  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.
