[Paper Review] Degree counting and shadow system for $SU(3)$ Toda system: one bubbling

Chang‐Shou Lin, Juncheng Wei+1 more|arXiv (Cornell University)|Aug 25, 2014

Advanced Differential Equations and Dynamical Systems45 references18 citations

TL;DR

This paper establishes a topological degree counting method for the $SU(3)$ Toda system on compact Riemann surfaces by analyzing concentration phenomena and constructing a shadow system. It proves the existence of bubbling solutions and computes the Leray-Schauder degree for $\rho_1 \in (0,4\pi) \cup (4\pi,8\pi)$ and $\rho_2 \notin 4\pi\mathbb{N}$, resolving a key step toward full degree computation in the $SU(3)$ Toda framework.

ABSTRACT

Here we initiate the program for computing the Leray-Schauder topological degree for $SU(3)$ Toda system. This program still contains a lot of challenging problems for analysts. The first step of our approach is to answer whether concentration phenomena holds or not. In this paper, we prove the concentration phenomena holds while $ρ_1$ crosses $4π$, and $ρ_2 otin 4π\mathbb{N}$. However, for $ρ_1\geq 8π$, the question whether concentration holds or not still remains open up to now. The second step is to study the corresponding shadow system and its degree counting formula. The last step is to construct bubbling solution of $SU(3)$ Toda system via a non-degenerate solution of the shadow system. Using this construction, we succeed to calculate the degree for $ρ_1\in(0,4π)\cup(4π,8π)$ and $ρ_2 otin 4π\mathbb{N}$.

Motivation & Objective

• To compute the Leray-Schauder topological degree for the $SU(3)$ Toda system on compact Riemann surfaces, a long-standing challenge in geometric analysis and gauge theory.
• To determine whether concentration phenomena occur when $\rho_1$ crosses $4\pi$, especially in the regime $\rho_1 \in (4\pi,8\pi)$.
• To develop a shadow system approach that reduces the full Toda system to a lower-dimensional problem amenable to degree counting.
• To construct bubbling solutions of the $SU(3)$ Toda system via non-degenerate solutions of the shadow system.
• To establish a complete degree counting formula for $\rho_1 \in (0,4\pi) \cup (4\pi,8\pi)$ and $\rho_2 \notin 4\pi\mathbb{N}$.

Proposed method

• Analyzes the $SU(3)$ Toda system using a variational framework and blow-up analysis to detect concentration of solutions near conical singularities.
• Introduces a shadow system derived from the original Toda system by decoupling the blow-up profiles and reducing the problem to a finite-dimensional Morse-type problem.
• Applies a perturbation method to construct solutions of the full Toda system from solutions of the shadow system, leveraging non-degeneracy of the shadow solution.
• Employs Green's function expansions and asymptotic analysis to control the behavior of solutions near concentration points, particularly using $G(x,p_j)$ and $R(x,p_j)$ for regular and singular parts.
• Uses the ansatz $u_1^* = w + 2\psi - 4\pi \sum (1+\alpha_j)a_j G(x,p_j)$ to model blow-up profiles and estimates error terms $\mathfrak{E}_2$ via logarithmic and exponential expansions.
• Establishes error bounds of order $O(e^{-\lambda(P)/(1+\alpha_{m+1})})$ to validate the asymptotic approximation and justify the shadow system reduction.

Experimental results

Research questions

• RQ1 Does concentration occur in the $SU(3)$ Toda system when $\rho_1$ crosses $4\pi$, particularly for $\rho_1 \in (4\pi,8\pi)$?
• RQ2 Can the Leray-Schauder degree for the $SU(3)$ Toda system be computed in the regime $\rho_1 \in (0,4\pi) \cup (4\pi,8\pi)$ and $\rho_2 \notin 4\pi\mathbb{N}$?
• RQ3 Is the shadow system approach valid for constructing bubbling solutions of the $SU(3)$ Toda system from non-degenerate solutions of the reduced problem?
• RQ4 What is the precise asymptotic behavior of the solution profiles near blow-up points, and how do error terms in the approximation scale with the blow-up parameters?
• RQ5 How does the degree counting formula for the $SU(3)$ Toda system relate to the generating function $\Xi_1(x)$, and what is its topological significance?

Key findings

• Concentration of solutions is proven to occur in the $SU(3)$ Toda system when $\rho_1$ crosses $4\pi$, provided $\rho_2 \notin 4\pi\mathbb{N}$, confirming a key step in the degree computation program.
• The Leray-Schauder topological degree is computed for $\rho_1 \in (0,4\pi) \cup (4\pi,8\pi)$ and $\rho_2 \notin 4\pi\mathbb{N}$, establishing a complete formula in this parameter range.
• The degree is expressed as a sum of coefficients $\mathfrak{c}_j$ from the generating function $\Xi_0(x)$, with $d_\rho = \sum_{j=0}^k \mathfrak{c}_j$ for $8\pi\mathfrak{a}_k < \rho < 8\pi\mathfrak{a}_{k+1}$.
• The shadow system is rigorously justified as a reduction mechanism, and its non-degenerate solutions yield actual solutions to the full Toda system via a constructive perturbation method.
• Error estimates for the approximation are shown to be $O(e^{-\lambda(P)/(1+\alpha_{m+1})})$, validating the asymptotic expansion and the shadow system approach.
• The method provides a foundational framework for extending degree counting to higher $\rho_1$ values, though the case $\rho_1 \geq 8\pi$ remains open.