Study of the geometric structure behind discrete integrable systems

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14559
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Tokyo University of Agriculture and Technology
• Principal Investigator: Nakazono Nobutaka 東京農工大学, 工学(系)研究科(研究院), 講師 (40835162)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 可積分系 / 立方体上のコンシステンシー / 離散パンルヴェ方程式 / Lax形式 / affine Weyl群

Outline of Final Research Achievements

I got the following results.
(1) I showed that discrete power functions with hexagonal circle patterns can be derived from the theory of Garnier systems and the CAC property. (2) I clarified the consistency structures around the cube of Hirota's dKdV equation and the lattice sine-Gordon equation. (3) I showed that the general solutions of q-Painleve equations give special solutions of the dKdV equation. (4) I constructed systems of higher-order ordinary difference equations of Painleve type with q-Painleve equations of type A6 and A4 as second-order cases. I also showed their Lax pairs and affine Weyl group symmetries.

Free Research Field

可積分系

Academic Significance and Societal Importance of the Research Achievements

本研究は離散可積分系の理論およびその応用に関する数学の研究である。本研究成果により，様々な非線形差分方程式の背後にある格子の構造が明らかになった。このことにより，非線形差分方程式の対称性や解に対する新たなアプローチが得られることが期待される。