173 citations · 228 across the 28 of their papers we have counts for
30 papers
Geometric analysis on rhombus torus: Green function with two singularities
Zhijie Chen, Erjuan Fu, Chang-Shou Lin +1
Let be the Green function on the flat torus with the singularity at . Lin and Wang (Ann. Math. 2010) proved that has at m…
Sovability of curvature equations with multiple singular sources on torus via Painleve VI equations
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
We study the curvature equation with multiple singular sources on a torus \[Δu+e^{u}=8π\sum_{k=0}^{3}n_{k}δ_{\frac{ω_{k}}{2}}% +4π\left( δ_{p}+δ_{-p}\right) \quad \text{ on }\;E_τ:…
Generic non-degeneracy of critical points of multiple Green functions on torus and applications to curvature equations
Zhijie Chen, Erjuan Fu, Chang-Shou Lin
Let with be a flat torus and be the Green function on with the singularity at . Consider the mul…
Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law
Zhijie Chen, Chang-Shou Lin
The Darboux-Treibich-Verdier (DTV) potential is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Tr…
Sharp estimates and non-degeneracy of low energy nodal solutions for the Lane-Emden equation in dimension two
Zhijie Chen, Zetao Cheng, Hanqing Zhao
We study the Lane-Emden problem \[\begin{cases} -Δu_p =|u_p|^{p-1}u_p&\text{in}\quad Ω, u_p=0 &\text{on}\quad\partialΩ, \end{cases}\] where is a smooth bounde…
Asymptotic behavior of least energy nodal solutions for biharmonic Lane-Emden problems in dimension four
Zhijie Chen, Zetao Cheng, Hanqing Zhao
In this paper, we study the asymptotic behavior of least energy nodal solutions to the following fourth-order elliptic problem \[ \begin{cases} Δ^2 u =|u|^{p-1}u \quad &\h…
Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains
Zhijie Chen, Houwang Li, Wenming Zou
We study the Lane-Emden system where $Ω\subset…
Co-Axial Metrics on the Sphere and Algebraic Numbers
Zhijie Chen, Chang-Shou Lin, Yifan Yang
In this paper, we consider the following curvature equation $$Δu+{\rm e}^u=4π\biggl((θ_0-1)δ_0+(θ_1-1)δ_1 +\sum_{j=1}^{n+m}\bigl(θ_j'-1\bigr)δ_{t_j}\biggr)\qquad \text{in}\ \mathbb…
Monodromy of a generalized Lame equation of third order
Zhijie Chen, Chang-Shou Lin
We study the monodromy of the following third order linear differential equation \[y'''(z)-(α\wp(z;τ)+B)y'(z)+β\wp'(z;τ)y(z)=0, \] where is a parameter, …
On number and evenness of solutions of the Toda system on flat tori with non-critical parameters
Zhijie Chen, Chang-Shou Lin
We study the Toda system with singular sources \[ \begin{cases} Δu+2e^{u}-e^v=4π\sum_{k=0}^m n_{1,k}δ_{p_k}\quad\text{ on }\; E_τ,\\ Δv+2e^{v}-e^u=4π\sum_{k=0}^m n_{2,k}δ_{…
Modular Ordinary Differential Equations on of Third Order and Applications
Zhijie Chen, Chang-Shou Lin, Yifan Yang
In this paper, we study third-order modular ordinary differential equations (MODE for short) of the following form , $z\in\mathbb{H}=\{z\in\mathbb{C} \,|\,…
Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lamé equations with index modulo scalar equivalence with the monodromy group dihedral …
The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In this paper, the third in a series, we continue to study the generalized Lamé equation H with the Darboux-Treibich-Verdier potential \begin{equation*} y^{\pr…
A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in
Zhijie Chen, Erjuan Fu, Chang-Shou Lin
In this paper, we study the spectrum of the complex Hill operator in with the Darboux-Treibich-Verdier potential \[q(x;τ):=…
The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k…
Sharp nonexistence results for curvature equations with four singular sources on rectangular tori
Zhijie Chen, Chang-Shou Lin
In this paper, we prove that there are no solutions for the curvature equation \[ Δu+e^{u}=8πnδ_{0}\text{ on }E_τ, \quad n\in\mathbb{N}, \] where is a flat rectangular torus…
The geometry of generalized Lamé equation, I
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In this paper, we prove that the spectral curve of the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[…
Critical points of the classical Eisenstein series of weight two
Zhijie Chen, Chang-Shou Lin
In this paper, we completely determine the critical points of the normalized Eisenstein series of weight . Although is not a modular form, our result shows tha…
Simple zero property of some holomorphic functions on the moduli space of tori
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter , we introd…
Real-root property of the spectral polynomial of the Treibich-Verdier potential and related problems
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin +1
We study the spectral polynomial of the Treibich-Verdier potential. Such spectral polynomial, which is a generalization of the classical Lame polynomial, plays fundamental roles in…
Non-existence of solutions for a mean field equation on flat tori at critical parameter
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
It is known from \cite{LW} that the solvability of the mean field equation with on a flat torus essentially depends on the geome…
Unitary monodromy implies the smoothness along the real axis for some Painlevé VI equation, I
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In this paper, we study the Painlevé VI equation with parameter . We prove (i) An explicit formula to count the number of poles…
Hamiltonian system for the elliptic form of Painlevé VI equation
Zhijie Chen, Ting-Jung Kuo, Chang-Shou Lin
In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on . In this paper…
A new type of non-topological bubbling solutions to a competitive Chern-Simons model
Zhijie Chen, Chang-Shou Lin
We study a non-Abelian Chern-Simons system in , including the simple Lie algebras and . In a previous work, we proved the existence of radial non-topologic…