Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 057, 28 pages      arXiv:1910.13639      https://doi.org/10.3842/SIGMA.2024.057
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

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

Yuqi Li
School of Mathematical Sciences, Key Laboratory of MEA & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P.R. China

Received December 31, 2023, in final form June 13, 2024; Published online June 29, 2024

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.

Key words: tt* equation; fine asymptotics; truncation structure; numerical study.

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