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Meisen Chen

Publications and source records attributed to Meisen Chen.

6 recordsLinked to original sources
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↗
arXiv2025

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

We investigate the soliton resolution and Painlevé 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 Painlevé-type asymptotics which can be expressed in terms of the solution of the second Painlevé transcendents.

math.AP↗
arXiv2022

$L^2$ Sobolev space bijectivity of the scattering-inverse scattering transforms related to defocusing Ablowitz-Ladik systems

In this paper, we establish $L^2$-Sobolev space bijectivity of the inverse scattering transform related to the defocusing Ablowitz-Ladik system. On the one hand, in the direct problem, based on the spectral problem, we establish the reflection coefficient and the corespondent Riemann-Hilbert problem. And we also prove that if the potential belongs to $l^{2,k}$ space, then the reflection coefficient belongs to $H^k_θ(Σ)$. On the other hand, in the inverse problem, based on the Riemann-Hilbert problem, we obtain the corespondent reconstructed formula and recover potentials from reflection coefficients. And we also confirm that if reflection coefficients are in $H^k_θ(Σ)$, then we show that potentials also belong to $l^{2,k}$. This study also confirm that for the initial-valued problem of defocusing Ablowitz-Ladik equations, it the initial potential belongs to $l^{2,k}$ and satisfying $\parallel q\parallel_\infty<1$, then the solution for $t\ne0$ also belongs to $l^{2,k}$.

math.AP↗
arXiv2021

Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spectrum

In this paper, we use the $\bar{\partial}$ steepest descent method to study the initial value problem for focusing nonlinear Schrödinger (fNLS) equation with non-generic weighted Sobolev initial data that allows for the presence of high-order discrete spectrum. More precisely, we shall characterize the properties of the eigenfunctions and scattering coefficients in the presence of high-order poles; further we formulate an appropriate enlarged RH problem; after a series of deformations, the RH problem is transformed into a solvable model. Finally, we obtain the asymptotic expansion of the solution of the fNLS equation in any fixed space-time cone: %as $t \to \infty$, \begin{equation*} \mathcal{S}(x_1,x_2,v_1,v_2):=\left\lbrace (x,t)\in \mathbb{R}^2: x=x_0+vt, \ x_0\in[x_1,x_2]\text{, }v\in[v_1,v_2]\right\rbrace. \end{equation*} Observing the result indicates that the solution of fNLS equation in this case satisfies the soliton resolution conjecture. The leading order term of this solution includes a high-order pole-soliton whose parameters are affected by soliton-soliton interactions through the cone and soliton-radiation interactions on continuous spectrum. The error term of this result is up to $\mathcal{O}(t^{-3/4})$ which comes from the corresponding $\bar{\partial}$ equation.

math.AP↗
arXiv2021

Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with initial potential in weighted sobolev space

In this paper, we are going to investigate Cauchy problem for nonlocal nonlinear Schrödinger equation with the initial potential $q_0(x)$ in weighted sobolev space $H^{1,1}(\mathbb{R})$, \begin{align*} iq_t(x,t)&+q_{xx}(x,t)+2σq^2(x,t)\bar q(-x,t)=0,\quadσ=\pm1,\\ q(x,0)&=q_0(x). \end{align*} We show that the solution can be represented by the solution of a Riemann-Hilbert problem (RH problem), and assuming no discrete spectrum, we majorly apply $\bar\partial$-steepest cescent descent method on analyzing the long-time asymptotic behavior of it.

math.AP↗