Soliton resolution for the Harry Dym equation with weighted Sobolev initial data

The Harry Dym equation was first discovered by H.Dym in 1973-1974, and its first appearance in the literature occurred in a 1975 paper of Kruskal [1] where it was named after its discoverer, then it was rediscovered in more general form in [2] within the classical string problem. In 1978, the Harry Dym equation, the KdV equation, the mKdV equation and the nonlinear Schrödinger equation were used as examples to verify a general model of integrable Hamiltonian equation [3], which gives its bi-Hamiltonian structure. The equation is proved to satisfy the properties: infinite number of conservation laws and infinitely many symmetries [4]. The recursion operators and complete Lie-Bäcklund symmetry of this equation were given by [5].

Direct links were found between the KdV equation and Harry Dym equation [6], or mKdV equation and Harry Dym equation [7, 8]. The Lax pair of the Harry Dym equation is associated with the Sturm-Liouville operator. The Liouville transformation transforms this operator isospectrally into the Schrödinger operator [9]. The Dym equation is an important nonlinear partial differential equation which is integrable and finds applications in several physical systems, which is related to such physical problems as the Hele-Shaw problem [10], the Saffmae-Taylor problem and the chiral dynamics of closed curves on the complex plane [11].

Harry Dym equation, as a completely integrable non-linear evolution equation, admits a cusp solitary wave solution in implicit expression by the inverse scattering transform (IST) in [12]. It has various forms of solutions: the algebraic geometric solution [13], the exact solution [14], the elliptic solution [5] and N-loop solitons are constructed by means of N-cusp soliton solutions of Harry Dym equation [1].

In 1974, Manakov first studied the long time behavior of the nonlinear wave equation solvable by the inverse scattering method [15]. Then Zakharov and Manakov use this method and give the first result of the large time asymptoticity of the NLS equation [16]. The inverse scattering method also apply to KdV, Landau-Lifshitz and the reduced Maxwell-Bloch system [17, 18, 19]. In 1993, Deift and Zhou proposed the nonlinear steepest descent method which can obtain the long-term asymptotic behavior of the solution for the MKdV equation by deforming contours to reduce the original Riemann-Hilbert problem (RHP) to a model one whose solution is calculated in terms of parabolic cylinder functions [20]. Since then, this method has been widely used for focusing NLS equation [21], KdV equation [22], Fokas-Lenells equation [23], derivative NLS equation [24], short pulse equation [25], Camassa-Holm equations [26] and the Harry Dym equation [27].

In recent years, McLaughlin and Miller further presented a $\bar{\partial}$ steepest descent method which combine steepest descent with $\bar{\partial}$-problem rather than the asymptotic analysis of singular integrals on contours to analyze asymptotic of orthogonal polynomials with non-analytical weights [28]. When it is applied to integrable systems, the $\bar{\partial}$ steepest descent method also has displayed some advantages, such as avoiding delicate estimates involving $L^{p}$ estimates of Cauchy projection operators, and leading the non-analyticity in the RHP reductions to a $\bar{\partial}$-problem in some sectors of the complex plane which can be solved by being recast into an integral equation and by using Neumann series. Dieng and McLaughin use it to study the defocusing NLS equation under essentially minimal regularity assumptions on finite mass initial data [29]. This $\bar{\partial}$ steepest descent method also was successfully applied to prove asymptotic stability of N-soliton solutions to focusing NLS equation [30]. Jenkins studied soliton resolution for the derivative nonlinear NLS equation for generic initial data in a weighted Sobolev space [31]. Their work provided the soliton resolution property for derivative NLS equation, which decomposes the solution into the sum of a finite number of separated solitons and a radiative parts when $t\rightarrow\infty$. And the dispersive part contains two components, one coming from the continuous spectrum and another from the interaction of the discrete and continuous spectrum.

In this paper, the main purpose is to study the long time asymptotic behavior for the initial value problem of the Harry Dym equation, we apply $\bar{\partial}$ steepest descent method [30] to study the Cauchy problem for the equation

$$q_{t}-2\left(\frac{1}{\sqrt{1+q}}\right)_{xxx}=0$$

(1.1)

with the initial value

$$q(x,0)=q_{0}(x)\in H^{1,1}(\mathbb{R}),$$

(1.2)

where

$$H^{1,1}(\mathbb{R})=\left\{f\in L^{2}(\mathbb{R}):f^{\prime},xf\in L^{2}(\mathbb{R})\right\}$$

(1.3)

and $q(x,t)\geq-1$ for $x\in R,t\geq 0.$

In general, the matrix Riemann-Hilbert problem is defined in the $\lambda$ plane and has explicit $(x,t)$ dependence, while for the Harry-Dym equation (1.1), we need to construct a new matrix Riemann-Hilbert problem with explicit $(y,t)$ dependence, where $y(x,t)$ is a function unknown from the initial value condition. For this purpose, let $u=\sqrt{1+q}$, the problem of Harry Dym equation (1.1) transforms into

$$\left(u^{2}\right)_{t}-2\left(\frac{1}{u}\right)_{xxx}=0,$$

(1.4)

$$u(x,0)=u_{0}(x)=\sqrt{1+q_{0}(x)}.$$

In addition, assume that the initial value $q_{0}(x)$ satisfy two conditions:

$$\int_{-\infty}^{+\infty}q_{0}(x)dx=0,\quad\int_{-\infty}^{+\infty}\int_{\pm\infty}^{x}q_{0}\left(x^{\prime}\right)dx^{\prime}dx=0,\quad x\in R.$$

(1.5)

The assumptions in (1.5) imply the following conditions which are needs for the spectral analysis:

$$\int_{-\infty}^{+\infty}q(x,t)dx=0,\quad\int_{-\infty}^{+\infty}\int_{\pm\infty}^{x}q\left(x^{\prime},t\right)dx^{\prime}dx=0,x\in R,t\geq 0.$$

(1.6)

Organization of the paper: In section 2, by deformation of Lax pairs, we analyze eigenfunctions at spectral parameter $\lambda\rightarrow\infty$ and $\lambda\rightarrow 0$ with the initial data. In section 3, the solution of (1.1-1.2) can be expressed by associated matrix RHP for $M(\lambda)$ dependent on the new parameter $(y,t)$ when $\lambda\rightarrow 0$. In section 4, by introducing a function $T(\lambda)$, we get a new RHP for $M^{(1)}(\lambda)$, which admits a regular discrete spectrum distribution and two triangular decompositions of the jump matrix near the phase point $\pm\lambda_{0}$. In section 5, we construct a mixed $\bar{\partial}$-RH problem for $M^{(2)}(\lambda)$ by introducing a transformation caused by the function $\mathcal{R}(\lambda)$, which make continuous extension off the real axis to the jump matrix. In section 6, we decompose $M^{(2)}(\lambda)$ into two parts: $M^{(2)}_{RHP}(\lambda)$ and a pure $\bar{\partial}$ problem for $M^{(3)}(\lambda)$. And $M^{(2)}_{RHP}(\lambda)$ can be divided into outer model $M^{(2)}_{out}$ be solved in section 7 and the solvable RH model $M^{(\pm\lambda_{0})}$ which can be approximated by a solvable model in [27] in section 8. In section 9, we study the small norm problem of $E(\lambda)$. In section 10, we analyze the pure $\bar{\partial}$ problem for $M^{(3)}$. Finally, based on the series of transformations we have done, we obtain (11.2), which contributes to the soliton resolution and long-time asymptotic behavior for the Harry Dym equation.

Let

$$\sigma_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}.$$

The Harry Dym equation (1.4) admits the following Lax pair

$$\left\{\begin{aligned} &\psi_{xx}=-\lambda^{2}u^{2}\psi,\\ &\psi_{t}=2\lambda^{2}\left[\frac{2}{u}\psi_{x}-\left(\frac{1}{u}\right)_{x}\psi\right].\end{aligned}\right.$$

(2.1)

Write $\varphi=\left(\begin{array}[]{c}\psi\\ \psi_{x}\end{array}\right)$, then (2.1) can be written as

$$\left\{\begin{aligned} &\varphi_{x}=M\varphi,\\ &\varphi_{t}=N\varphi.\end{aligned}\right.$$

(2.2)

where

$$M=\begin{pmatrix}0&1\\[6.0pt] -\lambda^{2}u^{2}&0\end{pmatrix},\ N=\begin{pmatrix}-2\lambda^{2}\left(\frac{1}{u}\right)_{x}&4\lambda^{2}\frac{1}{u}\\[6.0pt] -4\lambda^{4}u-2\lambda^{2}\left(\frac{1}{u}\right)_{xx}&2\lambda^{2}\left(\frac{1}{u}\right)_{x}\end{pmatrix}.$$

we analyze the eigenfunctions at spectral parameter $\lambda\rightarrow 0$ and $\lambda\rightarrow\infty$ with the initial data. (1) The case of $\lambda=0$

The Harry Dym equation (1.4) has also the compatibility condition of the following Lax pair

$$\left\{\begin{aligned} &\Phi_{x}^{0}+i\lambda\sigma_{3}\Phi^{0}=U_{0}\Phi^{0},\\ &\Phi_{t}^{0}+4i\lambda^{3}\sigma_{3}\Phi^{0}=V_{0}\Phi^{0},\end{aligned}\right.$$

(2.3)

where

		$\displaystyle\Phi^{0}=P\varphi,\quad P^{-1}=\begin{pmatrix}1&1\\ -i\lambda&i\lambda\end{pmatrix},$
		$\displaystyle U_{0}(x,t)=\frac{\lambda q}{2}(\sigma_{2}-i\sigma_{3}),$
		$\displaystyle V_{0}(x,t,\lambda)=2\lambda^{3}[\frac{q}{u}\sigma_{2}-i(\frac{1}{u}+u-2)\sigma_{3}]-2\lambda^{2}(\frac{1}{u})_{x}\sigma_{1}+(\frac{1}{u})_{xx}(\sigma_{2}-i\sigma_{3}).$

Introducing the following transformation

$$\Psi^{0}=\Phi^{0}\exp[i(\lambda x+4\lambda^{3}t)\sigma_{3}],$$

then we get the Lax pair of $\Psi^{0}$

$$\left\{\begin{aligned} &\Psi_{x}^{0}+i\lambda[\sigma_{3},\Psi^{0}]=U_{0}\Psi^{0},\\ &\Psi_{t}^{0}+4i\lambda^{3}[\sigma_{3},\Psi^{0}]=V_{0}\Psi^{0}.\end{aligned}\right.$$

(2.4)

As $|x|\rightarrow\infty,U_{0}\rightarrow 0$. The two eigenfunctions $\Psi_{\pm}^{0}$ of equation (2.4) satisfy the Volterra integral equations

$$\Psi_{+}^{0}(x,t,\lambda)=I-\int_{x}^{\infty}e^{-i\lambda\left(x-x^{\prime}\right)\hat{\sigma}_{3}}U_{0}\left(x^{\prime},t\right)\Psi_{+}^{0}\left(x^{\prime},t,\lambda\right)dx^{\prime},$$

$$\Psi_{-}^{0}(x,t,\lambda)=I+\int_{-\infty}^{x}e^{-i\lambda\left(x-x^{\prime}\right)\hat{\sigma}_{3}}U_{0}\left(x^{\prime},t\right)\Psi_{-}^{0}\left(x^{\prime},t,\lambda\right)dx^{\prime}.$$

Denote these vectors with superscripts $(i)$ to indicate the $i-th$ column of vectors, then

$$\Psi_{\pm}^{0}=(\Psi_{\pm}^{0,(1)},\Psi_{\pm}^{0,(2)}).$$

From the above definition, we find that

$$\Psi_{+}^{0,(2)}(x,t,\lambda),\Psi_{-}^{0,(1)}(x,t,\lambda)\text{ are bounded and analytic for }\lambda\in\mathbb{C}_{+},$$

$$\Psi_{+}^{0,(1)}(x,t,\lambda),\Psi_{-}^{0,(2)}(x,t,\lambda)\text{ are bounded and analytic for }\lambda\in\mathbb{C}_{-},$$

where $\mathbb{C}_{+}=\{\lambda\in\mathbb{C}\mid\operatorname{Im}\lambda>0\},\mathbb{C}_{-}=\{\lambda\in\mathbb{C}\mid\operatorname{Im}\lambda<0\}.$

Introducing the gauge transformation

$$\Phi=\mathrm{PD}\varphi,$$

(2.6)

where

$$P^{-1}=\begin{pmatrix}1&1\\ -i\lambda&i\lambda\end{pmatrix},\quad D=\begin{pmatrix}\sqrt{u}&0\\ 0&\frac{1}{\sqrt{u}}\end{pmatrix}.$$

Then we have the Lax pair of $\varphi$ (2.2) becomes

$$\left\{\begin{aligned} &\Phi_{x}+i\lambda u\sigma_{3}\Phi=U\Phi,\\ &\Phi_{t}+i\left[\lambda\left[\frac{1}{u}\left(\frac{1}{u}\right)_{xx}-\frac{1}{2}\left(\left(\frac{1}{u}\right)_{x}\right)^{2}\right]+4\lambda^{3}\right]\sigma_{3}\Phi=V\Phi.\end{aligned}\right.$$

(2.7)

where

		$\displaystyle U(x,t)=\frac{1}{4}\frac{q_{x}}{u^{2}}\sigma_{1},$
		$\displaystyle V(x,t,\lambda)=-2\lambda^{2}\left(\frac{1}{u}\right)_{x}\sigma_{1}+\lambda\left[\frac{1}{u}\left(\frac{1}{u}\right)_{xx}\sigma_{2}-\frac{i}{2}\left(\left(\frac{1}{u}\right)_{x}\right)^{2}\sigma_{3}\right]+\frac{1}{2u^{2}}\left(\frac{1}{u}\right)_{xxx}\sigma_{1}.$

It is clear that as $|x|\rightarrow\infty,U(x,t)\rightarrow 0,V(x,t,\lambda)\rightarrow 0$.

In order to dedicate solutions via integral Volterra equation more conveniently, we make a transformation

$$\Psi=\Phi\exp\left[i\lambda p(x,t,\lambda)\sigma_{3}\right],$$

(2.8)

where

$$p(x,t,\lambda)=x+\int_{x}^{\infty}(1-u(\xi,t))d\xi+4\lambda^{2}t.$$

(2.9)

From (2.9), we have

$$p_{x}=u(x,t),\quad p_{t}=-\int_{x}^{\infty}u_{t}(\xi,t)d\xi+4\lambda^{2},$$

and the conservation law

$$u_{t}-\left(-\frac{1}{2}\left(\left(\frac{1}{u}\right)_{x}\right)^{2}+\frac{1}{u}\left(\frac{1}{u}\right)_{xx}\right)_{x}=0$$

implies that

$$p_{t}=-\frac{1}{2}\left(\left(\frac{1}{u}\right)_{x}\right)^{2}+\frac{1}{u}\left(\frac{1}{u}\right)_{xx}+4\lambda^{2}.$$

Then the Lax pair (2.7) is changed into a new Lax pair

$$\left\{\begin{aligned} &\Psi_{x}+i\lambda p_{x}\left[\sigma_{3},\Psi\right]=U\Psi,\\ &\Psi_{t}+i\lambda p_{t}\left[\sigma_{3},\Psi\right]=V\Psi.\end{aligned}\right.$$

(2.10)

Because the eigenfunctions $\Psi_{\pm}$ are both the solutions of (2.11), define the spectral function $s(\lambda)$ by

$$\Psi_{-}(x,t,\lambda)=\Psi_{+}(x,t,\lambda)e^{-i\lambda p(x,t,\lambda)\hat{\sigma}_{3}}s(\lambda).$$

(2.13)

From (2.10), we can deduce that

$$det\left(\Psi_{\pm}(x,t,\lambda)\right)=1.$$

(2.14)

According the definition of $U(x,t)$, it is obvious that $U(x,t)=\sigma_{1}U(x,t)\sigma_{1},$ we get the $\Psi_{\pm}(x,t,\lambda)$ satisfy the symmetry properties

$$\left\{\begin{aligned} &\Psi_{\pm 11}(x,t,\lambda)=\overline{\Psi_{\pm 22}(x,t,\bar{\lambda})},\quad\Psi_{\pm 21}(x,t,\lambda)=\overline{\Psi_{\pm 12}(x,t,\bar{\lambda})},\\ &\Psi_{\pm 11}(x,t,-\lambda)=\Psi_{\pm 22}(x,t,\lambda),\quad\Psi_{\pm 12}(x,t,-\lambda)=\Psi_{\pm 21}(x,t,\lambda).\end{aligned}\right.$$

(2.15)

Then the matrix $s(\lambda)$ has the form

$$s(\lambda)=\begin{pmatrix}a(\lambda)&\overline{b(\bar{\lambda})}\\ b(\lambda)&\overline{a(\bar{\lambda})}\end{pmatrix},$$

(2.16)

where $a(-\bar{\lambda})=\overline{a(\lambda)}$ and $b(-\bar{\lambda})=\overline{b(\lambda)}$, from (2.14) we get $det(s(\lambda))=1$, combining (2.13) and (2.14) gives

$$a(\lambda)=det(\Psi_{-}^{(1)},\Psi_{+}^{(2)}),$$

(2.17)

then we can get $a(\lambda)$ and $b(\lambda)$ have the following properties:

• •

  $a(\lambda)$ is analytic in $\mathbb{C}_{+}$ and continuous for $\lambda\in\overline{\mathbb{C}}_{+}$, $b(\lambda)$ is continuous for $\lambda\in R.$

• •

  $a(\lambda)=1+O\left(\frac{1}{\lambda}\right),\lambda\rightarrow\infty$; $a(\lambda)=1+O(\lambda),\lambda\rightarrow 0,$ for $\lambda\in\mathbb{C}_{+}$.

• •

  $b(\lambda)=O\left(\frac{1}{\lambda}\right),\lambda\rightarrow\infty$; $b(\lambda)=O(\lambda),\lambda\rightarrow 0,$ for $\lambda\in R.$

We introduce the reflection coefficient

$$r(\lambda)=\frac{b(\lambda)}{a(\lambda)}$$

with symmetry $r(-\bar{\lambda})=\overline{r(\lambda)}$. And for $a(\lambda)\neq 0$ for all $\lambda$ such that $\operatorname{Im}\lambda\geq 0$ [27], we assume our initial data satisfy the following assumption.

In the following analysis, we formulate a RH problem by defining the matrix function $M(x,t,z)$ with eigenfunctions $\Psi_{\pm}$, while the reconstruction formula between the solution $q(x,t)$ and the RH problem can be found from the asymptotic of $\Psi_{\pm}$ as $\lambda\rightarrow 0$. So we need to calculate the relation between $\Psi_{\pm}$ and $\Psi_{\pm}^{0}$.

The eigenfunctions $\Psi_{\pm}$ and $\Psi_{\pm}^{0}$ can be related to each other as

$$\Psi_{\pm}=G(x,t)\Psi_{\pm}^{0}e^{-i\left(\lambda x+4\lambda^{3}t\right)\sigma_{3}}C_{\pm}(\lambda)e^{i\lambda p(x,t,\lambda)\sigma_{3}},$$

(2.18)

where $C_{\pm}(\lambda)$ is independent of $x$ and $t$,

$$G(x,t)=\mathrm{PDP}^{-1}=\frac{1}{2}\begin{pmatrix}\sqrt{u}+\frac{1}{\sqrt{u}}&\sqrt{u}-\frac{1}{\sqrt{u}}\\ \sqrt{u}-\frac{1}{\sqrt{u}}&\sqrt{u}+\frac{1}{\sqrt{u}}\end{pmatrix}.$$

(2.19)

Considering (2.18) when $x\rightarrow\pm\infty$, we have $C_{+}(\lambda)=I$ and $C_{-}(\lambda)=e^{ic\lambda\sigma_{3}}$, where
$c=\int_{-\infty}^{\infty}(u(\xi,t)-1)d\xi$ is a quantity conserved under the dynamics governed by (1.1).

Note that (2.18) can be written as

		$\displaystyle\Psi_{+}(x,t,\lambda)=G(x,t)\Psi_{+}^{0}(x,t,\lambda)e^{-i\lambda F(x,t)\sigma_{3}},$		(2.20)
		$\displaystyle\Psi_{-}(x,t,\lambda)=G(x,t)\Psi_{-}^{0}(x,t,\lambda)e^{i\lambda\int_{-\infty}^{x}(\rho(\xi,t)-1)d\xi\sigma_{3}},$

where $F(x,t)=\int_{x}^{\infty}(u(\xi,t)-1)d\xi$, then we have (2.18) and (2.20) imply that as $\lambda\rightarrow 0$, coefficients of the expansions for $\Psi_{\pm}(x,t,\lambda)$ can be expressed by $q(x,t)$. In addition, according to (2.5), (2.17) and (2.20), we find

$$a(\lambda)=1+i\lambda c+(i\lambda)^{2}\frac{c^{2}}{2}+O\left(\lambda^{3}\right),\quad\text{ as }\lambda\rightarrow 0.$$

(2.21)

Suppose that $\Lambda=\{\lambda_{n},n=1,\cdots,N\}$ are simple zeros of $a(\lambda)$, which implies that $\overline{\lambda}_{n}\in\mathbb{C}_{-}$ are simple zeros of $\overline{a(\bar{\lambda})}$, denote $\bar{\Lambda}=\{\overline{\lambda}_{n},n=1,\cdots,N\}$.

Define a sectionally analytic matrix $M(x,t,\lambda)$

$$M(x,t,\lambda)=\left\{\begin{aligned} \left(\frac{\Psi_{-}^{(1)}}{a(\lambda)},\Psi_{+}^{(2)}\right),\quad\lambda\in\mathbb{C}^{+},\\ \left(\Psi_{+}^{(1)},\frac{\Psi_{-}^{(2)}}{\overline{a(\bar{\lambda})}}\right),\quad\lambda\in\mathbb{C}^{-}.\end{aligned}\right.$$

(3.1)

we can get the symmetry of $M(x,t,\lambda)$ by (2.15)

$$\overline{M(x,t,\bar{\lambda})}=M(x,t,-\lambda)=\sigma_{1}M(x,t,\lambda)\sigma_{1}.$$

(3.2)

The function $M$ defined by (3.1) satisfies the following Riemann-Hilbert problem.
RHP 1 Find a matrix-valued function $M(x,t,\lambda)$ with the following properties

1. 1.

   Analyticity: $M(x,t,\lambda)$ is meromorphic in $\mathbb{C}\backslash\mathbb{R}.$

2. 2.

   Jump condition: The boundary values $M_{\pm}$ of $M$ as $\lambda\rightarrow R$ from $\mathbb{C}_{\pm}$ are related by

   $$M_{+}(x,t,\lambda)=M_{-}(x,t,\lambda)J(x,t,\lambda),\quad\lambda\in R$$

   (3.3)

   where

   $$J(x,t,\lambda)=\begin{pmatrix}1-|r(\lambda)|^{2}&-\overline{r(\bar{\lambda})}e^{-2i\lambda p(x,t,\lambda)}\\[6.0pt] r(\lambda)e^{2i\lambda p(x,t,\lambda)}&1\end{pmatrix}.$$

   (3.4)

3. 3.

   Normalization: As $\lambda\rightarrow\infty,\quad M(x,t,\lambda)\rightarrow I.$

4. 4.

   Residues: At $\lambda=\lambda_{n}$ and $\lambda=\overline{\lambda}_{n}$, $M(x,t,\lambda)$ has simple poles and the residues satisfy the conditions

   		$\displaystyle\mathop{\operatorname{Res}}\limits_{\lambda=\lambda_{n}}M(x,t,\lambda)=\lim\limits_{\lambda\rightarrow\lambda_{n}}M(x,t,\lambda){\begin{pmatrix}0&0\\[6.0pt] c_{n}e^{2i\lambda_{n}p(\lambda_{n})}&0\end{pmatrix}},$		(3.5)
   		$\displaystyle\mathop{\operatorname{Res}}\limits_{\lambda=\overline{\lambda}_{n}}M(x,t,\lambda)=\lim\limits_{\lambda\rightarrow\overline{\lambda}_{n}}M(x,t,\lambda)\begin{pmatrix}0&\overline{c}_{n}e^{-2i\overline{\lambda}_{n}p(\overline{\lambda}_{n})}\\[6.0pt] 0&0\end{pmatrix},$

   where $c_{n}=\frac{b(\lambda_{n})}{a^{\prime}(\lambda_{n})}.$

Considering the asymptotic behavior for $M(x,t,\lambda)$ when $\lambda\rightarrow 0$, we can solve solution of (1.1)-(1.2) by RHP1. It follows from (2.5), (2.20) and (2.21), for $\lambda\rightarrow 0$ we have

	$\displaystyle M(x,t,\lambda)$	$\displaystyle=G(x,t)\left\{I-i\lambda\left[\frac{A(x,t)}{2}\left(i\sigma_{2}+\sigma_{3}\right)+F(x,t)\sigma_{3}\right]\right.$		(3.6)
		$\displaystyle\left.+(i\lambda)^{2}\left[A_{1}(x,t)\sigma_{1}+\frac{A(x,t)F(x,t)}{2}\left(I-\sigma_{1}\right)+\frac{F^{2}(x,t)}{2}I\right]+O\left(\lambda^{3}\right)\right\}.$

We introduce a row vector function

$$\left(\mu_{1},\mu_{2}\right)(x,t,\lambda)=(1,1)M(x,t,\lambda).$$

(3.7)

From (3.6), then it follows that as $\lambda\rightarrow 0$,

		$\displaystyle\mu_{1}(x,t,\lambda)=\sqrt{\rho}\left\{1-i\lambda F(x,t)+(i\lambda)^{2}\left[A_{1}(x,t)+\frac{F^{2}(x,t)}{2}\right]+O\left(\lambda^{3}\right)\right\},$		(3.8)
		$\displaystyle\mu_{2}(x,t,\lambda)=\sqrt{\rho}\left\{1+i\lambda F(x,t)+(i\lambda)^{2}\left[A_{1}(x,t)+\frac{F^{2}(x,t)}{2}\right]+O\left(\lambda^{3}\right)\right\},$

which yield

		$\displaystyle\frac{\mu_{1}}{\mu_{2}}(x,t,\lambda)=1-2i\lambda F(x,t)+O\left(\lambda^{2}\right),$		(3.9)
		$\displaystyle q(x,t)=\left(\mu_{1}\mu_{2}\right)^{2}(x,t,0)-1.$

Equations (3.9) show that we can reconstruct the solution of the initial value problem of (1.1)-(1.2) by the matrix-valued function $M(x,t,\lambda)$. However, due to the jump matrix $J$ are determined by $q_{0}(x)$, $e^{-2i\lambda p(x,t,\lambda)}$, while $p(x,t,\lambda)$ cannot be given by initial data $q_{0}(x)$. Thus, we introduce the new scale

$$y(x,t)=x-\int_{x}^{\infty}(u(\xi,t)-1)d\xi=x-F(x,t),$$

(3.10)

which makes the jump matrix $J$ explicitly dependent on the parameters $(y,t)$:

$$J(x,t,\lambda)=\hat{J}(y,t,\lambda)=\begin{pmatrix}1-|r(\lambda)|^{2}&-\overline{r(\bar{\lambda})}e^{-2i\left(\lambda y+4\lambda^{3}t\right)}\\[6.0pt] r(\lambda)e^{2i\left(\lambda y+4\lambda^{3}t\right)}&1\end{pmatrix},\quad\lambda\in R.$$

(3.11)

By the definition of the new sacle $y(x,t)$, then we can transform this RH problem into a new RH problem parametrized by $(y,t)$. Define

$$\begin{array}[]{l}\hat{M}(y,t,\lambda)=M(x(y,t),t,\lambda),\\[6.0pt] (\hat{\mu}_{1},\hat{\mu}_{2})(y,t,\lambda)=(1,1)\hat{M}(y,t,\lambda).\end{array}$$

(3.12)

It follows from (3.13) that $\hat{M}(y,t,\lambda)$ has the symmetry property

$$\overline{\hat{M}(y,t,\bar{\lambda})}=\hat{M}(y,t,-\lambda)=\sigma_{1}\hat{M}(y,t,\lambda)\sigma_{1}.$$

(3.13)

We construct the row function $\hat{M}(y,t,\lambda)$ as a solution of the following $2\times 2$ matrix Riemann-Hilbert problem.
RHP 2 Find a matrix function $\hat{M}(y,t,\lambda)$ with the following properties

1. 1.

   Analyticity: $\hat{M}(y,t,\lambda)$ is meromorphic in $\mathbb{C}\backslash\mathbb{R}.$

2. 2.

   Jump condition:

   $$\hat{M}_{+}(y,t,\lambda)=\hat{M}_{-}(y,t,\lambda)\hat{J}(y,t,\lambda),\quad\lambda\in R,$$

   (3.14)

   where $\hat{J}(y,t,\lambda)$ is defined by (3.11).

3. 3.

   Normalization: $\hat{M}(y,t,\lambda)\rightarrow I$, as $\lambda\rightarrow\infty$.

4. 4.

   Residues: At $\lambda=\lambda_{n}$ and $\lambda=\overline{\lambda}_{n}$, $\hat{M}(y,t,\lambda)$ has simple poles and the residues satisfy the conditions

   		$\displaystyle\mathop{\operatorname{Res}}\limits_{\lambda=\lambda_{n}}\hat{M}(y,t,\lambda)=\lim\limits_{\lambda\rightarrow\lambda_{n}}\hat{M}(y,t,\lambda){\begin{pmatrix}0&0\\[6.0pt] c_{n}e^{2i(\lambda_{n}y+4\lambda_{n}^{3}t)}&0\end{pmatrix}},$		(3.15)
   		$\displaystyle\mathop{\operatorname{Res}}\limits_{\lambda=\overline{\lambda}_{n}}\hat{M}(y,t,\lambda)=\lim\limits_{\lambda\rightarrow\overline{\lambda}_{n}}\hat{M}(y,t,\lambda)\begin{pmatrix}0&\overline{c}_{n}e^{-2i(\bar{\lambda}_{n}y+4\bar{\lambda}_{n}^{3}t)}\\[6.0pt] 0&0\end{pmatrix}.$

Then, we have

• •

  The solution $\hat{M}(y,t,\lambda)$ exists and is unique.

• •

  The solution of the initial value problem of (1.1)-(1.2) can be expressed in parametric form of $(y,t)$ in terms of the solution $\hat{M}(y,t,\lambda)$ of above RHP 2 as follows,

  $$q(x,t)=\hat{q}(y,t),$$

  (3.16)

  where

  		$\displaystyle\hat{q}(y,t)=\left(\hat{\mu}_{1}\hat{\mu}_{2}\right)^{2}(y,t,0)-1,$		(3.17)
  		$\displaystyle x(y,t)=y-\lim\limits_{\lambda\rightarrow 0}\frac{1}{2i\lambda}\left(\frac{\hat{\mu}_{1}}{\hat{\mu}_{2}}(y,t,\lambda)-1\right),$

  and the $\hat{\mu}_{1},\hat{\mu}_{2}$ defined by (3.12).