Smooth Solutions of the tt* Equation: A Numerical Aided Case Study

TL;DR

This paper uses numerical methods to verify and discover fine asymptotics of solutions to the tt*-Toda equations, enhancing understanding of these complex equations and their solution classes.

Contribution

It provides the first numerical verification of known asymptotics and explores new asymptotic formulas for an enlarged class of solutions.

Findings

Numerical verification of asymptotics for the tt*-Toda equations.

Discovery of new asymptotic formulas via numerical study.

Confirmation of the truncation structures' role in solution behavior.

Abstract

An important special class of the tt* equations are the tt*-Toda equations. Guest et al. have given comprehensive studies on the tt*-Toda equations in a series of papers. The fine asymptotics for a large class of solutions of a special tt*-Toda equation, the case 4a in their classification, have been obtained in the paper [Comm. Math. Phys. 374 (2020), 923-973] in the series. Most of these formulas are obtained with elaborate reasoning and the calculations involved are lengthy. There are concerns about these formulas if they have not been verified by other methods. The first part of this paper is devoted to the numerical verification of these fine asymptotics. In fact, the numerical studies can do more and should do more. A natural question is whether we can find more such beautiful formulas in the tt* equation via numerical study. The second part of this paper is devoted to the numerical study of the fine asymptotics of the solutions in an enlarged class defined from the Stoke data side. All the fine asymptotics of the solutions in the enlarged class are found by the numerical study. The success of the numerical study is largely due to the truncation structures of the tt* equation.