Paper
A flow approach to the Toda system
Authors
Yong Luo, Linlin Sun, Guofang Wang
Abstract
In this paper we introduce a flow to study the Toda system, which we call {\it Toda flow.} More generally, we introduce a flow of the Liouville systems, formulated as a coupled parabolic system with nonlocal interactions. Finite-time singularities are characterized and both necessary and sufficient conditions for convergence are provided in this general setting, even when the prescribed functions are allowed to change sign. As an application, we prove a global existence for the Toda flow in the critical case without restricting the sign of the prescribed functions. We provide a detailed description of blow-up behavior at infinity and obtain a sharp lower bound for the functional in cases where global convergence fails. By constructing appropriate test functions, we further establish a sufficient condition for the global convergence of the flow. These results are not affected by the sign-changing nature of the prescribed functions, and extend the theorem of Jost, Lin and Wang (Comm. Pure Appl. Math. 59, 526-558, 2006) to systems of multiple equations under this more general and physically relevant condition.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2602.20707v1</id>\n <title>A flow approach to the Toda system</title>\n <updated>2026-02-24T09:12:33Z</updated>\n <link href='https://arxiv.org/abs/2602.20707v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2602.20707v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>In this paper we introduce a flow to study the Toda system, which we call {\\it Toda flow.} More generally, we introduce a flow of the Liouville systems, formulated as a coupled parabolic system with nonlocal interactions. Finite-time singularities are characterized and both necessary and sufficient conditions for convergence are provided in this general setting, even when the prescribed functions are allowed to change sign.\n As an application, we prove a global existence for the Toda flow in the critical case without restricting the sign of the prescribed functions. We provide a detailed description of blow-up behavior at infinity and obtain a sharp lower bound for the functional in cases where global convergence fails. By constructing appropriate test functions, we further establish a sufficient condition for the global convergence of the flow. These results are not affected by the sign-changing nature of the prescribed functions, and extend the theorem of Jost, Lin and Wang (Comm. Pure Appl. Math. 59, 526-558, 2006) to systems of multiple equations under this more general and physically relevant condition.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.DG'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.AP'/>\n <published>2026-02-24T09:12:33Z</published>\n <arxiv:primary_category term='math.DG'/>\n <author>\n <name>Yong Luo</name>\n </author>\n <author>\n <name>Linlin Sun</name>\n </author>\n <author>\n <name>Guofang Wang</name>\n </author>\n </entry>"
}