arXiv subjects

Mengyao Chen

Publications and source records attributed to Mengyao Chen.

8 recordsLinked to original sources

arXiv2026

Why does AI unlock new possibilities in STEM education? A Bibliometric Analysis of Trends and Future Agenda

STEM education faces challenges in personalization and interdisciplinary integration. AI technology has brought new possibilities, but the mechanisms by which AI reshapes the STEM education ecosystem require systematic investigation. This study employs bibliometric methods to analyze 242 publications from 2015-2025, constructing knowledge maps to reveal the evolutionary trajectory. The findings show that the field has transformed from intelligent tutoring systems to inquiry-based learning and computational thinking cultivation driven by LLMs. AI's key contribution lies in providing intelligent scaffolding that lowers the threshold for understanding knowledge. In this sense, AI is a core driving force promoting its shift from knowledge transmission to capability development.

cs.CY↗

arXiv2026

Uniqueness and nondegeneracy of positive solutions to elliptic equations with Robin boundary conditions

In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -Δu = u^p, & \text{in } Ω,\\ u > 0, & \text{in } Ω,\\ \frac{\partial u}{\partial ν} + βu = 0, & \text{on } \partial Ω, \end{cases} $$ where $β> 0$, $p$ is subcritical, and $Ω$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $Ω$ is a ball, the problem is open for arbitrary $β>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $β>0$ provided that $p$ and $Ω$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.

math.AP↗

arXiv2026

Asymptotic behavior of solutions to elliptic problems with Robin boundary conditions

In this paper, we investigate the asymptotic behavior, as $β\to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -Δu = u^p, & \text{in } Ω,\\ u > 0, & \text{in } Ω,\\ \frac{\partial u}{\partial ν} + βu = 0, & \text{on } \partial Ω, \end{cases} \end{equation*} where $p \ge 0$, $β> 0$, and $Ω$ is a bounded smooth domain. We will prove that, for all $p\ge0$, the solution $u_β$ behaves like a constant as $β\to0$. However, the value of this constant is strongly influenced by the value of $p$. Indeed, \begin{itemize} \item if $0 \le p < 1$, $u_β$ blows up uniformly in $Ω$ as $β\to 0$. \item if $p=1$ (eigenvalue problem), $u_β$ converge to a constant. \item if $p>1$ $u_β$ converge uniformly to zero. \end{itemize} In the critical and supercritical regime $p \ge \frac{N+2}{N-2}$, the existence of solutions is no longer guaranteed a priori. In this case, when $Ω$ is a ball and $0<β<\frac{2}{p-1}$ we prove the existence of a radial positive solution.

math.AP↗

arXiv2025

Transformations of the 2-component BKP tau functions

The 2-component BKP (2-BKP) hierarchy is an important integrable system corresponding to the infinite dimensional Lie algebras $b_{\infty}$ and $d_{\infty}$, which contains Novikov-Veselov equation and can be used to describe the total descendent potential of D type singularity. Here we firstly introduce the projections of the mixed pseudo-differential operators to rewrite the 2-BKP Lax equation in the Shiota construction, where the scalar Lax operators involving two differential operators $\partial_1$ and $\partial_2$ are used. Based upon this, the $(M_1,M_2)$-reduction of the 2-BKP hierarchy is given. After that, we give the most important result of this paper, i.e., the transformations of the 2-BKP tau functions, which are in fact the 2-BKP Darboux transformations. Here we further give the corresponding changes in the 2-BKP Lax operators. Also the corresponding results are investigated for the reduction case. Finally, the additional symmetries can be viewed as the special cases of the transformations of the 2-BKP tau functions. Besides, we discuss the Pfaffian identities of the 2-BKP tau functions by successive applications of the above transformations, which are closely related with the 2-BKP addition formulae.

nlin.SI↗

arXiv2025

One reduction of the modified Toda hierarchy

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP hierarchies share the same fermionic form, we firstly construct the reduction of the mToda hierarchy $L_1(n)^M=L_2(n)^N+\sum_{l\in\mathbb{Z}}\sum_{i=1}^{m}q_{i,n}Λ^lr_{i,n+1}Δ$ and $(L_1(n)^M+L_2(n)^N)(1)=0$, called the generalized bigraded modified Toda hierarchy, which can be viewed as a new two-component generalization of the constrained mKP hierarchy $\mathfrak{L}^k=(\mathfrak{L}^k)_{\geq 1}+\sum_{i=1}^m \mathfrak{q}_i\partial^{-1}\mathfrak{r}_i\partial$. Next the relation with the Toda reduction $\mathcal{L}_1(n)^M=\mathcal{L}_2(n)^{N}+\sum_{l\in \mathbb{Z}}\sum_{i=1}^{m}\tilde{q}_{i,n}Λ^l\tilde{r}_{i,n}$ is discussed. Finally we give equivalent formulations of the Toda and mToda reductions in terms of tau functions.

nlin.SI↗

arXiv2024

Toda Darboux transformations and vacuum expectation values

Determinant formulas for vacuum expectation values $\langle s+k+n-m,-s|e^{H(\mathbf{t})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ are given by using Toda Darboux transformations. Firstly notice that 2--Toda hierarchy can be viewed as the 2--component bosonizations of fermionic KP hierarchy, then two elementary Toda Darboux transformation operators $T_{+}(q)=Λ(q)\cdotΔ\cdot q^{-1}$ and $T_{-}(r)=Λ^{-1}(r)^{-1}\cdotΔ^{-1}\cdot r$ are constructed from the changes of Toda (adjoint) wave functions by using 2--component boson--fermion correspondence. Based on this, the above vacuum expectation values now can be realized as the successive applications of Toda Darboux transformations. So the corresponding determinant formulas can be derived from the determinant representations of Toda Darboux transformations. Finally by similar methods, we also give the determinant formulas for $\langle n-m|e^{\mathcal{H}(\mathbf{x})}β_m^{*}\cdotsβ_1^{*}β_n\cdotsβ_1g|k\rangle $ related with KP tau functions.

nlin.SI↗

arXiv2024

Solutions of generalized constrained discrete KP hierarchy

Solutions of a generalized constrained discrete KP (gcdKP) hierarchy with constraint on Lax operator $L^k=(L^k)_{\geq m}+\sum_{i=1}^lq_iΔ^{-1}Λ^mr_i$, are invesitigated by Darboux transformations $T_D(f)=f^{[1]}\cdotΔ\cdot f^{-1}$ and $T_I(g)=(g^{[-1]})^{-1}\cdotΔ^{-1}\cdot g$. Due to this special constraint on Lax operator, it is showed that the generating functions $f$ and $g$ of the corresponding Darboux transformations, can only be chosen from (adjoint) wave functions or $(L^k)_{<m}=\sum_{i=1}^lq_iΔ^{-1}Λ^mr_i$. Then successive applications of Darboux transformations for gcdKP hierarchy are discussed. Finally based upon above, solutions of gcdKP hierarchy are obtained from $L^{\{0\}}=Λ$ by Darboux transformations.

nlin.SI↗

arXiv2022

A coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent: existence and multiplicity of high energy positive solutions

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -Δu+(V_1(x)+λ_1)u=μ_1(|x|^{-4}*u^{2})u+β(|x|^{-4}*v^{2})u, \ \ &x\in R^N, -Δv+(V_2(x)+λ_2)v=μ_2(|x|^{-4}*v^{2})v+β(|x|^{-4}*u^{2})v, \ \ &x\in R^N, \end{cases} \end{equation*} where $N\geq 5$, $λ_1$, $λ_2\geq 0$ with $λ_1+λ_2\neq 0$, $V_1(x), V_{2}(x)\in L^{\frac{N}{2}}(R^N)$ are nonnegative functions and $μ_1$, $μ_2$, $β$ are positive constants. Such system arises from mathematical models in Bose-Einstein condensates theory and nonlinear optics. By variational methods combined with degree theory, we prove some results about the existence and multiplicity of high energy positive solutions under the hypothesis $β>\max\{μ_1,μ_2\}$

math.AP↗