Blow-up Solutions for General Toda Systems on Riemann Surfaces

TL;DR

This paper constructs and analyzes blow-up solutions for general Toda systems on Riemann surfaces with Neumann boundary conditions, revealing asymmetric concentration phenomena and precise blow-up point locations under symmetry assumptions.

Contribution

It introduces a singular perturbation approach to construct bubbling solutions with distinct concentration rates and identifies blow-up points at symmetric centers for the $SU(3)$ Toda system.

Findings

Existence of bubbling solutions with multiple concentration scales

Blow-up points located at $k$-symmetric centers

Construction of asymmetric blow-up solutions for $SU(3)$ Toda system

Abstract

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the $SU(3)$ Toda system. Furthermore, the blow-up points are precisely located at the ``$k$-symmetric'' centers of the surface. Keywords: Toda system, Neumann boundary condition, Blow-up solutions, $k$-symmetry, Finite-dimensional reduction