Weyl chambers, asymptotic expansions, and blowup analysis of explicit Toda solutions
Abstract
Blowup analysis is a central part of geometric analysis, and systems of PDEs such as Toda systems exhibit particularly rich blowup behavior. In [Nie25], the author constructed explicit families of radial Toda solutions whose blowup masses correspond to arbitrary elements of the Weyl group of the underlying Lie algebra. In this paper, we study these families within the framework of blowup analysis developed by C.-S. Lin and his coauthors. The key is an asymptotic analysis of the solution family that distinguishes Cartan elements in the interior of a Weyl chamber from those lying on its boundary faces. We prove a uniform upper bound for uiλ,H(r)+2log r, so that the selection estimate holds with the center at the origin. We show that the successive bubbling scales for slow decay are determined by the intersections of the ray H+tqμ with root hyperplanes. At each crossing, rescaling and the face asymptotics identify the limiting profiles as explicit solutions of the corresponding Levi-Toda subsystems. The masses of these successive bubbles, computed using [KLNW24], account for the entire blowup mass. Examples in type A2 demonstrate this identification and exhibit both separated scalar bubbles and a simultaneous full-system bubble.
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BibTeX
@misc{nie2026weyl,
title = {{Weyl chambers, asymptotic expansions, and blowup analysis of explicit Toda solutions}},
author = {Zhaohu Nie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30767},
eprint = {2610.30767},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30767},
}
RIS
TY - UNPB
TI - Weyl chambers, asymptotic expansions, and blowup analysis of explicit Toda solutions
AU - Zhaohu Nie
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30767
UR - https://arxiv.org/abs/2610.30767
AB - Blowup analysis is a central part of geometric analysis, and systems of PDEs such as Toda systems exhibit particularly rich blowup behavior. In [Nie25], the author constructed explicit families of radial Toda solutions whose blowup masses correspond to arbitrary elements of the Weyl group of the underlying Lie algebra. In this paper, we study these families within the framework of blowup analysis developed by C.-S. Lin and his coauthors. The key is an asymptotic analysis of the solution family that distinguishes Cartan elements in the interior of a Weyl chamber from those lying on its boundary faces. We prove a uniform upper bound for u_i^λ,H(r)+2log r, so that the selection estimate holds with the center at the origin. We show that the successive bubbling scales for slow decay are determined by the intersections of the ray H+tq_μ with root hyperplanes. At each crossing, rescaling and the face asymptotics identify the limiting profiles as explicit solutions of the corresponding Levi-Toda subsystems. The masses of these successive bubbles, computed using [KLNW24], account for the entire blowup mass. Examples in type A_2 demonstrate this identification and exhibit both separated scalar bubbles and a simultaneous full-system bubble.
ER -