E. Fan

Painlevé \uppercase\expandafter{\romannumeral34\relax} and collisionless shock in the defocusing NLS equation with step-like initial data in the transition regions

Engui Fan, Zhaoyu Wang, Yidan Zhang

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation with step-like initial data. Using the nonlinear steepest descent method, we derive the long-…

Painleve XXXIV asymptotics for the defocusing mKdV equation with step-like initial data in transition regions

Engui Fan, Zhaoyu Wang, Yidan Zhang

In this paper, we consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with a step-like initial data. Based on the Riemann-Hilbert problem asso…

Local and global well-posedness for the nonlinear Schrödinger equation with nonhomogeneous boundary conditions

Engui Fan, Yuan Li, Xinhan Liu

In this paper, we study the initial-boundary value problem for the nonlinear Schrödinger equation in $R_{+}^{n}$ \begin{equation*} i\partial_{t}u+Δu+λ|u|^pu=0, \qquad (x,…

Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background

Engui Fan, Gaozhan Li, Yiling Yang +1

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr $o ¨$ dinger equation with a finite genus algebro-geometric background. Long-time asymptot…

math.AP2025

Existence of global solutions to the Fokas-Lenells equation with arbitrary spectral singularities

Yuan Li, Qiaoyuan Cheng, Engui Fan

We establish the global existence of solutions to the Fokas-Lenells equation for any initial data in a weighted Sobolev space $H^{3} (R) \cap H^{2, 1} (R)$ .This resul…

The Cauchy problem for the Degasperis-Procesi Equation: Painlevé Asymptotics in Transition Zones

Zhaoyu Wang, Xuan Zhou, Engui Fan

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3κu_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as an asymptotic approximation for the unidirectiona…

$L^{2}$ -Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schrödinger equations

Yuan Li, Xinhan Liu, Engui Fan

We consider the Cauchy problem to the general defocusing and focusing $p \times q$ matrix nonlinear Schrödinger (NLS) equations with initial data allowing arbitrary-order poles and…

Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons

Meisen Chen, Engui Fan, Zhaoyu Wang

We investigate the soliton resolution and Painlevé asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $ℓ^{2}$ space. First, we establi…

Orbital Stability of Soliton for the Derivative Nonlinear Schrödinger Equation in the $L^{2}$ Space

Yiling Yang, Engui Fan, Yue Liu

In this paper, we establish the orbital stability of the 1-soliton solution for the derivative nonlinear Schrödinger equation under perturbations in $L^{2} (R)$ . We demonstra…

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

Zhaoyu Wang, Xuan Zhou, Engui Fan

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}…

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

Ruihong Ma, Engui Fan

In this paper, the $\partial$ -steepest descent method and Bäcklund transformation are used to study the asymptotic stability of solitons to the Cauchy problem of focusing…

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

Yuan Li, Xinhan Liu, Engui Fan

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}(\ma…

Long-time asymptotic behavior of the Hunter-Saxton equation

Luman Ju, Kai Xu, Engui Fan

With $\partial ˉ$ -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…

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

Yiling Yang, Engui Fan, Yue Liu

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 prop…

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

Anran Liu, Engui Fan

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 d…

math.AP2022★ 1 cited

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

Qiaoyuan Cheng, Engui Fan

In this paper, we prove the existence of global solutions in $H^{3} (R) \cap H^{2, 1} (R)$ to the Fokas-Lenells (FL) equation on the line when the initial data includes…

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

Zhaoyu Wang, Kai Xu, Engui Fan

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 s…

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

Kai Xu, Yiling Yang, Engui Fan

Based on the $\partial$ -generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to th…

Long-time asymptotic analysis for defocusing Ablowitz-Ladik system with initial value in lower regularity

Chen Meisen, Fan Engui, He Jingsong

Recently, we have given the $l^{2}$ bijectivity for defocusing Ablowitz-Ladik systems in the discrete Sobolev space $l^{2, 1}$ by inverse spectral method. Based on these results, the…