arXiv subjects

Qingxuan Wang

Publications and source records attributed to Qingxuan Wang.

9 recordsLinked to original sources

arXiv2026

Condensation and Collapse in the Mean-Field Limit of Rotating 2D Bose Gases with Two-Body and Three-Body Interactions

We consider a system of $N$ interacting bosons in a rotating harmonic trap in $\mathbb{R}^2$, where the two-body interaction is attractive and scaled as $N^{2α}U(N^αx)$ with $0<α<1/12$, and the three-body interaction is repulsive and scaled as $N^{4β}W(N^βx,N^βy)$ with $0<β<1/24$. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schrödinger functional. We analyze the collapse regime where the two-body coupling $a$ approaches the critical value $a_*$ and the three-body coupling $b$ tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies $E^{\mathrm{NLS}} = (1-ζ/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$ with $ζ\geqslant0$ determined by the relative rates of $a_n\to a_*$ and $b_n\searrow0$. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.

math-ph↗

arXiv2025

Thermodynamic limit and $L^\infty$-convergence rate for the cubic-quintic Schrödinger model

We investigate the thermodynamic limit for the cubic-quintic Schrödinger model as the size of the domain tends to infinity with fixed density $ρ= N/|\mathcal{D}|$, where $N$ denotes particle number and $|\mathcal{D}|$ denotes the volume of the bounded domain $\mathcal{D}\subset\mathbb{R}^d$ ($d=1,2,3$). We firstly prove the existence of thermodynamic limit, which is equal to $-\frac{3}{32}$ for $0<ρ\leq \frac{3}{4}$, while $-\left(\frac{1}{2}-\fracρ{3}\right)\fracρ{2}$ for $\frac{3}{4}< ρ\leq 1$. When $0<ρ<1$ and $\mathcal{D}$ is a spherical domain, we further show that, up to a scaling, the ground state of the cubic-quintic Schrödinger energy will converge strongly to a Thomas-Fermi ground state in $L^2\cap L^6$. Finally, we obtain the $L^\infty$-convergence rate of ground states for $0<ρ<3/4$ by developing a novel method, including some iterative techniques, uniform energy estimates and gradient estimates. We believe this method is applicable to other general nonlinearities.

math.AP↗

arXiv2025

The Ground State of a Cubic-quintic Nonlinear Schrödinger Equation with Radial Potential in the Thomas-Fermi Limit

We focus on the ground state of the cubic-quintic nonlinear Schrödinger energy functional \begin{gather*} \begin{aligned} {E}(φ)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla φ|^2+V(x)|φ|^2\right)\,dx \pm\frac{1}{4}\int_{\mathbb{R}^d}|φ|^4\,dx +\frac{1}{6}\int_{\mathbb{R}^d}|φ|^6\,dx, (d=1,2,3) \end{aligned} \end{gather*} under the mass constraint $\int_{\mathbb{R}^d}|φ|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $φ_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[μ^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $μ^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $φ_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities.

math.AP↗

arXiv2024

Optimal convergence rate to the nonrelativistic limit of Chandrasekhar variational model for Neutron stars

In this paper, we consider the nonrelativistic limit of Chandrasekhar variational model for neutron stars. We show that the minimizer $ρ_{c}$ of Chandrasekhar energy $E_c(N)$ converges strongly to the minimizer $ρ_{\infty}$ of limit energy $E_{\infty}(N)$ in $L^1\cap L^{\frac{5}{3}}(\mathbb{R}^3)$ as the speed of light $c\rightarrow\infty$, this is a limit between two free boundary problems. Moreover, we develop a novel approach to obtain the convergence rates, we show that the above nonrelativistic limit has the optimal convergence rate $\frac{1}{c^2}$. For the radius $R_c$ of the compact support of $ρ_c(x)$ and the radius $R_\infty$ of the compact support of $ρ_\infty(x)$, we also get the optimal convergence rate $\frac{1}{c^2}$, this means that $R_\infty-R_c=O(\frac{1}{c^2})$ as $c\rightarrow\infty$. Moreover, we also obtain the optimal uniform bounds of $R_c$ and $L^\infty$-norm of $ρ_c$ with respect to $N$ as $c\rightarrow \infty$.

math.AP↗

arXiv2024

Traveling waves of NLS System arising in optical material without Galilean symmetry

We consider a system of NLS with cubic interactions arising in nonlinear optics without Galilean symmetry. The absence of Galilean symmetry can lead to many difficulties, such as global existence and blowup problems; see [Comm. Partial Differential Equations 46, 11 (2021), 2134-2170]. In this paper, we mainly focus on the influence of the absence of this symmetry on the traveling waves of the NLS system. Firstly, we obtain the existence of traveling solitary wave solutions that are non-radial and complex-valued. Secondly, using the asymptotic analysis method, when the frequency is sufficiently large, we establish the high frequency limit of the traveling solitary wave solution. Finally, for the mass critical case, we provide a novel condition for the existence of global solutions which is significantly different from the classical. In particular, this new condition breaks the traditional optimal assumption about initial data.

math.AP↗

arXiv2023

DocTrack: A Visually-Rich Document Dataset Really Aligned with Human Eye Movement for Machine Reading

The use of visually-rich documents (VRDs) in various fields has created a demand for Document AI models that can read and comprehend documents like humans, which requires the overcoming of technical, linguistic, and cognitive barriers. Unfortunately, the lack of appropriate datasets has significantly hindered advancements in the field. To address this issue, we introduce \textsc{DocTrack}, a VRD dataset really aligned with human eye-movement information using eye-tracking technology. This dataset can be used to investigate the challenges mentioned above. Additionally, we explore the impact of human reading order on document understanding tasks and examine what would happen if a machine reads in the same order as a human. Our results suggest that although Document AI models have made significant progress, they still have a long way to go before they can read VRDs as accurately, continuously, and flexibly as humans do. These findings have potential implications for future research and development of Document AI models. The data is available at \url{https://github.com/hint-lab/doctrack}.

cs.HC↗

arXiv2019

Boson Stars with Long-range Perturbations

We consider the Boson star equation with long-range perturbation given by $$i\partial_t ψ=\sqrt{-\triangle+m^2}\,ψ+β(\frac{1}{|x|^α}\ast |ψ|^2)ψ-(\frac{1}{|x|}\ast |ψ|^2)ψ \ \ \text{on $\mathbb{R}^3$,}$$ where $\frac{1}{|x|^α} (0<α<1)$ denotes the long-range potential. In contrast to the well known fact that for $β=0$ no maximal ground state solitary wave exists when the partical number $N=N_c$ (Chandrasekhar limiting mass) [E.H. Lieb, H.T. Yau, \emph{Commun. Math. Phys.}, 112 (1987), pp: 147-174 ], we show that for $β>0$ and small enough, there exists at least one maximal ground state at $N=N_c$. Moreover, for $β>0$, we find that for initial value $\|ψ_0\|^2_2=N_c$, the solution $ψ(t)$ is global well-posedness, and we obtain an "orbital stability" of those maximal ground state solitary waves in some sense, which implies that such long-range perturbation pushes the Boson star system more stable. Finally, we analyse blow-up behaviours of maximal ground states when $β\rightarrow 0^+$.

math.AP↗

arXiv2019

Normalized ground states for the fractional nonlinear Schrödinger equations

In this paper, we study the existence and instability of standing waves with a prescribed $L^2$-norm for the fractional Schrödinger equation \begin{equation} i\partial_{t}ψ=(-Δ)^{s}ψ-f(ψ), \qquad (0.1)\end{equation} where $0<s<1$, $f(ψ)=|ψ|^{p}ψ$ with $\frac{4s}{N}<p<\frac{4s}{N-2s}$ or $f(ψ)=(|x|^{-γ}\ast|ψ|^2)ψ$ with $2s<γ<\min\{N,4s\}$. To this end, we look for normalized solutions of the associated stationary equation \begin{equation} (-Δ)^s u+ωu-f(u)=0. \qquad (0.2) \end{equation} Firstly, by constructing a suitable submanifold of a $L^2$-sphere, we prove the existence of a normalized solution for (0.2) with least energy in the $L^2$-sphere, which corresponds to a normalized ground state standing wave of(0.1). Then, we show that each normalized ground state of (0.2) coincides a ground state of (0.2) in the usual sense. Finally, we obtain the sharp threshold of global existence and blow-up for (0.1). Moreover, we can use this sharp threshold to show that all normalized ground state standing waves are strongly unstable by blow-up.

math.AP↗

arXiv2018

Concentration Behavior of Nonlinear Hartree-type Equation with almost Mass Critical Exponent

We study the following nonlinear Hartree-type equation \begin{equation*} -Δu+V(x)u-a(\frac{1}{|x|^γ}\ast |u|^2)u=λu,~\text{in}~\mathbb{R}^N, \end{equation*} where $a>0$, $N\geq3$, $γ\in(0,2)$ and $V(x)$ is an external potential. We first study the asymptotic behavior of the ground state of equation for $V(x)\equiv1$, $a=1$ and $λ=0$ as $γ\nearrow2$. Then we consider the case of some trapping potential $V(x)$, and show that all the mass of ground states concentrate at a global minimum point of $V(x)$ as $γ\nearrow2$, which leads to symmetry breaking. Moreover, the concentration rate for maximum points of ground states will be given.

math.FA↗