Geodesic
Smooth Solutions of the tt* Equation: A Numerical Aided Case Study
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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.
Keywords: fine asymptotics, truncation structure, numerical study.
Mots-clés : tt* equation 
@article{SIGMA_2024_20_a56,
     author = {Yuqi Li},
     title = {Smooth {Solutions} of the tt* {Equation:} {A} {Numerical} {Aided} {Case} {Study}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
     volume = {20},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a56/}
}
TY  - JOUR
AU  - Yuqi Li
TI  - Smooth Solutions of the tt* Equation: A Numerical Aided Case Study
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2024
VL  - 20
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a56/
LA  - en
ID  - SIGMA_2024_20_a56
ER  - 
%0 Journal Article
%A Yuqi Li
%T Smooth Solutions of the tt* Equation: A Numerical Aided Case Study
%J Symmetry, integrability and geometry: methods and applications
%D 2024
%V 20
%U http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a56/
%G en
%F SIGMA_2024_20_a56
Yuqi Li. Smooth Solutions of the tt* Equation: A Numerical Aided Case Study. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a56/

[1] Cecotti S., Vafa C., “Topological–anti-topological fusion”, Nuclear Phys. B, 367 (1991), 359–461 | DOI | MR | Zbl

[2] Cecotti S., Vafa C., “Exact results for supersymmetric $\sigma$ models”, Phys. Rev. Lett., 68 (1992), 903–906 | DOI | MR | Zbl

[3] Cecotti S., Vafa C., “On classification of $N=2$ supersymmetric theories”, Comm. Math. Phys., 158 (1993), 569–644, arXiv: hep-th/9211097 | DOI | MR | Zbl

[4] Dubrovin B., “Geometry and integrability of topological-antitopological fusion”, Comm. Math. Phys., 152 (1993), 539–564, arXiv: hep-th/9206037 | DOI | MR | Zbl

[5] Guest M.A., Its A.R., Lin C.S., “Isomonodromy aspects of the tt* equations of Cecotti and Vafa I Stokes data”, Int. Math. Res. Not. IMRN, 2015 (2015), 11745–11784, arXiv: 1209.2045 | DOI | MR | Zbl

[6] Guest M.A., Its A.R., Lin C.S., “Isomonodromy aspects of the tt* equations of Cecotti and Vafa II: Riemann–Hilbert problem”, Comm. Math. Phys., 336 (2015), 337–380, arXiv: 1312.4825 | DOI | MR | Zbl

[7] Guest M.A., Its A.R., Lin C.S., “Isomonodromy aspects of the tt* equations of Cecotti and Vafa III: Iwasawa factorization and asymptotics”, Comm. Math. Phys., 374 (2020), 923–973, arXiv: 1707.00259 | DOI | MR | Zbl

[8] Guest M.A., Its A.R., Lin C.S., The tt*-Toda equations of $A_n$ type, arXiv: 2302.04597

[9] Guest M.A., Lin C.S., “Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa”, J. Reine Angew. Math., 689 (2014), 1–32, arXiv: 1010.1889 | DOI | MR | Zbl

[10] Mochizuki T., Harmonic bundles and Toda lattices with opposite sign, arXiv: 1301.1718

[11] Mochizuki T., “Harmonic bundles and Toda lattices with opposite sign II”, Comm. Math. Phys., 328 (2014), 1159–1198 | DOI | MR | Zbl