger equation with step-like initial value, Math. Phys. Anal. Geometry, 16(2013), 253-288.
  * [45] H. Krüger and G. Teschl,  Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21(2009), 61-109.
  * [46] A. B. de Monvel, D. Shepelsky,  Riemann-Hilbert approach for the Camassa-Holm equation on the line, Comptes Rendus Mathematique, 343(2006), 627-632.
  * [47] A. Boutet de Monvel, D. Shepelsky,  Riemann-Hilbert problem in the inverse scattering for the Camassa-Holm equation on the line, Math. Sci. Res. Inst. Publ., 55(2007), 53-75.
  * [48] A. Boutet de Monvel, A. Its, D. Shepelsky, Painleve-type asymptotics for Camassa-Holm equation, SIAM J. Math. Anal. 42(2010), 1854-1873.
  * [49] A. Minakov, Riemann-Hilbert problem for Camassa-Holm equation with step-like initial data, J. Math. Anal. Appl. 429 (2015), 81-104.
  * [50] A. Boutet de Monvel, A. Its, V. Kotlyarov, Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition on the half-line. Commun. Math. Phys. 290(2009), 479-522.
  * [51] K. T. R. McLaughlin, P. D. Miller,  The $\bar{\partial}$ steepest descent method and the asymptotic behavior of polynomials orthogonal on the unit circle with fixed and exponentially varying non-analytic weights, Int. Math. Res. Not., (2006), Art. ID 48673.
  * [52] K. T. R. McLaughlin, P. D. Miller,  The $\bar{\partial}$ steepest descent method for orthogonal polynomials on the real line with varying weights, Int. Math. Res. Not., (2008), Art. ID 075.
  * [53] M. Dieng, K. D. T. McLaughlin,  Dispersive asymptotics for linear and integrable equations by the Dbar steepest descent method,  Nonlinear dispersive partial differential equations and inverse scattering, 253-291, Fields Inst. Commun., 83, Springer, New York, 2019.
  * [54] M. Borghese, R. Jenkins, K. T. R. McLaughlin, Miller P,  Long-time asymptotic behavior of the focusing nonlinear Schr$\ddot{o}$dinger equation, Ann. I. H. Poincar$\acute{e}$ Anal, 35(2018), 887-920.
  * [55] R. Jenkins, J. Liu, P. Perry, C. Sulem,  Soliton resolution for the derivative nonlinear Schr$\ddot{o}$dinger equation, Commun. Math. Phys., 363(2018), 1003-1049.
  * [56] S. Cuccagna, R. Jenkins,  On asymptotic stability of N-solitons of the defocusing nonlinear Schr$\ddot{o}$dinger equation, Comm. Math. Phys, 343(2016), 921-969.
  * [57] Y. L. Yang, E. G. Fan, Soliton resolution for the short-pulse equation, J. Differential Equations, 280(2021), 644-689.
  * [58] Y. L. Yang, E. G. Fan, On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions, Adv. Math., 402(2022), 108340.
  * [59] Q. Y. Cheng, E. G. Fan, Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time, J. Differential Equations, 309(2022), 883-948.


There exist a $t_{0}\in\mathbb{R}$ such that $y$ belongs to the circle
$\Phi_{t_{0}}(q)$. Note that this great circle splits $\mathbb{CP}^{1}$ in two
parts, one containing $x$ and $s_{x}$ and one containing $z$ and $s_{z}$. In
particular $(x_{t})_{t\geq t_{0}}$ must lie in $U_{z}$ and $(x_{t})_{t\leq
t_{0}}$ must lie in $U_{x}$, leading to a contradiction. Hence no such map $u$
can exist.

$x$$s_{x}$$s_{z}$$z$$y$ Figure 7. A Jordan curve in $\mathbb{CP}^{1}$

∎

##  Appendix B Fitting immersions and spacelike immersions for
$\operatorname{PSp}(4,\mathbb{R})$

In this section we explain how the data of a maximal immersion into the
Pseudo-Riemannian space $\mathbb{H}^{2,2}$ with principal curvature in
$(-1,1)$ induces a fitting immersion. Combining this with a result from
Collier-Tholozan-Toulisse we show that our characterization of maximal
representations can be improved in $\operatorname{Sp}(4,\mathbb{R})$.

Note first that one has the following exceptional isomorphism:

$\operatorname{PSp}(4,\mathbb{R})\simeq\operatorname{SO}_{o}(2,3).$

This isomorphism comes form the fact that $\operatorname{PSp}(4,\mathbb{R})$
preserves a subspace of dimension $5$ of $\Lambda^{2}\mathbb{R}^{4}$, as well
as a symmetric bilinear form of signature $(2,3)$ on this subspace. Hence
$\operatorname{PSp}(4,\mathbb{R})$ acts naturally on the pseudo-Riemannian
symmetric space with constant negative sectional curvature $\mathbb{H}^{2,2}$,
which consists of vectors of norm $-1$ in $\mathbb{R}^{2,3}$. The space of
Lagrangians $\mathcal{L}$ in $\mathbb{R}^{4}$ is naturally identified with the
space of isotropic lines $\partial\mathbb{H}^{2,2}$ in $\mathbb{R}^{2,3}$.

To a pointed totally geodesic spacelike plane $(p,P)$ in $\mathbb{H}^{2,2}$
one can associate an element in a special $G$-orbit of
$\operatorname{Gr}^{\max}_{2}(\mathcal{Q})$. For every geodesic in this plane
passing though the base point, we consider the endpoints
$\ell_{1},\ell_{2}\in\mathcal{L}\simeq\partial\mathbb{H}^{2,2}$, and the space
generated by all such quadrics $q_{\ell_{1},\ell_{2}}$ forms a plane which is
a an element of $\operatorname{Gr}^{\omega}_{2}(\mathcal{Q})$. Indeed for some
symplectic basis $(x_{1},x_{2},y_{1},y_{2})$ the Lagrangians corresponding to
the boundary of the spacelike plane $P$ are for $\theta\in[0,2\pi]$:

$\ell(\theta)=\langle\cos\left(\frac{\theta}{2}\right)x_{1}+\sin\left(\frac{\theta}{2}\right)y_{1},\cos\left(\frac{\theta}{2}\right)x_{2}+\sin\left(\frac{\theta}{2}\right)y_{2}\rangle.$

The corresponding quadric $q_{\ell(0),\ell(\pi)}$ for $\theta=0$ in the basis
$(x_{1},y_{1},x_{2},y_{2})$ is equal to:

$\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\ 0&0&1&0\end{pmatrix}.$

Let $R_{\theta}$ be the following rotation matrix:

$\begin{pmatrix}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\end{pmatrix}.$

The corresponding quadric $q_{\ell(\theta),\ell(\theta+\pi)}$ in the basis
$(x_{1},y_{1},x_{2},y_{2})$ is equal to:

$\begin{pmatrix}R_{\frac{\theta}{2}}&0\\\
0&R_{\frac{\theta}{2}}\end{pmatrix}q_{\ell(0),\ell(\pi)}\begin{pmatrix}R_{-{\frac{\theta}{2}}}&0\\\
0&R_{-{\frac{\theta}{2}}}\end{pmatrix}.$

Hence $q_{\ell(\theta),\ell(\theta+\pi)}$ is equal to:

$\begin{pmatrix}\sin(\theta)&\cos(\theta)&0&0\\\
\cos(\theta)&-\sin(\theta)&0&0\\\ 0&0&\sin(\theta)&\cos(\theta)\\\
0&0&\cos(\theta)&-\sin(\theta)\end{pmatrix}.$

These quadrics span a plane in $\mathcal{Q}$ when $\theta$ varies, and these
quadrics are $\omega$-regular so this plane is in
$\operatorname{Gr}_{2}^{\omega}(\mathcal{Q})$.

We define the Gauss map
$\mathcal{G}u:S\to\operatorname{Gr}^{\max}_{2}(\mathcal{Q})$ of a spacelike
immersion $u:S\to\mathbb{H}^{2,2}$ the map that associates to $x\in S$ the
pencils corresponding to the pointed totally geodesic spacelike plane
$(u(x),P)$ where $T_{u(x)}P=\mathrm{d}u(T_{x}S)$.

######  Theorem B.1.

Let $u:S\to\mathbb{H}^{2,2}$ be a spacelike immersion such that for all $v\in
TS$, $\lVert I\\!\\!I_{u}(v,v)\rVert<\lVert v\rVert$. The Gauss map
$\mathcal{G}u:S\to\operatorname{Gr}^{\max}_{2}(\mathcal{Q})$ is a fitting
immersion.

Let $S$ be a spacelike surface in $\mathbb{H}^{2,n}$. Let $\gamma:[0,1]\to S$
be a geodesic for the induced metric on $S$ parametrized with unit speed an
let $V:[0,1]\to TS$ be the unit orthogonal vector field to $\gamma^{\prime}$
in $S$ along $\gamma$. We denote by
$V^{+},V^{-}:[0,1]\to\partial\mathbb{H}^{2,2}$ the endpoints of the geodesic
rays starting respectively at $V$ and $-V$. Up to changing the sign of $V$ one
can assume that $(V^{+},\gamma^{+},V^{-})$ is a maximal triple where
$\gamma^{+}$ is the endpoint of the geodesic ray srarting at
$\gamma^{\prime}$.

######  Lemma B.2.

Suppose that $\lVert I\\!\\!I_{u}(V,\gamma^{\prime})\rVert\leq 1$, then the
curves $V^{+},V^{-}:[0,1]\to\partial\mathbb{H}^{2,n}$ are spacelike.

Note that in this lemma $I\\!\\!I_{u}(V,\gamma^{\prime})$ is timelike, so its
norm is the timelike norm that we see as a positive number.

###### Proof.

We fix a orthogonal basis $(e_{i})$ of $\mathbb{R}^{2,n}$ such that $e_{1}$
and $e_{2}$ have norm $1$ and $e_{i}$ for $i\geq 3$ have norm $-1$. Without
any loss of generality we suppose that $\gamma(0)=e_{3}$,
$\gamma^{\prime}(0)=e_{1}$ and $V(0)=e_{2}$. Let $d$ be the flat connection on
$\mathbb{R}^{2,n}$ and let $\nabla$ be the Levi-civita connection on $u(S)$
for the induced spacelike metric. For $t$ close to $0$, $V(t)=V(0)+tdV(0)$ and
$dV=\nabla
V+I\\!\\!I_{\mathbb{H}^{2,n}}(V,\gamma^{\prime})+I\\!\\!I_{u}(V,\gamma^{\prime})$.
Here the second fundamental form of $\mathbb{H}^{2,n}$ inside
$\mathbb{R}^{2,n+1}$ is equal to
$I\\!\\!I_{\mathbb{H}^{2,n}}(v_{1},v_{2})=\langle v_{1},v_{2}\rangle v_{0}$
for $v_{1},v_{2}\in T_{v_{0}}\mathbb{H}^{2,n}$. Note also that since $\gamma$
is a geodesic and $V$ a orthogonal unit vector field along $\gamma$, $\nabla
V=0$.

Hence $V(t)=V(0)+tI\\!\\!I_{u}(V,\gamma^{\prime})+o(t)$. Since $V$ has norm
$1$, we can write a representative of the isotropic line $V^{+}(t)$ as
$v^{+}(t)=V(t)+\gamma(t)$. Therefore:

$v^{+}(t)=v^{+}(0)+t\left(I\\!\\!I_{u}(V,\gamma^{\prime})+e_{2}\right)+o(t).$

This curve is spacelike since $e_{2}$ is spacelike of norm $1$ and the
(timelike) norm of $I\\!\\!I_{u}(V,\gamma^{\prime})$ is strictly less than
$1$. The same holds for $V^{-}$. ∎

###### Proof of Theorem B.1.

The Gauss map $\mathcal{G}u:S\to\operatorname{Gr}^{\max}_{2}(\mathcal{Q})$
comes with an identification between $u^{*}\mathcal{E}$ and $TS$. We consider
the geodesic flow $\Phi$ on $\mathbb{S}TS$ for the metric induced by $u$.
Lemma B.2 implies that the Lagrangians $\ell^{+},\ell^{-}$ in $\mathbb{R}^{4}$
corresponding to the isotropic lines $V^{+}$ and $V^{-}$ in $\mathbb{R}^{2,3}$
satisfy the hypothesis of the second part of Lemma 6.12. Hence the derivative
of the associated quadrics along this flow is positive so $\Phi$ is a fitting
flow. Since $u$ admits a fitting flow it is in particular a well-fitting
immersion.

∎

The existence of a maximal spacelike immersion was proven by Collier-Tholozan-
Toulisse and a bound of its second fundamental form is a consequence of a
result from Cheng.

######  Theorem B.3 ([CTT19],[Che93]).

Every maximal representation $\rho:\Gamma_{g}\to\operatorname{SO}_{o}(2,3)$
admits a unique $\rho$-equivariant maximal spacelike immersion
$u:\widetilde{S_{g}}\to\mathbb{H}^{2,2}$. Moreover it is an embedding and for
all $v\in TS$, $\lVert I\\!\\!I_{u}(v,v)\rVert<\lVert v\rVert$.

The bound on the second fundamental from is a consequence of a maximal
principle, see [LT22, Corollary 5.2]. Note that this reference it is written
that the square norm of $I\\!\\!I_{u}$ is at most $2$, but since $u$ is
maximal it implies that the principal values are at most equal to $1$.

Putting together Theorem B.1, Theorem B.3 and Theorem 6.5 we obtain:

######  Corollary B.4.

Every maximal representation
$\rho:\Gamma_{g}\to\operatorname{Sp}(4,\mathbb{R})$ admits a fitting immersion
$u:\widetilde{S_{g}}\to\operatorname{Gr}^{\max}_{2}(\mathcal{Q})$ that is
$\rho$-equivariant. This characterizes representations which are maximal for
some orientation of $S_{g}$.

######  Remark B.5.

Note that a maximal spacelike surface in $\mathbb{H}^{2,2}$ always defines a
fibration of the space of photons and of the symmetric space of
$\operatorname{Sp}(4,\mathbb{R})$, without assuming any bound on the second
fundamental form. In this case however the bound is useful to deduce that the
fibration extends to a fibration of the symmetric space of
$\operatorname{SL}(4,\mathbb{R})$.

## References

  * [BG09] Jairo Bochi and Nicolas Gourmelon, _Some characterizations of domination_ , Math. Z. 263 (2009), no. 1, 221–231.
  * [BILW05] Marc Burger, Alessandra Iozzi, François Labourie, and Anna Wienhard, _Maximal representations of surface groups: symplectic Anosov structures_ , Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 543–590.
  * [BIW03] Marc Burger, Alessandra Iozzi, and Anna Wienhard, _Surface group representations with maximal Toledo invariant_ , Comptes Rendus Mathematique 336 (2003), no. 5, 387–390.
  * [BIW11] Marc Burger, Alessandra Iozzi, and Anna Wienhard, _Higher teichmüller spaces: from sl(2,r) to other lie groups_ , 2011, 1004.2894.
  * [BPS19] Jairo Bochi, Rafael Potrie, and Andrés Sambarino, _Anosov representations and dominated splittings_ , J. Eur. Math. Soc. (JEMS) 21 (2019), no. 11, 3343–3414.
  * [BT17] Jean-Philippe Burelle and Nicolaus Treib, _Schottky groups and maximal representations_ , Geometriae Dedicata 195 (2017), no. 1, 215–239.
  * [Che93] Qing Cheng, _Space-like surfaces in an anti-de sitter space_ , Colloquium Mathematicum 66 (1993), 201–208.
  * [CTT19] Brian Collier, Nicolas Tholozan, and Jérémy Toulisse, _The geometry of maximal representations of surface groups into ${\rm SO}_{0}(2,n)$_, Duke Math. J. 168 (2019), no. 15, 2873–2949.
  * [Dav23] Colin Davalo, _Nearly geodesic immersions and domains of discontinuity_ , 2023, 2303.11260.
  * [DR24] Colin Davalo and Max Riestenberg, _Finite-sided dirichlet domains for anosov representations_ , 2024.
  * [Eps86] Charles L. Epstein, _The hyperbolic Gauss map and quasiconformal reflections_ , J. Reine Angew. Math. 372 (1986), 96–135.
  * [ES22] Christian El Emam and Andrea Seppi, _On the gauss map of equivariant immersions in hyperbolic space_ , 2022, 2008.07390.
  * [GW12] Olivier Guichard and Anna Wienhard, _Anosov representations: domains of discontinuity and applications_ , Invent. Math. 190 (2012), no. 2, 357–438.
  * [HW15] Zheng Huang and Biao Wang, _Counting minimal surfaces in quasi-fuchsian three-manifolds_ , Transactions of the American Mathematical Society 367 (2015), no. 9, 6063–6083.
  * [LT22] François Labourie and Jérémy Toulisse, _Quasicircles and quasiperiodic surfaces in pseudo-hyperbolic spaces_ , 2022, 2010.05704.
  * [Wig98] Mark Wiggerman, _The fundamental group of a real flag manifold_ , Indag. Math. (N.S.) 9 (1998), no. 1, 141–153.