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Engui Fan

Publications and source records attributed to Engui Fan.

At least 37 records · Page 2Linked to original sources
arXiv2024

Soliton resolution and asymptotic stability of $N$-loop-soliton solutions for the Ostrovsky-Vakhnenko equation

The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3κu_x+3u_xu_{xx}+uu_{xxx}=0 \end{align*} is a short wave model of the well-known Degasperis-Procesi equation and admits a $3\times 3$ matrix Lax pair. In this paper, we study the soliton resolution and asymptotic stability of $N$-loop soliton solutions for the OV equation with Schwartz initial data that supports soliton solutions. It is shown that the solution of the Cauchy problem can be characterized via a $3\times 3$ matrix Riemann-Hilbert (RH) problem in a new scale. Further by deforming the RH problem into solvable models with $\bar\partial$-steepest descent method, we obtain the soliton resolution to the OV equation in two space-time regions $x/t>0$ and $x/t<0$. This result also implies that $N$-loop soliton solutions of the OV equation are asymptotically stable.

math-ph↗
arXiv2023

Long-time asymptotic behavior of the Hunter-Saxton equation

With $\bar{\partial}$-generalization of the Deift-Zhou steepest descent method, we investigate the long-time asymptotics of the solution to the Cauchy problem for the Hunter-Saxton (HS) equation \begin{eqnarray} &&u_{txx}-2ωu_x+2u_xu_{xx}+uu_{xxx}=0,\quad x\in \mathbb{R},\ t>0,\nonumber\\ &&u(x,0)=u_0(x), \nonumber \end{eqnarray} where $u_0\in H^{3,4}(\mathbb{R})$ and $ω>0$ is a constant. Using the new scale $(y,t)$ and a series of deformations to a Riemann-Hilbert problem associated with the Cauchy problem, we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in two space-time regions: The solution of the HS equation decays as the speed of $\mathcal{O}(t^{-1/2})$ in the region $y/t >0$; While in the region $y/t<0$, the solution of the HS equation is depicted by a parabolic cylinder model with an residual error order $\mathcal{O}(t^{-1+\frac{1}{2p}})$ with $ p>2$.

math.AP↗
arXiv2023

Long time asymptotic behavior for the nonlocal nonlinear Schrödinger equation with weighted Sobolev initial data

In this paper, we extend $\overline\partial$ steepest descent method to study the Cauchy problem for the nonlocal nonlinear Schrödinger (NNLS) equation with weighted Sobolev initial data %and finite density initial data \begin{align*} &iq_{t}+q_{xx}+2σq^2(x,t)\overline{q}(-x,t)=0, & q(x,0)=q_0(x), \end{align*} where $ q_0(x)\in L^{1,1}(\mathbb{R})\cap L^{2,1/2}(\mathbb{R})$. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem is expressed in terms of solutions of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, we obtain the asymptotic expansion of the Cauchy problem for the NNLS equation in solitonic region. The leading order term is soliton solutions, the second term is the error term is the interaction between solitons and dispersion, the error term comes from the corresponding $\bar{\partial}$ equation. Compared to the asymptotic results on the classical NLS equation, the major difference is the second and third terms in asymptotic expansion for the NNLS equation were affected by a function $ {\rm Im}ν(ξ)$ for the stationary phase point $ξ$.

math.AP↗
arXiv2023

he Cauchy problem for the Novikov equation under a nonzero background: Painlevé asymptotics in a transition zone

In this paper, we investigate the Painlevé asymptotics in a transition zone for the solutions to the Cauchy problem of the Novikov equation under a nonzero background \begin{align} &u_{t}-u_{txx}+4 u_{x}=3uu_xu_{xx}+u^2u_{xxx}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $u_0(x)\rightarrow κ>0, \ x\rightarrow \pm \infty$ and $u_0(x)-κ$ is assumed in the Schwarz space. This result is established by performing the $\overline\partial$-steepest descent analysis to a Riemann-Hilbert problem associated with the the Cauchy problem in a new spatial scale \begin{equation*} y = x - \int_{x}^{\infty} \left((u-u_{xx}+1)^{2/3}-1\right)ds, \end{equation*} for large times in the transition zone $y/t \approx -1/8 $. It is shown that the leading order term of the asymptotic approximation comes from the contribution of solitons, while the sub-leading term is related to the solution of the Painlevé \uppercase\expandafter{\romannumeral2} equation.n.

math.AP↗
arXiv2023

On the global existence for the modified Camassa-Holm equation via the inverse scattering method

In this paper, we address the existence of global solutions to the Cauchy problem of the modified Camassa-Holm (mCH) equation, which is known as a model for the unidirectional propagation of shallow water waves. Based on the spectral analysis of the Lax pair, we apply the inverse scattering transform to rigorously analyze the mCH equation with zero background. By connecting the Cauchy problem to the Riemann-Hilbert (RH) problem, we establish a bijective map between potential and reflection coefficients within the $L^2$-Sobolev space framework. Utilizing a reconstruction formula and estimates on the time-dependent RH problem, we obtain a unique global solution to the Cauchy problem for the mCH equation.

math.AP↗
arXiv2023

The asymptotic stability of solitons in the focusing Hirota equation on the line

In this paper, the $\overline\partial$-steepest descent method and Bäcklund transformation are used to study the asymptotic stability of solitons to the Cauchy problem of focusing Hirota equation. The solution of the RH problem is further decomposed into pure radiation solution and solitons solution obtained by using $\overline\partial$-techniques and Bäcklund transformation respectively. As a directly consequence, the asymptotic stability of solitons for the Hirota equation is obtained.

math.AP↗
arXiv2023

The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlevé asymptotics

Based on the $\overline\partial$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space $ H^{4,2}(\mathbb{R})$. With a new scale $(y,t)$ and a RH problem associated with the initial value problem,we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane $\{ (y,t): -\infty 0\}$ is divided into four asymptotic regions: 1. Fast decay region, $ y/t \in(-\infty,-1/4)$ with an error $\mathcal{O}(t^{-1/2})$; 2. Modulation-solitons region, $y/t \in(2,+\infty)$, the result can be characterized with an modulation-solitons with residual error $\mathcal{O}(t^{-1/2 })$; 3. Zakhrov-Manakov region,$y/t \in(0,2)$ and $y/t \in(-1/4,0)$. The asymptotic approximations is characterized by the dispersion term with residual error $\mathcal{O}(t^{-3/4})$; 4. Two transition regions, $|y/t|\approx 2$ and $|y/t| \approx -1/4$, the results are describe by the solution of Painlevé II equation with error order $\mathcal{O}(t^{-1/2})$.

math.AP↗
arXiv2023

Existence of global solutions for the nonlocal derivation nonlinear Schrödinger equation by the inverse scattering transform method

We address the existence of global solutions to the initial value problem for the integrable nonlocal derivative nonlinear Schrödinger equation in weighted Sobolev space $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$. The key to prove this result is to establish a bijectivity between potential and reflection coefficient by using the inverse scattering transform method in the form of the Riemann-Hilbert problem.

math.AP↗
arXiv2023

On the Cauchy problem of defocusing mKdV equation with finite density initial data: long time asymptotics in soliton-less regions

We investigate the long-time asymptotics for the solutions to the Cauchy problem of defocusing modified Kortweg-de Vries (mKdV) equation with finite density initial data. The present paper is the subsequent work of our previous paper [arXiv:2108.03650], which gives the soliton resolution for the defocusing mKdV equation in the central asymptotic sector $\{(x,t): \vert ξ\vert<6\}$ with $ξ:=x/t$. In the present paper, via the Riemann-Hilbert (RH) problem associated to the Cauchy problem, the long-time asymptotics in the soliton-less regions $\{(x,t): \vert ξ\vert>6, |ξ|=\mathcal{O}(1)\}$ for the defocusing mKdV equation are further obtained. It is shown that the leading term of the asymptotics are in compatible with the ``background solution'' and the error terms are derived via rigorous analysis.

math.AP↗
arXiv2023

Painlev\'e transcendents in the defocusing mKdV equation with non-zero boundary conditions

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlev\'{e} transcendent.

math-ph↗
arXiv2023

Existence of global solutions for the modified Camassa-Holm equation with a nonzero background

Consideration in the present paper is the existence of global solutions for the modified Camassa-Holm (mCH) equation with a nonzero background initial value. The mCH equation is completely integrable and can be considered as a model for the unidirectional propagation of shallow-water waves. By applying the inverse scattering transform with an application of the Cauchy projection operator, the existence of a unique global solution to the mCH equation in the line with a nonzero background initial value is established in the weighted Sobolev space $ H^{2, 1} (\mathbb{R})\cap H^{1, 2} (\mathbb{R})$ based on the representation of a Riemann-Hilbert (RH) problem associated with the Cauchy problem to the mCH equation. A crucial technique used is to derive the boundedness of the solution in the Sobolev space $ W^{1,\infty}(\mathbb{R}),$ then reconstruct a new RH problem for the Cauchy projection operator of reflection coefficients. The regularity of the global solution is achieved by the refined estimate arguments on those solutions of the corresponding RH problem.

math.AP↗
arXiv2023

Existence of global solutions to the nonlocal mKdV equation on the line

In this paper, we address the existence of global solutions to the Cauchy problem for the integrable nonlocal modified Korteweg-de vries (nonlocal mKdV) equation with the initial data $u_0 \in H^{3}(\mathbb{R}) \cap H^{1,1}(\mathbb{R}) $ with the $L^1(\mathbb{R})$ small-norm assumption. A Lipschitz $L^2$-bijection map between potential and reflection coefficient is established by using inverse scattering method based on a Riemann-Hilbert problem associated with the Cauchy problem. The map from initial potential to reflection coefficient is obtained in direct scattering transform. The inverse scattering transform goes back to the map from scattering coefficient to potential by applying the reconstruction formula and Cauchy integral operator. The bijective relation naturally yields the existence of a global solutions in a Sobolev space $H^3(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to the Cauchy problem.

math.AP↗
arXiv2023

The defocusing NLS equation with nonzero background: Painlevé asymptotics in two transition regions

In this paper, we address the Painlevé aymptotics in the transition region $|ξ|:=\big|\frac{x}{2t}\big| \approx 1$ to the Cauchy problem of the defocusing Schr$\ddot{\text{o}}$dinger equation with a nonzero background.With the $\bar\partial$-generation of the nonlinear steepest descent approach and double scaling limit to compute the long-time asymptotics of the solution in two transition regions defined as $$ \mathcal{P}_{\pm 1}(x,t):=\{ (x,t) \in \mathbb{R}\times\mathbb{R}^+, \ \ 0<|ξ-(\pm 1)|t^{2/3}\leq C\}, $$ we find that the long-time asymptotics in both transition regions $ \mathcal{P}_{\pm 1}(x,t)$ can be expressed in terms of the Painlevé II equation. We are also able to express the leading term explicitly in terms of the Ariy function.

math-ph↗
arXiv2022

The Fokas-Lenells equation on the line: Global well-posedness with solitons

In this paper, we prove the existence of global solutions in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the Fokas-Lenells (FL) equation on the line when the initial data includes solitons.A key tool in proving this result is a newly modified Darboux transformation, which adds or subtracts a soliton with given spectral and scattering parameters. In this way the inverse scattering transform technique is then applied to establish the global well-posedness of initial value problem with a finite number of solitons based on our previous results on the global well-posedness of the FL equation.

math.AP↗
arXiv2022

The complex mKdV equation with step-like initial data: Large time asymptotic analysis

In this paper, we study large-time asymptotics for the complex modified Korteveg-de Vries equation \begin{equation} u_t + \frac{1}{2}u_{xxx}+3|u|^2 u_x=0, \end{equation} with the step-like initial data \begin{equation} u(x,0)=u_0(x)= \begin{cases} 0, & {x \ge 0,}\\ A e^{iBx}, &{x < 0.} \end{cases} \end{equation} It is shown that the step-like initial problem can be described by a matrix Riemann-Hilbert problem. We apply the steepest descent method to obtain different large-time asymptotics in the the Zakharov-Manakov region, a plane wave region and a slow decay region.

math.AP↗
arXiv2022

Large-time asymptotics to the focusing nonlocal modified Kortweg-de Vries equation with step-like boundary conditions

We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, $u_0(x)\rightarrow A$ as $x\rightarrow+\infty$,where $A$ is an arbitrary positive real number. We firstly develop the direct scattering theory to establish the basic Riemann-Hilbert (RH) problem associated with step-like initial data. Thanks to the symmetries $x\rightarrow-x$, $t\rightarrow-t$ of nonlocal MKdV equation, we investigate the asymptotics for $t\rightarrow-\infty$ and $t\rightarrow+\infty$ respectively. Our main technique is to use the steepest descent analysis to deform the original matrix-valued RH problem to corresponded regular RH problem, which could be explicitly solved. Finally we obtain the different large-time asymptotic behaviors of the solution of the Cauchy problem for focusing nonlocal MKdV equation in different space-time sectors $\mathcal{R}_{I}$, $\mathcal{R}_{II}$, $\mathcal{R}_{III}$ and $\mathcal{R}_{IV}$ on the whole $(x,t)$-plane.

math-ph↗
arXiv2022

A Riemann-Hilbert approach to existence of global solutions to the Fokas-Lenells equation on the line

We obtain the the existence of global solutions to the Cauchy problem of the Fokas-Lenells (FL) equation on the line \begin{align} &u_{xt}+αβ^2u-2iαβu_x-αu_{xx}-iαβ^2|u|^2u_x=0,\nonumber \\ &u(x,t=0)=u_0(x), \nonumber \end{align} where without the small-norm assumption on initial data $u_0(x)\in H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$. Our main technical tool is the inverse scattering transform method based on the representation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem. The existence and the uniqueness of the RH problem is shown via a general vanishing lemma. The spectral problem associated with the FL equation is changed into an equivalent Zakharov-Shabat-type spectral problem to establish the RH problems on the real axis. By representing the solutions of the RH problem via the Cauchy integral protection and the reflection coefficients, the reconstruction formula is used to obtain a unique local solution of the FL equation. Further, the eigenfunctions and the reflection coefficients are shown Lipschitz continuous with respect to initial data, which provides a priori estimate of the solution to the FL equation. Based on the local solution and the uniformly priori estimate, we construct a unique global solution in $H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the FL equation.

math.AP↗