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Chang-Shou Lin

Publications and source records attributed to Chang-Shou Lin.

70 records · Page 4Linked to original sources
arXiv2013

Isomonodromy aspects of the tt* equations of Cecotti and Vafa II. Riemann-Hilbert problem

In Part I (arXiv:1209.2045) we computed the Stokes data, though not the "connection matrix", for the smooth solutions of the tt*-Toda equations whose existence we established by p.d.e. methods. Here we give an alternative proof of the existence of some of these solutions by solving a Riemann-Hilbert problem. In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data. We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions, clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, for the tt*-Toda equations, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S+S^t (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix.

math.DG↗
arXiv2013

Segregated Vector Solutions for linearly coupled Nonlinear Schrödinger Systems

We consider the following system linearly coupled by nonlinear Schrödinger equations in $\R^3$ $$ \left\{\begin{array}{ll} -Δu_j+u_j=u^3_j-\va\sum\limits_{i\neq j}^N u_i,\{1cm}& x\in \R^3, \{0.2cm}\\ u_j\in H^1(\R^3),\quad j=1,\cdots,N, \end{array} \right. $$ where $\va\in\R$ is a coupling constant. This type of system arises in particular in models in nonlinear $N$-core fiber. We examine the effect of the linear coupling to the solution structure. When $N=2,3$, for any prescribed integer $\ell\ge 2$, we construct a non-radial vector solutions of segregated type, with two components having exactly $\ell$ positive bumps for $\va>0$ sufficiently small. We also give an explicit description on the characteristic features of the vector solutions.

math.AP↗
arXiv2013

On the Entire Radial Solutions of the Chern-Simons SU(3) System

In this paper, we study the entire radial solutions of the self-dual equations arising from the relativistic SU(3) Chern-Simons model proposed by Kao-Lee and Dunne. Understanding the structure of entire radial solutions is one of fundamental issues for the system of nonlinear equations. In this paper, we prove any entire radial solutions must be one of topological, non-topological and mixed type solutions, and completely classify the asymptotic behaviors at infinity of these solutions. Even for radial solutions, this classification has remained an open problem for many years. As an application of this classification, we prove that the two components u and v have intersection at most finite times.

math.AP↗
arXiv2013

Chern--Simons Vortices in the Gudnason Model

We present a series of existence theorems for multiple vortex solutions in the Gudnason model of the ${\cal N}=2$ supersymmetric field theory where non-Abelian gauge fields are governed by the pure Chern--Simons dynamics at dual levels and realized as the solutions of a system of elliptic equations with exponential nonlinearity over two-dimensional domains. In the full plane situation, our method utilizes a minimization approach, and in the doubly periodic situation, we employ an-inequality constrained minimization approach. In the latter case, we also obtain sufficient conditions under which we show that there exist at least two gauge-distinct solutions for any prescribed distribution of vortices. In other words, there are distinct solutions with identical vortex distribution, energy, and electric and magnetic charges.

math.AP↗
arXiv2013

Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-Δu_1 +\la_1 u_1 = μ_1 u_1^3+βu_1 u_2^2, \quad x\in Ω,\\ -Δu_2 +\la_2 u_2 =μ_2 u_2^3+βu_1^2 u_2, \quad x\in \Om,\\ u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.\end{cases}\end{displaymath} Here $\Om\subset\RN (N=2, 3)$ is a smooth bounded domain, $\la_1, \la_2$, $μ_1, μ_2$ are all positive constants. We show that, for each $k\in\mathbb{N}$ there exists $\bb_k>0$ such that this system has at least $k$ sign-changing solutions (i.e., both two components change sign) and $k$ semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$.

math.AP↗
arXiv2013

Asymptotic analysis of solutions to a gauged O(3) sigma model

We analyze an elliptic equation arising in the study of the gauged O(3) sigma model with the Chern-Simons term. In this paper, we study the asymptotic behavior of solutions and apply it to prove the uniqueness of stable solutions. However, one of the features of this nonlinear equation is the existence of stable nontopological solutions in $\RN$, which implies the possibility that a stable solution which blows up at a vortex point exists. To exclude this kind of blow up behavior is one of the main difficulties which we have to overcome.

math.AP↗
arXiv2012

Some tt* structures and their integral Stokes data

In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpretation in quantum cohomology, and enumerate those solutions with integral Stokes data. This leads to a characterization of quantum D-modules for certain complete intersections of Fano type in weighted projective spaces.

math.DG↗
arXiv2012

Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter

We consider the mean field equation: (1) Δu+ρ\frac{e^u}{\int_Ωe^u}=0 & \hbox{in} \;Ω, u=0 & \hbox{on}\;\partialΩ, where $Ω\subset \mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper, Suzuki proved that if $Ω$ is a simply-connected domain, then equation (1) admits a unique solution for $ρ\in[0,8π)$. This result for $Ω$ a simply-connected domain has been extended to the case $ρ=8π$ by Chang, Chen and the second author. However, the uniqueness result for $Ω$ a multiply-connected domain has remained a long standing open problem which we solve positively here for $ρ\in[0,8π]$. To obtain this result we need a new version of the classical Bol's inequality suitable to be applied on multiply-connected domains. Our second main concern is the existence of solutions for (1) when $ρ=8π$. We a obtain necessary and sufficient condition for the solvability of the mean field equation at $ρ=8π$ which is expressed in terms of the Robin's function $γ$ for $Ω$. For example, if equation (1) has no solution at $ρ=8π$, then $γ$ has a unique nondegenerate maximum point. As a by product of our results we solve the long-standing open problem of the equivalence of canonical and microcanonical ensembles in the Onsager's statistical description of two-dimensional turbulence on multiply-connected domains.

math.AP↗
arXiv2012

Solutions to the Master Equations Governing Fractional Vortices

By means of variational methods, in this paper, we establish sharp existence results for solutions of the master equations governing `fractional multiple vortices.' In the doubly periodic situation, the conditions for existence are both necessary and sufficient and give the upper bounds for the vortex numbers in terms of the size of the periodic cell domain. In the planar situation, there is no restriction on the vortex numbers. In both situations, the solutions are uniquely determined by the prescribed locations and the local winding numbers of the vortices.

math-ph↗
arXiv2011

Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources

We consider the following Toda system Δu_i + \D \sum_{j = 1}^n a_{ij}e^{u_j} = 4πγ_{i}δ_{0} \text{in}\mathbb R^2, \int_{\mathbb R^2}e^{u_i} dx < \infty, \forall 1\leq i \leq n, where $γ_{i} > -1$, $δ_0$ is Dirac measure at 0, and the coefficients $a_{ij}$ form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result: $$\sum_{j=1}^n a_{ij}\int_{\R^2}e^{u_j} dx = 4π(2+γ_i+γ_{n+1-i}), \;\;\forall\; 1\leq i \leq n.$$ This generalizes the classification result by Jost and Wang for $γ_i=0$, $\forall \;1\leq i\leq n$. (ii) We prove that if $γ_i+γ_{i+1}+...+γ_j \notin \mathbb Z$ for all $1\leq i\leq j\leq n$, then any solution $u_i$ is \textit{radially symmetric} w.r.t. 0. (iii) We prove that the linearized equation at any solution is \textit{non-degenerate}. These are fundamental results in order to understand the bubbling behavior of the Toda system.

math.AP↗
arXiv2011

Tropical Atmospheric Circulations with Humidity Effects

The main objective of this article is to study the effect of the moisture on the planetary scale atmospheric circulation over the tropics. The modeling we adopt is the Boussinesq equations coupled with a diffusive equation of humidity and the humidity dependent heat source is modeled by a linear approximation of the humidity. The rigorous mathematical analysis is carried out using the dynamic transition theory. In particular, we obtain the same types of transitions and hence the scenario of the El Niño mechanism as described in \cite{MW2,MW3}. The effect of the moisture only lowers slightly the magnitude of the critical thermal Rayleigh number.

physics.ao-ph↗
arXiv2011

A Nonlinear Elliptic PDE with Two Sobolev-Hardy Critical Exponents

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& Δu + λ\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} Ω, & u=0 \qquad \text{on} Ω, where $0 \le s_2 < s_1 \le 2$, $0 \ne λ\in \Bbb R$ and $0 \in \partial Ω$. The existence (or nonexistence) for least-energy solutions has been extensively studied when $s_1=0$ or $s_2=0$. In this paper, we prove that if $0< s_2 < s_1 <2$ and the mean curvature of $\partial Ω$ at 0 $H(0)<0$, then \eqref{0.1} has a least-energy solution. Therefore, this paper has completed the study of \eqref{0.1} for the least-energy solutions. We also prove existence or nonexistence of positive entire solutions of \eqref{0.1} with $Ω=\rn$ under different situations of $s_1, s_2$ and $λ$.

math.AP↗
arXiv2010

Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa

Using nonlinear pde techniques, we construct a new family of globally smooth tt* structures. This includes tt* structures associated to the (orbifold) quantum cohomology of a finite number of complex projective spaces and weighted projective spaces. The existence of such "magical solutions" of the tt* equations, namely smooth solutions characterized by asymptotic boundary conditions, was predicted by Cecotti and Vafa. In our situation, the tt* equations belong to a class of equations which we call the tt*-Toda lattice. Solutions of the tt*-Toda lattice are harmonic maps which have dual interpretations as Frobenius structures or variations of (semi-infinite) Hodge structures.

math-ph↗
arXiv2009

On the Best Constant in the Moser-Onofri-Aubin Inequality

Let $S^2$ be the 2-dimensional unit sphere and let $J_α$ denote the nonlinear functional on the Sobolev space $H^{1,2}(S^2)$ defined by $$ J_α(u) = \fracα{4}\int_{S^2}|\nabla u|^2 dω+ \int_{S^2} u dω-\ln \int_{S^2} e^{u} dω, $$ where $dω$ denotes Lebesgue measure on $S^2$, normalized so that $\int_{S^2} dω= 1$. Onofri had established that $J_α$ is non-negative on $H^1(S^2)$ provided $α\geq 1$. In this note, we show that if $J_α$ is restricted to those $u\in H^1(S^2)$ that satisfy the Aubin condition: \int_{S^2}e^u x_j dw=0\quad\text{for all}1\leq j\leq 3, then the same inequality continues to hold (i.e., $J_α(u)\geq0$) whenever $α\geq {2/3}-ε_0$ for some $ε_0>0$. The question of Chang-Yang on whether this remains true for all $α\geq {1/2}$ remains open.

math.AP↗
arXiv2008

Elliptic functions, Green functions and the mean field equations on tori

We show that the Green functions on flat tori can have either 3 or 5 critical points only. There does not seemto be any directmethod to attack this problem. Instead, we have to employ sophisticated non-linear partial differential equations to study it. We also study the distribution of number of critical points over the moduli space of flat tori through deformations. The functional equations of special theta values provide important inequalities which lead to a solution for all rhombus tori.

math.AP↗