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Engui Fan

Publications and source records attributed to Engui Fan.

At least 19 recordsLinked to original sources
arXiv2026

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

We consider the Cauchy problem for the defocusing nonlinear Schr\"odinger (NLS) equation with step-like initial data. Using the nonlinear steepest descent method, we derive the long-time asymptotic expansion of the solution to the Cauchy problem in three distinct transition regions. In the first two transition regions, the leading-order asymptotics are characterized by Painlev\'e \uppercase\expandafter{\romannumeral34\relax}-type formula, while in the third one is a collisionless shock region, the leading-order asymptotics is describedin terms of Riemann theta functions. Our analysis is based on the Riemann-Hilbert formulation associated with the Cauchy problem of the defocusing NLS equation.

math.AP↗
arXiv2026

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

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 associated with the mKdV equation, we derive the long-time asymptotic expansion of the solution to the defocusing mKdV equation in two transition regions using the nonlinear steepest descent method. It comes out that the leading term in the expansion is shown to match the corresponding background constants. The subleading term, however, decays at the order $\mathcal{O}(t^{-2/3})$, and its coefficient is derived from the associated Painlev\'e \uppercase\expandafter{\romannumeral34\relax} model.

math.AP↗
arXiv2026

Local and global well-posedness for the nonlinear Schr\"{o}dinger equation with nonhomogeneous boundary conditions

In this paper, we study the initial-boundary value problem for the nonlinear Schr\"{o}dinger equation in $\mathbb{R}^{n}_{+}$ \begin{equation*} i\partial_{t}u+\Delta u+\lambda |u|^pu=0, \qquad (x, t) \in \mathbb{R}_{+}^{n} \times \mathbb{R}_{+},\ \ p\in\mathbb{R}_{+} \end{equation*} with nonhomogeneous Dirichlet boundary conditions. For the corresponding linear problem, endpoint Strichartz estimates are derived. For the nonlinear problem, we prove local well-posedness in $H^{s}(\mathbb{R}^{n}_{+})$ with $s\in[0,\frac{5}{2})$ and $p<\frac{4}{n-2s}$. Moreover, global well-posedness is established in the same regularity range. For $s\in[1,\frac{5}{2})$, the one-dimensional global theory of \cite{figment} in $H^{s}(\mathbb{R}_{+})$ is extended to $H^{s}(\mathbb{R}^{n}_{+})$. Additionally, we obtain global solutions in the lower regularity setting $s\in[0,1)$ for the first time. It is noteworthy that for $s=0$, we overcome the lack of mass conservation resulting from the nonzero boundary data and derive the pivotal $L^{2}(\mathbb{R}^{n}_{+})$ a priori estimates.

math.AP↗
arXiv2026

Painlev\'e XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum

We investigate the Cauchy problem for the focusing modified Korteweg--de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift--Zhou to the associated Riemann--Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce with the endpoints of the finite-genus branch cuts. The collision is resolved via a local Painlev\'{e} XXXIV parametrix, and the discrete spectrum (breathers) is incorporated into the analysis. The resulting expansion is valid uniformly up to an error of order $\mathcal{O}(t^{-1/2})$. In this critical region, the leading-order term comprises the finite-genus algebro-geometric background together with breathers, whose parameters are slowly modulated by the background solution.

math-ph↗
arXiv2026

Painlev\'e Asymptotics of the Focusing Nonlinear Schr\"odinger Equation with a Finite-Genus Algebro-Geometric Background

We investigate the Cauchy problem for the focusing nonlinear Schr\"odinger (NLS) equation \begin{equation} iq_t(x,t)+q_{xx}(x,t)+2|q(x,t)|^2q(x,t)=0,\quad x\in\mathbb{R},\quad t\ge0,\nonumber \end{equation} subject to initial data $ q(x,0)$ satisfying the asymptotic boundary conditions \begin{equation}\label{eq:boundary} q(x,0) \sim q^{alg}(x,0) \quad \text{as} \quad x \to \pm\infty,\nonumber \end{equation} where $q^{alg}(x,t)$ denote finite-genus algebro-geometric quasi-periodic solutions of the focusing NLS equation. Employing the Riemann--Hilbert (RH) approach combined with the Deift--Zhou nonlinear steepest descent method, we analyze the long-time asymptotic behavior of solutions to this Cauchy problem. Our analysis distinguishes between two cases based on the genus $n$ of the underlying hyperelliptic Riemann surface: (i) Odd genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of odd genus $n = 2s+1$ $(s \in \mathbb{N}_0)$, we identify distinct asymptotic regions in the $(x,t)$-plane characterized by the variable $\xi = x/t$, within which the leading-order asymptotics is expressed in terms of the second Painlev\'e transcendent. (ii)Even genus backgrounds: When the background solutions $q^{alg}(x,0)$ correspond to hyperelliptic curves of even genus $n = 2s$ $(s \in \mathbb{N})$, the asymptotic behavior in regions selected by $\xi$ is described in terms of parabolic cylinder functions. Specifically, we derive the leading-order asymptotics and establish explicit error bounds for the solution $q(x,t)$ as $t \to +\infty$, uniformly for $x \in \mathbb{R}$.

math-ph↗
arXiv2026

A dense focusing Ablowitz-Ladik soliton gas and its asymptotics

In this paper, we propose a soliton gas solution for the focusing Ablowitz-Ladik system. This solution is defined as the large N limit of the N-soliton solution, and arises from a continuous spectrum of poles that accumulate within two disjoint intervals on the imaginary axis. We show that this gas solution admits a Fredholm determinant representation. By further exploring its Riemann-Hilbert characterization, we are able to establish the large-space asymptotics at t = 0 and large-time asymptotics of the gas solution.

math-ph↗
arXiv2026

Painlev\'{e} XXXIV asymptotics for the defocusing nonlinear Schr\"odinger equation with a finite-genus algebro-geometric background

In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlev\'e XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems.

math.AP↗
arXiv2025

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

We establish the global existence of solutions to the Fokas-Lenells equation for any initial data in a weighted Sobolev space $H^{3}(\mathbb{R})\cap H^{2,1}(\mathbb{R})$.This result removes all spectral restrictions on the initial data required in our previous work. The proof primarily relies on the inverse scattering transform formulated as new Riemann-Hilbert problems and Zhou's $L^{2}$-Sobolev bijectivity theory.

math.AP↗
arXiv2025

Soliton resolution, asymptotic stability and Painlev\'e transcendents in the combined Wadati-Konno-Ichikawa and short-pulse equation

In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the combined Wadati-Konno-Ichikawa and short-pulse (WKI-SP) equation. The solution of the Cauchy problem is first expressed in terms of the solution of a RH problem with direct scattering transform based on the Lax pair. Further through a series of deformations to the RH problem by using the $\bar{\partial}$-generalization of Deift-Zhou steepest descent method, we obtain the long-time asymptotic approximations to the solution of the WKI-SP equation under a new scale $(y,t)$ in three kinds of space-time regions. The first asymptotic result from the space-time regions $ \xi:=y/t <-2\sqrt{3\alpha\beta}, \alpha\beta>0$ and $|\xi|<\infty,\alpha\beta<0$ with saddle points on $\mathbb{R}$, is characterized with solitons and soliton-radiation interaction with residual error $\mathcal{O}(t^{-3/4})$. The second asymptotic result from the region $ \xi >-2\sqrt{3\alpha\beta}, \alpha\beta>0$ without saddle point on $\mathbb{R}$, is characterized with modulation-solitons with residual error $\mathcal{O}(t^{-1})$; These two results above are a verification of the soliton resolution conjecture for the WKI-SP equation. The third asymptotic result from a transition region $\xi \approx -2\sqrt{3\alpha\beta},\alpha\beta>0$ can be expressed in terms of the solution of the Painlev\'{e} \uppercase\expandafter{\romannumeral2} equation with error $\mathcal{O}(t^{-1/2})$. This is a new phenomena that the long-time asymptotics for the solution to the Cauchy problem of the WKI equation and SP equation don't possesses.

math-ph↗
arXiv2025

Riemann-Hilbert approach to the Algebro-Geometric solution of the modified Camassa-Holm equation with linear dispersion term

This paper aims at providing an exact algebro-geometric solution of the modified Camassa-Holm (mCH) equation derived from hyperelliptic curves in $4(p+q)-1$ genus. To achieve this goal, we construct the Riemann-Hilbert problems cosponsoring to the mCH equation, which can be solved exactly by the Baker-Akhiezer function. Then the precise expression of the algebro-geometric solution of the mCH equation can be obtained through reconstructed formula.

math-ph↗
arXiv2024

Soliton Shielding of the focusing modified KdV equation

We consider soliton gas solutions of the modified Korteweg-de Vries (mKdV) equation, where the point spectrum of the condensate is located within a bounded domain in the upper half-plane. We first demonstrate that when the domain is a quadrature and the soliton density is an analytic function, the corresponding deterministic soliton gas coincides with a finite number of solitons, which we call this effect soliton shielding. When the domain is an ellipse and the soliton density is analytic, the corresponding deterministic soliton gas reduces the spectral data to the segment joining the foci. The initial datum of this Cauchy problem is asymptotically step-like oscillatory, described by a periodic elliptic function as \( x \to +\infty \), and it vanishes exponentially fast as \( x \to -\infty \).

math-ph↗
arXiv2024

The Cauchy problem for the Degasperis-Procesi Equation: Painlev\'e Asymptotics in Transition Zones

The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as an asymptotic approximation for the unidirectional propagation of shallow water waves, is an integrable model of the Camassa-Holm type and admits a $3\times3$ matrix Lax pair. In our previous work, we obtained the long-time asymptotics of the solution $u(x,t)$ to the Cauchy problem for the DP equation in the solitonic region $\{(x,t): \xi>3 \} \cup \{(x,t): \xi<-\frac{3}{8} \}$ and the solitonless region $\{(x,t): -\frac{3}{8}<\xi< 0 \} \cup \{(x,t): 0\leq \xi <3 \}$ where $\xi:=\frac{x}{t}$. In this paper, we derive the leading order approximation to the solution $u(x,t)$ in terms of the solution for the Painlev\'{e} \uppercase\expandafter{\romannumeral2} equation in two transition zones $\left|\xi+\frac{3}{8}\right|t^{2/3} 0$ lying between the solitonic region and solitonless region. Our results are established by performing the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method and applying a double scaling limit technique to an associated vector Riemann-Hilbert problem.

math.AP↗
arXiv2024

$L^{2}$-Sobolev space bijectivity and existence of global solutions for the matrix nonlinear Schr\"{o}dinger equations

We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schr\"{o}dinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the $L^{2}$-Sobolev space bijectivity of the direct and inverse scattering transforms associated with a $(p+q)\times(p+q)$ matrix spectral problem, we prove that both defocusing and focusing matrix NLS equations are globally well-posed in the weighted Sobolev space $H^{1,1}(\mathbb{R})$.

math.AP↗
arXiv2024

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

We investigate the soliton resolution and Painlev\'e asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value problem, which is further reformulated as a Riemann-Hilbert problem with higher-order poles. Using Fredholm theory, the Riemann-Hilbert problem with the jump contour consisting of three circles centered around the origin is uniquely solved. Then, by performing a $\bar\partial$-nonlinear steepest descent method to the Riemann-Hilbert problem, we obtain the asymptotic approximation to the solution of the focusing Ablowitz-Ladik system for large time in different space-time regions of the $(n,t)$-half plane. In the sectors $\{(n,t): n /(2t) <-M_0 \}$ and $\{(n,t): n /(2t) >M_0 \}$, where $M_0$ is a positive constant, the leading order asymptotics is dominated by the solitons; while in the sector $\{(n,t): |n /(2t) -1 <M_0^{-1} \}$, the long-time asymptotics is influenced by both the solitons and the oscillations; In the two transition zones $\{(n,t): |n /(2t)+1|t^{2/3} <C \}$ and $\{(n,t): |n /(2t)-1|t^{2/3} <C \}$ with $C$ being a positive constant, we find the Painlev\'e-type asymptotics which can be expressed in terms of the solution of the second Painlev\'e transcendents.

math.AP↗
arXiv2024

A Riemann-Hilbert approach to the two-component modified Camassa-Holm equation

In this paper, we develop a Riemann-Hilbert (RH) approach to the Cauchy problem for the two-component modified Camassa-Holm (2-mCH) equation based on its Lax pair. Further via a series of deformations to the RH problem by using the $\bar{\partial}$-generalization of Deift-Zhou steepest descent method, we obtain the long-time asymptotic approximations to the solutions of the 2-mCH equation in four kinds of space-time regions. Especially we introduce a technique to unify multi-jump matrix factorizations into one form which can greatly simplify the calculation of the $\bar{\partial}$-steepest descent method.

math-ph↗
arXiv2023

he Cauchy problem for the Novikov equation under a nonzero background: Painlev\'e asymptotics in a transition zone

In this paper, we investigate the Painlev\'e 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 $u_0(x)\rightarrow \kappa>0, \ x\rightarrow \pm \infty$ and $u_0(x)-\kappa$ 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 Painlev\'e \uppercase\expandafter{\romannumeral2} equation.n.

math.AP↗
arXiv2023

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

The Ostrovsky-Vakhnenko (OV) equation \begin{align*} &u_{txx}-3\kappa u_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↗