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

Yuqi Li

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.

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

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.

Paper Structure

This paper contains 25 sections, 1 theorem, 68 equations, 9 figures, 23 tables.

Table of Contents

  1. Introduction
  2. Fine asymptotics of (\ref{['TT-2']}) at $\boldsymbol{r=0}$ of the class of solutions defined by Theorem \ref{['thm-GIL-1']}
  3. Verifying numerically the fine asymptotics of (\ref{['TT-2']}) at $\boldsymbol{r=0}$ of the class of solutions defined by Theorem \ref{['thm-GIL-1']}
  4. Preliminary for the numerical experiments: an approximation proper for calculations near $\boldsymbol{r=\infty}$
  5. The general case: in the triangular
  6. Numerical integration from $\boldsymbol{r=45}$ to $\boldsymbol{r=1}$
  7. Near $\boldsymbol{r=0}$
  8. Case E1
  9. Numerical integration from $\boldsymbol{r=45}$ to $\boldsymbol{r=1}$
  10. Near $\boldsymbol{r=0}$
  11. Case E2
  12. Case E3
  13. Numerical integration from $\boldsymbol{r=45}$ to $\boldsymbol{r=1}$
  14. Near $\boldsymbol{r=0}$
  15. Case V1
  16. ...and 10 more sections

Key Result

Theorem 1.1

For $a,b>0$ and any $(\gamma,\delta)$ in the triangular region $\gamma \ge -\frac{2}{a}$, $\delta \le \frac{2}{b}$, ${\gamma-\delta \le 2}$, the system TT has a unique smooth solution that satisfies the boundary condition BC. Furthermore, the unique solution is real and radially-invariant.

Figures (9)

  • Figure 1: The triangular region for $(\gamma,\delta)$.
  • Figure 2: The region map of the connection formula (\ref{['ConnectFormula']}).
  • Figure 3: Regions of $\Omega_i$, $i=0,1,2,3,4,5,6$, edges of E1, E2, E3, $E_1^U$, $E_2^U$, $E_1^D$, and vertex of V1, V2, V3.
  • Figure 4: Contour in the complex plane of $s$ to compute $v_0(s)$ and $v_1(s)$.
  • Figure 5: ${\rm e}^{-\gamma_0 s-\rho_0} v_0(s)$ (red); $\frac{1}{2} {\rm e}^{-\operatorname{Re}(\gamma_1)s-\operatorname{Re}(\rho_1)} v_1(s)$ (green).
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1.1: GIL-1
  • Definition 1.2
  • Remark 3.1
  • Conjecture 4.1