Classification of Solutions with Polynomial Energy Growth for the SU (n + 1) Toda System on the Punctured Complex Plane

TL;DR

The paper classifies solutions of the ${\mathrm{SU}}(n+1)$ Toda system on the punctured plane $\mathbb{C}^*$ that exhibit polynomial energy growth near $0$ and $\infty$, using Nevanlinna theory to connect solutions with unitary curves. It proves that the associated holomorphic components have finite local growth orders at both singularities, and that these components form a basis of solutions to a linear differential equation of order $n+1$ with coefficients expressible as sums of a polynomial in $z$ and a polynomial in $1/z$. This establishes a precise link between the nonlinear PDE, complex-analytic growth control, and linear ODE theory, with implications for the monodromy and structure of solutions. The results extend prior work by Eremenko, Jost–Wang, and Mu, and provide a framework for further investigation of monodromy constraints and higher-order Toda-type systems on punctured domains.

Abstract

This paper investigates the classification of solutions satisfying the polynomial energy growth condition near both the origin and infinity to the ${\mathrm SU}(n+1)$ Toda system on the punctured complex plane $\mathbb{C}^*$. The ${\mathrm SU}(n+1)$ Toda system is a class of nonlinear elliptic partial differential equations of second order with significant implications in integrable systems, quantum field theory, and differential geometry. Building on the work of A. Eremenko (J. Math. Phys. Anal. Geom., Volume 3 p.39-46), Jingyu Mu's thesis, and others, we obtain the classification of such solutions by leveraging techniques from the Nevanlinna theory. In particular, we prove that the unitary curve corresponding to a solution with polynomial energy growth to the ${\mathrm SU}(n+1)$ Toda system on $\mathbb{C}^*$ gives a set of fundamental solutions to a linear homogeneous ODE of $(n+1)^{th}$ order, and each coefficient of the ODE can be written as a sum of a polynomial in $z$ and another one in $\frac{1}{z}$.