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.Comment: To appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienz

Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: {displaymath} {cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega, -\Delta u_2 +\la_2 u_2 =\mu_2 u_2^3+\beta u_1^2 u_2, \quad x\in \Om, u_1=u_2=0 \,\,\,\hbox{on \,\partial\Om}.{cases}{displaymath} Here \Om is a smooth bounded domain in $\R^N (N=2, 3)$ or \Om=\RN, \la_1,\, \la_2, $\mu_1,\,\mu_2$ are all positive constants and the coupling constant \bb<0. We show that this system has infinitely many sign-changing solutions. We also obtain infinitely many semi-nodal solutions in the following sense: one component changes sign and the other one is positive. The crucial idea of our proof, which has never been used for this system before, is turning to study a new problem with two constraints. Finally, when \Om is a bounded domain, we show that this system has a least energy sign-changing solution, both two components of which have exactly two nodal domains, and we also study the asymptotic behavior of solutions as $\beta\to -\infty$ and phase separation is expected.Comment: Final version, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Si

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 $$\Delta u+e^{u}=8\pi n\delta_{0}\text{ on }E_{\tau}, \quad n\in\mathbb{N},$$ where $E_{\tau}$ is a flat rectangular torus and $\delta_{0}$ 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 $8\pi n$ in the RHS is replaced by $2\pi \rho$ with $0<\rho\notin 4\mathbb{N}$. Geometrically, our result implies that a rectangular torus $E_{\tau}$ admits a metric with curvature $+1$ acquiring a conic singularity at the lattice points with angle $2\pi\alpha$ if and only if $\alpha$ 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.Comment: 28 page

Removable singularity of positive solutions for a critical elliptic system with isolated singularity

We study qualitative properties of positive singular solutions to a two-coupled elliptic system with critical exponents. This system is related to coupled nonlinear Schrodinger equations with critical exponents for nonlinear optics and Bose-Einstein condensates. We prove a sharp result on the removability of the same isolated singularity for both two components of the solutions. We also prove the nonexistence of positive solutions with one component bounded near the singularity and the other component unbounded near the singularity. These results will be applied in a subsequent work where the same system in a punctured ball will be studied.Comment: 18 pages, comments are welcom

Critical points of the classical Eisenstein series of weight two

In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(\tau)$ of weight $2$. Although $E_2(\tau)$ is not a modular form, our result shows that $E_2(\tau)$ has at most one critical point in every fundamental domain of $\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\tau)$. Furthermore, under the M\"obius transformation of $\Gamma_{0}(2)$ 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.Comment: 38pages, 3figure

The geometry of generalized Lam\'{e} equation, I

In this paper, we prove that the spectral curve $\Gamma_{\mathbf{n}}$ of the generalized Lam\'{e} 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{% \omega_{k}}{2}|\tau)+B\bigg] y(z),\text{ \ }n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} can be embedded into the symmetric space Sym$^{N}E_{\tau}$ of the $N$-th copy of the torus $E_{\tau}$, where $N=\sum n_{k}$. This embedding induces an addition map $\sigma_{\mathbf{n}}(\cdot|\tau)$ from $\Gamma_{\mathbf{n}}$ onto $E_{\tau}$. The main result is to prove that the degree of $\sigma _{\mathbf{n}}(\cdot|\tau)$ 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 Lam\'{e} equation

Hamiltonian system for the elliptic form of Painlev\'{e} VI equation

In literature, it is known that any solution of Painlev\'{e} VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lam\'{e} equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlev\'{e} VI equation for generic parameters. For Painlev\'{e} VI equation with some special parameters, the isomonodromy theory of the generalized Lam\'{e} equation greatly simplifies the computation of the monodromy group in $\mathbb{CP}^{1}$. This is one of the advantages of the elliptic form.Comment: 39 pages. Any comment is welcom

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 $\tau$, we introduce a conceptual proof by applying Painlev\'{e} 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.Comment: 20 pages, 1 figur

The geometry of generalized Lam\'{e} equation, II: Existence of pre-modular forms and application

In this paper, the second in a series, we continue to study the generalized Lam\'{e} 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{ \omega_{k}}{2}|\tau)+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 $Z_{r,s}^{\mathbf{n}}(\tau)$ of weight $\frac{1}{2}\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\mathbf{n}}(\tau)=0$. This generalizes the result in \cite{LW2}, where the Lam\'{e} 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 $$\Delta u+e^u=16\pi\delta_{0}\quad\text{and}\quad \Delta u+e^u=8\pi\sum_{k=1}^3\delta_{\frac{\omega_k}{2}}$$ on a flat torus $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ has the same number of even solutions. This result is quite surprising from the PDE point of view.Comment: 23page

SAGNet:Structure-aware Generative Network for 3D-Shape Modeling

We present SAGNet, a structure-aware generative model for 3D shapes. Given a set of segmented objects of a certain class, the geometry of their parts and the pairwise relationships between them (the structure) are jointly learned and embedded in a latent space by an autoencoder. The encoder intertwines the geometry and structure features into a single latent code, while the decoder disentangles the features and reconstructs the geometry and structure of the 3D model. Our autoencoder consists of two branches, one for the structure and one for the geometry. The key idea is that during the analysis, the two branches exchange information between them, thereby learning the dependencies between structure and geometry and encoding two augmented features, which are then fused into a single latent code. This explicit intertwining of information enables separately controlling the geometry and the structure of the generated models. We evaluate the performance of our method and conduct an ablation study. We explicitly show that encoding of shapes accounts for both similarities in structure and geometry. A variety of quality results generated by SAGNet are presented. The data and code are at https://github.com/zhijieW-94/SAGNet.Comment: Accepted by SIGGRAPH 201