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

Abstract

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 ${\gamma }_i>-1$, ${\delta }_0$ 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: $$\sum _{j=1}^n{a}_{ij}{\int }_{{\mathbb{R}}^2}{e}^{u_j}dx=4\pi (2+{\gamma }_i+{\gamma }_{n+1-i}),\forall 1\le i\le n.$$ This generalizes the classification result by Jost and Wang for ${\gamma }_i=0$, $\forall 1\le i\le n$. (ii) We prove that if ${\gamma }_i+{\gamma }_{i+1}+...+{\gamma }_j\notin \mathbb{Z}$ for all $1\le i\le j\le n$, 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.

28 pages

Cited by in corpus (31)

• Classification of blowup limits for SU(3) singular Toda systems
• Chern--Simons Vortices in the Gudnason Model
• An improved geometric inequality via vanishing moments, with applications to singular Liouville equations
• On Rank Two Toda System with Arbitrary Singularities: Local Mass and New Estimates
• Degree counting and shadow system for $SU(3)$ Toda system: one bubbling
• On non-topological solutions for planar Liouville Systems of Toda-type
• Existence of mixed type solutions in the Chern-Simons gauge theory of rank two in ${\mathbb{R}}^2$
• New blow-up phenomena for SU(n+1) Toda system
• Non-degeneracy of Gauss curvature equation with negative conic singularity
• Total masses of solutions to general Toda systems with singular sources
• On Non-topological Solutions of the ${\mathbf{G}}_2$ Chern-Simons System
• On $SU(3)$ Toda system with multiple singular sources
• Nondegeneracy of the traveling lump solution to the $2+1$ Toda lattice
• Nondegeneracy of harmonic maps from ${\mathbb{R}}^2$ to ${\mathbb{S}}^2$
• Toda systems and hypergeometric equations
• On the general Toda system with multiple singular points
• On stable and finite Morse index solutions of the fractional Toda system
• Classifying solutions of $\mathrm{SU}(n+1)$ Toda system around a singular source
• Non-degeneracy and uniqueness of solutions to general singular Toda systems on bounded domains
• Toda field theories and integral curves of standard differential systems
• Classification of solutions to Toda systems of types $C$ and $B$ with singular sources
• On a general SU(3) Toda System
• The blow-up analysis of an affine Toda system corresponding to superconformal minimal surfaces in ${\mathbb{S}}^4$
• Blow up at infinity in the SU(3) Chern-Simons model, part I
• Modular Ordinary Differential Equations on $\mathrm{SL}(2,\mathbb{Z})$ of Third Order and Applications
• A new type of non-topological bubbling solutions to a competitive Chern-Simons model
• Blow-up Solutions for General Toda Systems on Riemann Surfaces
• Blowup solutions of elliptic systems in two dimensional spaces
• On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters
• Nonradial entire solutions for Liouville systems
• Wave equations associated to Liouville-type problems: global existence in time and blow up criteria