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Chang-Shou Lin

Publications and source records attributed to Chang-Shou Lin.

At least 37 records · Page 2Linked to original sources

arXiv2021

Proof of a conjecture of Dahmen and Beukers on counting integral Lamé equations with finite monodromy

In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lamé equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)Ψ(N)}{24}-\left( a_{n}% ϕ(N)+b_{n}ϕ(\tfrac{N}{2}) \right) \right) +\frac{2}% {3}\varepsilon_{n}(N).\] Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(τ)$ of weight $n(n+1)/2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(τ):=\prod_{(r,s)}Z_{r,s}^{(n)}(τ)$ of weight $Ψ(N)n(n+1)/{2}$, where the product runs over all $N$-torsion points $(r,s)$ of exact order $N$. We show that this conjecture is equivalent to the precise formula of the vanishing order of $M_{n,N}(τ)$ at infinity: \[v_{\infty}(M_{n,N}(τ))=a_{n}ϕ(N)+b_{n}ϕ( N/2).\] This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(τ)$ is not known for general $n$. Here we succeed to prove it by using certain Painlevé VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlevé VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.

math.NT↗

arXiv2021

Quasimodular forms and modular differential equations which are not apparent at cusps: I

In this paper, we explore a two-way connection between quasimodular forms of depth $1$ and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases $Γ=Γ_0^+(N)$ generated by $Γ_0(N)$ and the Atkin-Lehner involutions for $N=1,2,3$ ($Γ_0^+(1)=\mathrm{SL}(2,\mathbb Z)$). Firstly, we note that a quasimodular form of depth $1$, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups $Γ_0^+(N)$, $N=1,2,3$.

math.NT↗

arXiv2020

The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence

In this paper, the third in a series, we continue to study the generalized Lamé equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} and a related linear ODE with additional singularities $\pm p$ from the monodromy aspect.We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+2,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations $ρ:π_1(E_τ)\to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $τ=\frac12+i\frac{\sqrt{3}}{2}$, there are $B_1,B_2$ such that the monodromy representations of H$(1,0,0,0;B_1)$ and H$(4,0,0,0;B_2)$ are {\bf the same}, namely the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+3,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations is {\bf not} necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on $\mathbb{CP}^1$ with finite singularities.

math.CA↗

arXiv2020

Blow up at infinity in the SU(3) Chern-Simons model, part I

We consider non-topological solutions of a nonlinear elliptic system problem derived from the $SU(3)$ Chern-Simons models in $\mathbb{R}^2$. The existence of non-topological solutions even for radial symmetric case has been a long standing open problem. Recently, [Choe, Kim, Lin (2015, 2016)] showed the existence of radial symmetric non-topological solution when the vortex points collapse. However, the arguments in [Choe, Kim, Lin (2015, 2016)] cannot work for an arbitrary configuration of vortex points. In this paper, we develop a new approach by using different scalings for different components of the system to construct a family of non-topological solutions, which blows up at infinity.

math.AP↗

arXiv2020

A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in $\mathbb{R}$

In this paper, we study the spectrum of the complex Hill operator $L=\frac{d^2}{dx^2}+q(x;τ)$ in $L^2(\mathbb{R},\mathbb{C})$ with the Darboux-Treibich-Verdier potential \[q(x;τ):=-\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp \left( x+z_0+\tfrac{ω_{k}}{2};τ\right),\] where $n_k\in\mathbb{Z}_{\geq 0}$ with $\max n_k\geq 1$ and $z_0\in\mathbb{C}$ is chosen such that $q(x;τ)$ has no singularities on $\mathbb{R}$. For any fixed $τ\in i\mathbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $σ(L)$ is \[σ(L)=(-\infty, E_{2g}]\cup[E_{2g-1}, E_{2g-2}]\cup \cdots \cup[E_{1}, E_{0}],\quad E_j\in \mathbb{R},\] and hence generalizes Ince's remarkable result in 1940 for the Lamé potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1}$, $E_{2j-2})$, which generalizes the recent result in \cite{HHV} where the Lamé case $n_1=n_2=n_3=0$ was studied.

math.CA↗

arXiv2019

On $SU(3)$ Toda system with multiple singular sources

We consider the singular $SU(3)$ Toda system with multiple singular sources \begin{align*} \left\{\begin{array}{ll}-Δw_1=2e^{2w_1}-e^{w_2}+2π\sum_{\ell=1}^mβ_{1,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2\\ \rule{0cm}{.5cm} -Δw_2=2e^{2w_2}-e^{w_1}+2π\sum_{\ell=1}^mβ_{2,\ell}δ_{P_{\ell}}\quad\text{in }\mathbb{R}^2 \\ w_i(x)=-2\log|x|+O(1)\quad\text{as }|x|\to\infty,\, i=1,2, \end{array}\right.\end{align*} with $m\geq 3$ and $β_{i,\ell}\in [0,1)$. We prove the existence and non-existence results under suitable assumptions on $β_{i,\ell}$. This generalizes Luo-Tian's \cite{Luo-Tian} result for a singular Liouville equation in $\mathbb{R}^2$. We also study existence results for a higher order singular Liouville equation in $\mathbb{R}^n$.

math.AP↗

arXiv2018

Isomonodromy aspects of the tt* equations of Cecotti and Vafa III. Iwasawa factorization and asymptotics

This paper, the third in a series, completes our description of all (radial) solutions on C* of the tt*-Toda equations, using a combination of methods from p.d.e., isomonodromic deformations (Riemann-Hilbert method), and loop groups. We place these global solutions into the broader context of solutions which are smooth near 0. For such solutions, we compute explicitly the Stokes data and connection matrix of the associated meromorphic system, in the resonant cases as well as the non-resonant case. This allows us to give a complete picture of the monodromy data of the global solutions.

math.DG↗

arXiv2018

The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

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=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2}|τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} from the monodromy aspect. We prove the existence of a pre-modular form $Z_{r,s}^{\mathbf{n}}(τ)$ of weight $\frac{1}{2}\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\mathbf{n}}(τ)=0$. This generalizes the result in \cite{LW2}, where the Lamé case (i.e. $n_1=n_2=n_3=0$) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations \[Δu+e^u=16πδ_{0}\quad\text{and}\quad Δu+e^u=8π\sum_{k=1}^3δ_{\frac{ω_k}{2}}\] on a flat torus $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ has the same number of even solutions. This result is quite surprising from the PDE point of view.

math.CA↗

arXiv2018

Existence of bubbling solutions without mass concentration

The seminal work \cite{bm} by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the so-called "bubbling implies mass concentration". Recently, Lin and Tarantello in \cite{lt} pointed out that the "bubbling implies mass concentration" phenomena might not hold in general if the collapse of singularities occurs. In this paper, we shall construct the first concrete example of non-concentrated bubbling solution of the mean field equation with collapsing singularities.

math.AP↗

arXiv2018

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

The aim of this paper is to complete the program initiated in [50], [23] and then carried out by several authors concerning non-degeneracy and uniqueness of solutions to mean field equations. In particular, we consider mean field equations with general singular data on non-smooth domains. The argument is based on the Alexandrov-Bol inequality and on the eigenvalues analysis of linearized singular Liouville-type problems.

math.AP↗

arXiv2017

Sharp nonexistence results for curvature equations with four singular sources on rectangular tori

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 $E_τ$ is a flat rectangular torus and $δ_{0}$ is the Dirac measure at the lattice points. This confirms a conjecture in \cite{CLW2} and also improves a result of Eremenko and Gabrielov \cite{EG}. The nonexistence is a delicate problem because the equation always has solutions if $8πn$ in the RHS is replaced by $2πρ$ with $0<ρ\notin 4\mathbb{N}$. Geometrically, our result implies that a rectangular torus $E_τ$ admits a metric with curvature $+1$ acquiring a conic singularity at the lattice points with angle $2πα$ if and only if $α$ is not an odd integer. Unexpectedly, our proof of the nonexistence result is to apply the spectral theory of finite-gap potential, or equivalently the algebro-geometric solutions of stationary KdV hierarchy equations. Indeed, our proof can also yield a sharp nonexistence result for the curvature equation with singular sources at three half periods and the lattice points.

math.AP↗

arXiv2017

The geometry of generalized Lamé equation, I

In this paper, we prove that the spectral curve $Γ_{\mathbf{n}}$ of the generalized Lamé equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{% ω_{k}}{2}|τ)+B\bigg] y(z),\text{ \ }n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} can be embedded into the symmetric space Sym$^{N}E_τ$ of the $N$-th copy of the torus $E_τ$, where $N=\sum n_{k}$. This embedding induces an addition map $σ_{\mathbf{n}}(\cdot|τ)$ from $Γ_{\mathbf{n}}$ onto $E_τ$. The main result is to prove that the degree of $σ_{% \mathbf{n}}(\cdot|τ)$ is equal to% \begin{equation*} \sum_{k=0}^{3}n_{k}(n_{k}+1)/2. \end{equation*} This is the first step toward constructing the premodular form associated with this generalized Lamé equation.

math.CA↗

arXiv2017

Uniqueness for bubbling solutions with collapsing singularities

The seminal work \cite{bm} by Brezis and Merle showed that the bubbling solutions of the mean field equation have the property of mass concentration. Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass concentration" phenomena might not hold if there is a collapse of singularities. Furthermore, a sharp estimate \cite{llty} for the bubbling solutions has been obtained. In this paper, we prove that there exists at most one sequence of bubbling solutions if the collapsing singularity occurs. The main difficulty comes from that after re-scaling, the difference of two solutions locally converges to an element in the kernel space of the linearized operator. It is well-known that the kernel space is three dimensional. So the main technical ingredient of the proof is to show that the limit after re-scaling is orthogonal to the kernel space.

math.AP↗

arXiv2017

Critical points of the classical Eisenstein series of weight two

In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(τ)$ of weight $2$. Although $E_2(τ)$ is not a modular form, our result shows that $E_2(τ)$ has at most one critical point in every fundamental domain of $Γ_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(τ)$. Furthermore, under the Möbius transformation of $Γ_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric interpretation of these smooth curves is also given. It turns out that these smooth curves coincide with the degeneracy curves of trivial critical points of a multiple Green function related to flat tori.

math.NT↗

arXiv2017

Non-existence of solutions for a mean field equation on flat tori at critical parameter $16π$

It is known from \cite{LW} that the solvability of the mean field equation $Δu+e^{u}=8nπδ_{0}$ with $n\in\mathbb{N}_{\geq 1}$ on a flat torus $E_τ$ essentially depends on the geometry of $E_τ$. A conjecture is the non-existence of solutions for this equation if $E_τ$ is a rectangular torus, which was proved for $n=1$ in \cite{LW}. For any $n\in \mathbb{N}_{\geq2}$, this conjecture seems challenging from the viewpoint of PDE theory. In this paper, we prove this conjecture for $n=2$ (i.e. at critical parameter $16π$).

math.AP↗

arXiv2017

Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in $\mathbb{R}^2$

We consider the Chern-Simons gauge theory of rank $2$ such as $SU(3)$, $SO(5)$, and $G_2$ Chern-Simons model in $\mathbb{R}^2$. There may exist three types of solutions in these theories, that is, topological, nontopological, and mixed type solutions. Among others, mixed type solutions can only exist in non-Abelian Chern-Simons models. We show the existence of mixed type solutions with an arbitrary configuration of vortex points which has been a long-standing open problem. To show it, as the first step, we need to find when a priori bound would fail. For the purpose, we shall find partially blowing up mixed type solutions by using different scalings for different components. Due to the different scalings, we should control the mass contribution from infinity which is one of the important parts in this paper.

math.AP↗

arXiv2017

A new type of non-topological bubbling solutions to a competitive Chern-Simons model

We study a non-Abelian Chern-Simons system in $\mathbb{R}^2$, including the simple Lie algebras $A_2$ and $B_2$. In a previous work, we proved the existence of radial non-topological solutions with prescribed asymptotic behaviors via the degree theory. We also constructed a sequence of bubbling solutions with only one component blowing up partially at infinity. In this paper, we construct a sequence of radial non-topological bubbling solutions of another type via the shooting argument. One component of these bubbling solutions locally converge to a non-topological solution of the Chern-Simons-Higgs scalar equation, but both components blow up partially in different regions at infinity at the same time. This generalizes a recent work by Choe, Kim and the second author, where the $SU(3)$ case (i.e. $A_2$) was studied. Our result is new even for the $SU(3)$ case and also confirms the difference between the $SU(3)$ case and the $B_2$ case.

math.AP↗

arXiv2017

Simple zero property of some holomorphic functions on the moduli space of tori

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 $τ$, we introduce a conceptual proof by applying Painlevé VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.

math.CV↗