1. Introduction
We consider the Cauchy problem for the defocusing Kundu–Eckhaus (KE) equation with nonzero boundary conditions1.1 ) under the change of variables $q\to q{{\rm{e}}}^{-2{\rm{i}}(1-2{\beta }^{2})t}$. However, the advantage of (1.1 ) is that its solution satisfying (1.2 ) is asymptotically time-independent as x → ± ∞. The focusing KE equation takes the1.3 ) and (1.4 ) reduce to defocusing and focusing NLS equation respectively. In recent years, exact solutions for both KE equations (1.3 ) and (1.4 ) have been investigated via different methods. for example, the N-soliton solutions are constructed by using Darboux transformation and Riemann–Hilbert (RH) method [3, 4]. Guo et al constructed rogue wave and multi-pole solutions for the KE equation (1.4 ) with nonzero boundary conditions via the RH method [5]. The rogue waves and higher-order rogue wave solutions in a chaotic wave were obtained [6, 7].
$\begin{eqnarray}\begin{array}{l}{\rm{i}}{q}_{t}+{q}_{{xx}}-2(| q{| }^{2}-1)q+4{\beta }^{2}(| q{| }^{4}-1)q\\ \,\,\,\,+\ 4{\rm{i}}\beta {\left(| q{| }^{2}\right)}_{x}q=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}q(x,0)={q}_{0}(x)\sim \pm 1,\ x\to \pm \infty ,\end{eqnarray}$
where ${q}_{0}\in \tanh x+{H}^{\mathrm{4,4}}({\mathbb{R}})$ and β is a constant. The usual form of the defocusing KE equation is $\begin{eqnarray}{\rm{i}}{q}_{t}+{q}_{{xx}}-2| q{| }^{2}q+4{\beta }^{2}| q{| }^{4}q+4{\rm{i}}\beta {\left(| q{| }^{2}\right)}_{x}q=0,\end{eqnarray}$
which reduces to ( $\begin{eqnarray}{\rm{i}}{q}_{t}+{q}_{{xx}}+2| q{| }^{2}q+4{\beta }^{2}| q{| }^{4}q-4{\rm{i}}\beta {\left(| q{| }^{2}\right)}_{x}q=0,\end{eqnarray}$
which was first introduced by Kundu [1] and later by Calogero and Eckhaus [2] from different perspectives. For the special case β = 0, the KE equations (In this paper, we are especially concerned with long-time asymptotics for the KE equation. For the focusing case, Wang et al studied the long-time asymptotic for the KE equation (1.4 ) with nonzero boundary conditions [8]. Ma and Fan obtained the long-time asymptotics for the KE equation (1.4 ) where weighted Sobolev initial data was investigated by using $\bar{\partial }$-steepest descent method [9]. Guo and Liu obtained the long-time asymptotics for the focusing KE equation on the half-line [10]. For the defocusing case, Zhu et. al presented the long-time asymptotic for the KE equation (1.3 ) with Schwartz initial data [11]. As a special case of the KE equation (1.1 ) as β = 0, the soliton resolution and long-time asymptotics for the defocusing have been obtained [12–14]. In this paper, we hope to generalize these results to the KE equation so as to give complete approximation formulas in three space-time regions of (x,t) half-plane for the KE equation (1.3 ) with nonzero boundary conditions by $\bar{\partial }$-steepest descent method. For this purpose, we make an appropriate transformation such that the associated RH problem for the KE equation into the RH problem for NLS equation.
The organization of this paper is as follows. In section 2 , we consider inverse scattering transform for weighted Sobolev initial data ${q}_{0}\in \tanh x+{H}^{\mathrm{4,4}}({\mathbb{R}})$. Based on the analyticity, symmetries and asymptotics for the Jost functions and scattering data, we construct a basic RH problem associated with the Cauchy problem (1.1 )–(1.2 ). In section 3 , with the $\bar{\partial }$-steepest descent method and the results of the defocusing NLS equation, we give a complete leading order approximation formula for the defocusing KE equation on the whole (x,t) half-plane including the soliton resolution in the solitonic region, Zakharov-shabat asymptotics in the solitonless region and the Painlevé asymptotics in the transition region.
2. Inverse scattering transform
2.1. Spectral analysis
The KE equation (1.1 ) admits the following Lax pair
$\begin{eqnarray}{\psi }_{x}+{\rm{i}}k{\sigma }_{3}\psi =U\psi ,\end{eqnarray}$
$\begin{eqnarray}{\psi }_{t}+2{\rm{i}}{k}^{2}{\sigma }_{3}\psi =V\psi ,\end{eqnarray}$
where $U=\mathrm{diag}(q,\bar{q}){\sigma }_{1}$, and $\begin{eqnarray}\begin{array}{rcl}V & = & [4{\rm{i}}{\beta }^{2}({U}^{4}-I)-{\rm{i}}({U}^{2}-I)\\ & & -{\rm{i}}{U}_{x}]{\sigma }_{3}+2{kU}-2\beta ({U}^{3}-U)+\beta [{U}_{x},U].\end{array}\end{eqnarray}$
Under the boundary condition (1.2 ), we define the Jost solutions of Lax pair (2.1 )–(2.2 ) with asymptotics
$\begin{eqnarray*}{\psi }^{\pm }(z)\sim {Y}_{\pm }{{\rm{e}}}^{-{\rm{i}}t\theta (z){\sigma }_{3}},\quad x\to \pm \infty ,\end{eqnarray*}$
where we denote ψ±(z) = ψ±(z; x, t) and $\begin{eqnarray}{Y}_{\pm }=I\pm {\sigma }_{1}{z}^{-1},\quad \det {Y}_{\pm }=1-{z}^{-2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\theta (z)=\zeta (z)\left({{xt}}^{-1}-2\lambda (z)\right),\\ \,\quad k(z)=\displaystyle \frac{1}{2}(z+{z}^{-1}),\quad \zeta (z)=\displaystyle \frac{1}{2}(z-{z}^{-1}).\end{array}\end{eqnarray}$
Making a transformation
$\begin{eqnarray}{\phi }^{\pm }(z)={\psi }^{\pm }(z){{\rm{e}}}^{{\rm{i}}t\theta (z){\sigma }_{3}},\end{eqnarray}$
then $\begin{eqnarray}{\phi }^{\pm }(z;x,t)\sim {Y}_{\pm },\ x\to \pm \infty ,\end{eqnarray}$
and φ±(z) satisfy the Volterra integral equations$\begin{eqnarray*}\begin{array}{l}{\phi }^{\pm }(z)={Y}_{\pm }\\ \,+{\displaystyle \int }_{\pm \infty }^{x}{Y}_{\pm }{{\rm{e}}}^{-{\rm{i}}\zeta (z)(x-y){\widehat{\sigma }}_{3}}{Y}_{\pm }^{-1}{\rm{\Delta }}{U}_{\pm }(z;y){\phi }^{\pm }(z;y){\rm{d}}y,z\ne \pm 1,\\ {\phi }^{\pm }(z)={Y}_{\pm }\\ \,+{\displaystyle \int }_{\pm \infty }^{x}\left(I+(x-y){L}_{\pm }\right){\rm{\Delta }}{U}_{\pm }(\pm 1;y){\phi }^{\pm }(\pm 1;y){\rm{d}}y,z=\pm 1,\end{array}\end{eqnarray*}$
where ΔU± = U − U± = iΣ3(U ∓ Σ1), and ${\widehat{\sigma }}_{3}$ acts on a matrix A by ${{\rm{e}}}^{{\widehat{\sigma }}_{3}}A={{\rm{e}}}^{{\sigma }_{3}}A{{\rm{e}}}^{-{\sigma }_{3}}$.By denoting ${\phi }^{\pm }(z)=\left({\phi }_{1}^{\pm }(z),{\phi }_{2}^{\pm }(z)\right)$, then the existence, analyticity and differentiability of φ±(z) can be proven directly. Here we list their properties without proof.
Let ${q}_{0}\in \tanh (x)+{L}^{1}({\mathbb{R}})$ and $q{{\prime} }_{0}\in {W}^{\mathrm{1,1}}({\mathbb{R}})$, then we have
| i | (i) ${\phi }_{1}^{+}(z)$ and ${\phi }_{2}^{-}(z)$ can be analytically extended to $z\in {{\mathbb{C}}}^{-}$, while ${\phi }_{1}^{-}(z)$ and ${\phi }_{2}^{+}(z)$ can be analytically extended to $z\in {{\mathbb{C}}}^{+}$. |
| ii | (ii) ${\phi }^{\pm }(z)$ satisfy the symmetries $\begin{eqnarray}{\phi }^{\pm }(z)={\sigma }_{1}\overline{{\phi }^{\pm }(\bar{z})}{\sigma }_{1}=\pm {z}^{-1}{\phi }^{\pm }({z}^{-1}){\sigma }_{1}.\end{eqnarray}$ |
| iii | (iii) ${\phi }^{\pm }(z)$ admit the asymptotic properties $\begin{eqnarray*}\begin{array}{rcl}{\phi }^{\pm }(z) & = & {{\rm{e}}}^{{\rm{i}}\beta {\displaystyle \int }_{\pm \infty }^{x}(| q{| }^{2}-1){\rm{d}}y{\sigma }_{3}}+{ \mathcal O }\left({z}^{-1}\right),\ z\to \infty ,\\ {\phi }^{\pm }(z) & = & -{\rm{i}}{z}^{-1}{\sigma }_{1}{{\rm{e}}}^{{\rm{i}}\beta {\displaystyle \int }_{\pm \infty }^{x}(| q{| }^{2}-1){\rm{d}}y}+{ \mathcal O }(1),\ z\to 0.\end{array}\end{eqnarray*}$ |
To formulate a regular RH problem, we further define
$\begin{eqnarray}{c}^{\pm }(z)={{\rm{e}}}^{-{\rm{i}}\beta {\int }_{\infty }^{x}(| q{| }^{2}-1){\rm{d}}y{\sigma }_{3}}{\phi }^{\pm }(z),\end{eqnarray}$
which admits the asymptotics$\begin{eqnarray*}\begin{array}{rcl}{\mu }^{\pm }(z) & = & I+{ \mathcal O }\left({z}^{-1}\right),\ z\to \infty ,\\ {\mu }^{\pm }(z) & = & -{\rm{i}}{z}^{-1}{\sigma }_{1}+{ \mathcal O }(1),\ z\to 0.\end{array}\end{eqnarray*}$
The Jost functions μ±(z) admit a scattering relation
$\begin{eqnarray}\begin{array}{l}{\mu }^{-}(z)={\mu }^{+}(z)\left(\begin{array}{cc}a(z) & \overline{b(z)}{{\rm{e}}}^{-2{\rm{i}}t\theta (z)}\\ b(z){{\rm{e}}}^{2{\rm{i}}t\theta (z)} & \overline{a(z)}\end{array}\right),\\ \,\ z\in {\mathbb{R}}\setminus \{-1,0,1\},\end{array}\end{eqnarray}$
where a(z) and b(z) are the scattering coefficients, which can be described by the Jost functions $\begin{eqnarray}a(z)=\displaystyle \frac{\det \left[{\psi }_{1}^{-}(z),{\psi }_{2}^{+}(z)\right]}{1-{z}^{-2}},\quad b(z)=\displaystyle \frac{\det \left[{\psi }_{1}^{+}(z),{\psi }_{1}^{-}(z)\right]}{1-{z}^{-2}}.\end{eqnarray}$
We define a reflection coefficient
$\begin{eqnarray}r(z)=\displaystyle \frac{b(z)}{a(z)},\end{eqnarray}$
which satisfies the relation $\begin{eqnarray}| r(z){| }^{2}=1-| a(z){| }^{-2}\lt 1,\ z\in {\mathbb{R}}\setminus \{0,\pm 1\}.\end{eqnarray}$
It is shown that the scattering coefficients and the reflection coefficient have the following properties.
Let ${q}_{0}\in \tanh x+{H}^{\mathrm{2,2}}({\mathbb{R}})$, then
| i | (i)The scattering data a(z) can be analytically extended to $z\in {{\mathbb{C}}}^{+}$, while b(z) is defined for $z\in {\mathbb{R}}\setminus \{0,\pm 1\}$. Zeros of a(z) in ${{\mathbb{C}}}^{+}$ are simple, finite and distribute on the unit circle. |
| ii | (ii) $a(z)=a({z}^{-1})$ , $b(z)=-b({z}^{-1}),$ $r(z)=-r({z}^{-1}).$ |
| iii | (iii)The scattering data admit asymptotics $\begin{eqnarray*}| b(z)| ={ \mathcal O }(| z{| }^{-2}),\ | z| \to \infty ,\ | b(z)| ={ \mathcal O }(| z{| }^{2}),\ | z| \to 0.\end{eqnarray*}$ |
| iv | (iv)In the non-generic case, a(z) and $b(z)$ are continuous at $z=\pm 1$ and $| r(\pm 1)| \lt 1;$ In the generic case, a(z) and b(z) have the first order singularities at $z=\pm 1$, $\begin{eqnarray}a(z)=\displaystyle \frac{{c}_{\pm }}{z\mp 1}+{ \mathcal O }(1),\ b(z)=\mp \displaystyle \frac{{c}_{\pm }}{z\mp 1}+{ \mathcal O }(1),\end{eqnarray}$ which leads to ${\mathrm{lim}}_{z\to \pm 1}r(z)=\mp 1$, where ${c}_{\pm }\,=\det \left[{\psi }_{1}^{-}(\pm 1),{\psi }_{2}^{+}(\pm 1)\right].$ |
2.2. Set-up of the RH problem
Let ${\{{z}_{j}\}}_{j=0}^{N}$ and ${\{{\bar{z}}_{j}\}}_{j=0}^{N}$ be the zeros of a(z) in ${{\mathbb{C}}}^{+}$ and $\overline{a(z)}$ in ${{\mathbb{C}}}^{-}$, respectively. Denote
$\begin{eqnarray*}\begin{array}{rcl}{{ \mathcal Z }}^{+} & = & \{{z}_{j}| a({z}_{j})=0,{z}_{j}\in {{\mathbb{C}}}^{+},| {z}_{j}| =1,j=1,\cdots ,N\},\\ {{ \mathcal Z }}^{-} & = & \{{\bar{z}}_{j}| \overline{a({\bar{z}}_{j})}=0,{\bar{z}}_{j}\in {{\mathbb{C}}}^{-},| {\bar{z}}_{j}| =1,j=1,\cdots ,N\}.\end{array}\end{eqnarray*}$
For $z\in {\mathbb{C}}\setminus {\mathbb{R}}$, and the normalized Jost functions ${\mu }_{j}^{\pm }(z),j=1,2,$ we construct the function1 and 2 , it can be verified that M(z) satisfies the following RH problem.
This is an RH problem with jumps on ${\mathbb{R}}$ and poles distributed on $| z| =1$. See figure 1. The solution to the Cauchy problem (1.1 )–(1.2 ) can be given by the reconstruction formula
$\begin{eqnarray*}\begin{array}{l}M(z)=M(z;x,t):= \left\{\begin{array}{l}({\mu }_{1}^{-}(z)/a(z),{\mu }_{2}^{+}(z)),\ z\in {{\mathbb{C}}}^{+},\\ ({\mu }_{1}^{+}(z),{\mu }_{2}^{-}(z)/\overline{a(\bar{z})}),\ z\in {{\mathbb{C}}}^{-}.\end{array}\right.\end{array}\end{eqnarray*}$
By lemma Find a matrix-valued function M(z) which satisfies
| 1.Analyticity: M(z) is meromorphic in ${\mathbb{C}}\setminus {\mathbb{R}}$. | |
| 2.Jump condition: M(z) satisfies the jump condition $\begin{eqnarray*}\begin{array}{l}{M}_{+}(z)={M}_{-}(z)V(z),\ z\in {\mathbb{R}},\end{array}\end{eqnarray*}$ where $\begin{eqnarray}V(z)=\left(\begin{array}{cc}1-| r(z){| }^{2} & -\overline{r(z)}{{\rm{e}}}^{-2{\rm{i}}t\theta (z)}\\ r(z){{\rm{e}}}^{2{\rm{i}}t\theta (z)} & 1\end{array}\right),\end{eqnarray}$ with $\theta (z)=\zeta (z)\tfrac{x}{t}-2\zeta (z)\lambda (z).$ | |
| 3.Asymptotic behaviors: $\begin{eqnarray*}\begin{array}{rcl}M(z) & = & I+{ \mathcal O }({z}^{-1}),\quad z\to \infty ,\\ {zM}(z) & = & {\sigma }_{1}+{ \mathcal O }(z),\quad z\to 0.\end{array}\end{eqnarray*}$ | |
| 4.Residue conditions: M(z) has simple poles at each points zj in ${{ \mathcal Z }}^{+}\cup {{ \mathcal Z }}^{-}$ with the following residue conditions: $\begin{eqnarray}\mathop{\mathrm{Res}}\limits_{z={z}_{j}}M(z)=\mathop{\mathrm{lim}}\limits_{z\to {z}_{j}}M(z)\left(\begin{array}{cc}0 & 0\\ {c}_{j}{{\rm{e}}}^{2{\rm{i}}t\theta ({z}_{j})} & 0\end{array}\right),\end{eqnarray}$ $\begin{eqnarray}\mathop{\mathrm{Res}}\limits_{z={\bar{z}}_{j}}M(z)=\mathop{\mathrm{lim}}\limits_{z\to {\bar{z}}_{j}}M(z)\left(\begin{array}{cc}0 & {\bar{c}}_{j}{{\rm{e}}}^{-2{\rm{i}}t\theta ({\bar{z}}_{j})}\\ 0 & 0\end{array}\right),\end{eqnarray}$ where ${c}_{j}=\tfrac{4{\rm{i}}{z}_{j}}{{\int }_{{\mathbb{R}}}| {\psi }_{2}^{+}({z}_{j};x,0){| }^{2}{\rm{d}}x}={\rm{i}}{z}_{j}| {c}_{j}| .$ |
$\begin{eqnarray}q(x,t)=2{\rm{i}}m(x,t){{\rm{e}}}^{2{\rm{i}}\beta {\int }_{\infty }^{x}(| q{| }^{2}-1){\rm{d}}y},\end{eqnarray}$
where $\begin{eqnarray}m(x,t)=\mathop{\mathrm{lim}}\limits_{z\to \infty }{\left({zM}(z)\right)}_{21}.\end{eqnarray}$
Figure 1. The poles and jump lines of M(z). |
From 2.18 , we obtain ∣q(x, t)∣ = 2∣m(x, t)∣ and the reconstruction formula (2.18 ) can be written as2.19 ) of the solution M(z) of the RH problem 2.1 .
$\begin{eqnarray}q(x,t)=2{\rm{i}}m(x,t){{\rm{e}}}^{2{\rm{i}}\beta {\int }_{\infty }^{x}(4| m(y,t){| }^{2}-1){\rm{d}}y},\end{eqnarray}$
were m(x, t) is given by the limit (2.3. Classification of asymptotic regions
The long-time asymptotics of the RH problem 2.1 is affected by the growth and decay of the oscillatory terms e±2itθ(z) in the jump matrix V(z). Direct calculations show that
$\begin{eqnarray}\mathrm{Re}(2{\rm{i}}\theta (z))=2\mathrm{Re}z\mathrm{Im}z\left(1+| z{| }^{-4}\right)-2\xi \mathrm{Im}z\left(1+| z{| }^{-2}\right),\end{eqnarray}$
where $\xi =\tfrac{x}{2t}$. The signature table of $\mathrm{Re}(2{\rm{i}}\theta (z))$ is shown in figures 2–4
Figure 2. Solitonic regions: ∣ξ∣ < 1, there is no phase point on ${\mathbb{R}}$. |
Figure 3. Solitonless region: ∣ξ∣ > 1, there are two phase points on ${\mathbb{R}}$. |
Figure 4. Transition region: ∣ξ∣ = 1, there is one phase point on ${\mathbb{R}}$. |
According to the distribution of the phase points on the real axis, we divide the half-plane (x,t) into the following three kinds of asymptotic regions.
3. Long-time asymptotic behavior
In this section, we give the asymptotic behavior of the solution of the Cauchy problem (1.1 )–(1.2 ). Noting that we find the RH problem 2.1 corresponding to defocusing KE equation is the same with the defocusing NLS equation [12–14], then solving the RH problem 2.1 with $\bar{\partial }$-descent method as done in [12–14] and using the reconstruction formula (2.20 ), we obtain the following results for the KE equation (1.1 ). The proof is omitted.
3.1. Solitonic region
For the solitonic region ∣x/(2t)∣ < 1, we obtain the result on soliton resolution and stability of N-soliton solutions for the KE equation.
The initial data ${q}_{0}\in \tanh (x)+{H}^{\mathrm{4,4}}({\mathbb{R}})$, and $\{r(z),{\{{z}_{j},{c}_{j}\}}_{j\,=\,0}^{N-1}\}$ are associated with the scattering data. Then in the solitonic region $| x/(2t)| \lt 1$, the N-soliton solution of the Cauchy problem (
$\begin{eqnarray}| q(x,t)-{{\rm{e}}}^{{\rm{i}}\gamma (\xi )}{q}_{\mathrm{sol}}(x,t)| \leqslant {{ct}}^{-1},\ t\gt {T}_{0},| \xi | \leqslant {\xi }_{0}.\end{eqnarray}$
where $\xi =x/(2t)$, and $\begin{eqnarray}\begin{array}{rcl}\gamma (\xi ) & = & {\int }_{0}^{\infty }\nu (s)/s{\rm{d}}s+2\displaystyle \sum _{k:\mathrm{Re}{z}_{k}\gt \xi }\arg {z}_{k},\ \nu (z)\\ & = & -\,\displaystyle \frac{1}{2}\mathrm{log}(1-| r(z){| }^{2}),\end{array}\end{eqnarray}$
while ${q}_{\mathrm{sol}}(x,t)$ is the N-soliton with associated scattering data $\{\widetilde{r}=0,{\{{z}_{j},{\widetilde{c}}_{j}\}}_{j=0}^{N-1}\}$, with$\begin{eqnarray*}{\widetilde{c}}_{j}={c}_{j}\exp \left(2{\rm{i}}{\int }_{0}^{\infty }\nu (s)\left(\displaystyle \frac{1}{s-{z}_{j}}-\displaystyle \frac{1}{2s}\right){\rm{d}}s\right).\end{eqnarray*}$
Moreover, the N-soliton can be separated in the form $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & 2{\rm{i}}m(x,t){{\rm{e}}}^{4{\rm{i}}\beta {\displaystyle \int }_{\infty }^{x}(| m(y,t){| }^{2}-1){\rm{d}}y},\\ m(x,t) & = & {{\rm{e}}}^{{\rm{i}}\gamma (1)}\left[1+\displaystyle \sum _{k=0}^{N-1}(\displaystyle \prod _{j\lt k}{z}_{j}^{2})[\mathrm{sol}({z}_{k},x,t)-1]\right]\\ & & +{ \mathcal O }({t}^{-1}).\end{array}\end{eqnarray}$
where $\mathrm{sol}({z}_{k},x,t)$ is a single-soliton corresponding discrete spectral ${z}_{k}={\xi }_{k}+{\rm{i}}{\eta }_{k}$.3.2. Solitonless region
For the solitonless region ∣x/(2t)∣ > 1, we obtain the long-time asymptotic behavior for the solution of the KE equation.
Under the condition of the theorem
$\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & 2{\rm{i}}m(x,t){{\rm{e}}}^{4{\rm{i}}\beta {\displaystyle \int }_{\infty }^{x}(| m(y,t){| }^{2}-1){\rm{d}}y},\\ m(x,t) & = & {{\rm{e}}}^{-{\rm{i}}\delta }\left(1+{t}^{-1/2}h(x,t)\right)+{ \mathcal O }\left({t}^{-3/4}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\delta ={\int }_{I(\xi )}\nu (s)/s\,{\rm{d}}s,\ \nu (z)=-\displaystyle \frac{1}{2}\mathrm{log}(1-| r(z){| }^{2}),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}h(x,t)=\displaystyle \frac{\nu {\left({\xi }_{1}\right)}^{1/2}}{2{\rm{i}}\sqrt{t\pi }\left(1-{\xi }_{1}^{2}\right)}\left[\displaystyle \frac{{\xi }_{1}^{2}{{\rm{e}}}^{-{\rm{i}}{\alpha }_{1}}+{{\rm{e}}}^{{\rm{i}}{\alpha }_{1}}}{\sqrt{| \theta ^{\prime\prime} ({\xi }_{1})| }}\right.\\ \,+\,\left.\displaystyle \frac{{{\rm{e}}}^{-{\rm{i}}{\alpha }_{2}}+{\xi }_{1}^{2}{{\rm{e}}}^{{\rm{i}}{\alpha }_{2}}}{\sqrt{| \theta ^{\prime\prime} ({\xi }_{1}^{-1})| }}\right],\end{array}\end{eqnarray}$
with other parameters given by$\begin{eqnarray*}\begin{array}{rcl}{\alpha }_{1} & = & \displaystyle \frac{\pi }{4}+\arg {\rm{\Gamma }}({\rm{i}}v({\xi }_{1}))-\arg \left({r}_{{\xi }_{1}}\right),{\alpha }_{2}={\alpha }_{1}+\alpha -{\rm{i}}v({\xi }_{1}),\\ \alpha & = & \displaystyle \frac{\pi }{2}+4t\theta ({\xi }_{1})+v({\xi }_{1})\mathrm{log}\left(4{t}^{2}| \theta ^{\prime\prime} ({\xi }_{1})\theta ^{\prime\prime} ({\xi }_{1}^{-1})| \right)\\ & & \,+2\arg {\rm{\Gamma }}({\rm{i}}v({\xi }_{1})).\end{array}\end{eqnarray*}$
3.3. Two transition regions
Define two transition regions
$\begin{eqnarray*}{{ \mathcal A }}_{\pm }1:= \left\{(x,t):0\lt \left|\displaystyle \frac{x}{2t}-(\pm 1)\right|{t}^{2/3}\leqslant C\right\},\end{eqnarray*}$
then we obtain the Painlevé asymptotics for the KE equation.The initial data given in theorem
| 1.In the transition region ${{ \mathcal A }}_{-1}$, the solution to the Cauchy problem ( $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & 2{\rm{i}}m(x,t){{\rm{e}}}^{4{\rm{i}}\beta {\displaystyle \int }_{\infty }^{x}(| m(y,t){| }^{2}-1){\rm{d}}y},\\ m(x,t) & = & {{\rm{e}}}^{{\rm{i}}\varepsilon }\left(1+{\left(\displaystyle \frac{3}{4}t\right)}^{-1/3}\alpha (-1)\right)+{ \mathcal O }\left({t}^{-1/2}\right),\end{array}\end{eqnarray}$ where $\varepsilon $ and $\alpha (-1)$ are the associated parameters $\begin{eqnarray}\begin{array}{l}\varepsilon =\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\mathrm{log}(1-| r(z){| }^{2})}{z}\,{\rm{d}}z+2\displaystyle \sum _{j=0}^{N-1}\arg {z}_{j},\\ \alpha (-1)=\displaystyle \frac{{\rm{i}}}{2}\left({\displaystyle \int }_{s}^{\infty }{u}^{2}(y){\rm{d}}y-u(s){{\rm{e}}}^{\mathrm{iarg}(r(-1)}\right),\\ \,s=\displaystyle \frac{8}{3}\left(\displaystyle \frac{x}{2t}+1\right){\left(\displaystyle \frac{3}{4}t\right)}^{\tfrac{2}{3}},\end{array}\end{eqnarray}$ and $u(s)$ is the solution of the Painlevé equation $\begin{eqnarray}u^{\prime\prime} (s)=2{u}^{3}(s)+{su}(s).\end{eqnarray}$ | |
| 2.In the transition region ${{ \mathcal A }}_{+1}$, the solution to the Cauchy problem ( $\begin{eqnarray}\begin{array}{rcl}q(x,t) & = & 2{\rm{i}}m(x,t){{\rm{e}}}^{4{\rm{i}}\beta {\displaystyle \int }_{\infty }^{x}(| m(y,t){| }^{2}-1){\rm{d}}y},\\ m(x,t) & = & {{\rm{e}}}^{{\rm{i}}\varepsilon }\left(1+{\left(\displaystyle \frac{3}{4}t\right)}^{-1/3}\alpha (1)\right)+{ \mathcal O }\left({t}^{-1/2}\right),\end{array}\end{eqnarray}$ where $\begin{eqnarray}\begin{array}{rcl}\varepsilon & = & \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{0}^{\infty }\displaystyle \frac{\mathrm{log}(1-| r(z){| }^{2})}{z}\,{\rm{d}}z,\\ \alpha (1) & = & -\displaystyle \frac{{\rm{i}}}{2}\left({\displaystyle \int }_{s}^{\infty }{u}^{2}(y){\rm{d}}y-u(s){{\rm{e}}}^{\mathrm{iarg}(\overline{\tilde{r}(1)})}\right),\\ s & = & -\displaystyle \frac{8}{3}(\displaystyle \frac{x}{2t}-1){\left(\displaystyle \frac{3}{4}t\right)}^{\tfrac{2}{3}},\end{array}\end{eqnarray}$ and $u(s)$ is a real-value solution of the Painevé equation ( |