Construction of special solutions to Painleve systems and the quest for infinite-dimensional integrable systems

• Project/Area Number: 21740126
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Aoyama Gakuin University
• Principal Investigator: MASUDA Tetsu 青山学院大学, 理工学部, 准教授 (00335457)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 可積分系 / 離散羃函数 / 笹野系 / q-類似 / タウ函数 / 数理物理 / 離散冪函数 / 第6パンルヴェ方程式 / アフィンワイル群対称性 / 高次元可積分系 / q-超幾何函数 / q-パンルヴェ系

Research Abstract

We constructed several special solutions to the discrete Painleve systems. We determined the tau-functions of the solutions completely, and gave a natural description for some properties of the functions that appear as the solutions from the viewpoint of the affine Weyl group symmetry. We also constructed an explicit formula for the discrete power function in terms of the hypergeometric type solutions to the sixth Painlve equation. Moreover, we tried to construct a discrete analogue and special solutions of the higer-order Painleve systems to pave the way for discovering new infinite dimensional integrable systems

Research Products

• [Journal Article] 非線形波動研究の最前線 -構造と現象の多様性- (2013)
  • Author(s): 増田哲
  • Journal Title: 笹野系の q-類似について,九州大学応用力学研究所 研究集会報告 Volume: 24AO-S3 Pages: 71-76
  • Peer Reviewed
• [Journal Article] 笹野系の q-類似について (2013)
  • Author(s): 増田哲
  • Journal Title: 九州大学応用力学研究所 研究集会報告 Volume: 24AO-S3 Pages: 71-76
• [Journal Article] 非線形波動研究の進展 -現象と数理の相互作用- (2012)
  • Author(s): 岩尾慎介, 増田哲
  • Journal Title: 二項係数とSierpinski 三角形, 九州大学応用力学研究所 研究集会報告 Volume: 23AO-S7 Pages: 1-6
  • Peer Reviewed
• [Journal Article] 二項係数とSierpinski三角形 (2012)
  • Author(s): 岩尾慎介, 増田哲
  • Journal Title: 九州大学応用力学研究所研究集会報告 Volume: 23AO-S7 Pages: 1-6
• [Journal Article] Bilinearization and special solutions to the discrete Schwarzian KdV equation (2011)
  • Author(s): Mike Hay, Kenji Kajiwara and Tetsu Masuda
  • Journal Title: Journal of Math-for-Industry Volume: Vol. 3 Pages: 53-62
  • NAID: 40019161209
  • Peer Reviewed
• [Journal Article] Hypergeometric tau-functions of the q-Painleve system of type E_8^(1) (2011)
  • Author(s): Tetsu Masuda
  • Journal Title: The Ramanujan Journal Volume: Vol.24 Pages: 1-31
  • Peer Reviewed
• [Journal Article] 非線形波動研究の新たな展開現象とモデル化 (2011)
  • Author(s): 増田 哲
  • Journal Title: 離散羃函数の明示公式, 九州大学応用力学研究所研究集会報告 Volume: 22AO-S8 Pages: 56-61
• [Journal Article] Bilinearization and special solutions to the discrete Schwarzian KdV equation (2011)
  • Author(s): Mike Hay, Kenji Kajiwara, Tetsu Masuda
  • Journal Title: Journal of Math-for-Industry Volume: 3 Pages: 53-62
  • NAID: 40019161209
  • Peer Reviewed
• [Journal Article] Hypergeometric τ-functions of the q-Painlev'e system of type E_8^<(1)> (2011)
  • Author(s): Tetsu Masuda
  • Journal Title: The Ramanujan Journal Volume: 24 Pages: 1-31
  • Peer Reviewed
• [Journal Article] Hypergeometric tau-functions of the q-Painleve system of type E_7^(1) (2009)
  • Author(s): Tetsu Masuda
  • Journal Title: Symmetry, Integrability and Geometry: Methods and Applications Volume: Vol.5 Pages: 35-35
  • Peer Reviewed
• [Presentation] 笹野系の q-類似 (2013)
  • Author(s): 増田哲
  • Organizer: 日本数学会・無限可積分系セッション
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-22
• [Presentation] A q-analogue of the higher order Painleve type equations with W(D_l^(1))-symmetry (2013)
  • Author(s): Tetsu Masuda
  • Organizer: China-Japan Joint Workshop on Integrable Systems
  • Place of Presentation: Kyoto University, Japan
  • Year and Date: 2013-03-17
• [Presentation] D 型アフィン Weyl 群対称性をもつ高階 q-差分方程式 (2013)
  • Author(s): 増田哲
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学
  • Year and Date: 2013-03-16
• [Presentation] D 型アフィンワイル群対称性をもつ高階 q-パンルヴェ系 (2013)
  • Author(s): 増田哲
  • Organizer: 超幾何方程式研究会
  • Place of Presentation: 神戸大学
  • Year and Date: 2013-01-05
• [Presentation] D 型アフィンワイル群対称性をもつ高階 q-パンルヴェ系 (2013)
  • Author(s): 増田哲
  • Organizer: 超幾何方程式研究会 2013
  • Place of Presentation: 神戸大学大学院理学研究科
• [Presentation] D 型アフィン Weyl 群対称性をもつ高階 q-差分方程式 (2013)
  • Author(s): 増田哲
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学理学部
• [Presentation] A q-analogue of the higher order Painleve type equations with W(Dl(1))-symmetry (2013)
  • Author(s): Tetsu Masuda
  • Organizer: China-Japan Joint Workshop on Integrable Systems
  • Place of Presentation: Kyoto University
• [Presentation] 笹野系のq-類似 (2013)
  • Author(s): 増田哲
  • Organizer: 日本数学会・無限可積分系セッション
  • Place of Presentation: 京都大学
• [Presentation] 笹野系の q-類似 (2012)
  • Author(s): 増田哲
  • Organizer: 青山数理セミナー
  • Place of Presentation: 青山学院大学
  • Year and Date: 2012-12-19
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2012)
  • Author(s): Tetsu Masuda
  • Organizer: Various Aspects on the PainleveEquations
  • Place of Presentation: RIMS, Kyoto University, Japan
  • Year and Date: 2012-11-29
• [Presentation] 笹野系の q-類似について (2012)
  • Author(s): 増田哲
  • Organizer: 非線形波動研究の最前線 ‐構造と現象の多様性‐
  • Place of Presentation: 九州大学応用力学研究所
  • Year and Date: 2012-11-03
• [Presentation] 笹野系の q-類似について (2012)
  • Author(s): 増田哲
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: KKR蔵王 白銀荘
  • Year and Date: 2012-08-10
• [Presentation] 笹野系の q-類似について (2012)
  • Author(s): 増田哲
  • Organizer: 非線形波動研究の最前線－構造と現象の多様性－
  • Place of Presentation: 九州大学応用力学研究所
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2012)
  • Author(s): Tetsu Masuda
  • Organizer: Various Aspects on the Painleve Equations
  • Place of Presentation: RIMS, Kyoto University
• [Presentation] 二項係数とSierpinski 三角形 (2011)
  • Author(s): 岩尾慎介, 増田哲
  • Organizer: 非線形波動研究の進展-現象と数 理の相互作用-
  • Place of Presentation: 九州大学応用力学研究所
  • Year and Date: 2011-10-27
• [Presentation] 二項係数とSierpinski三角形 (2011)
  • Author(s): 岩尾慎介, 増田哲
  • Organizer: 非線形波動研究の進展-現象と数理の相互作用-
  • Place of Presentation: 九州大学応用力学研究所
  • Year and Date: 2011-10-27
• [Presentation] D 型高階 q-Painleve 系について (2011)
  • Author(s): 増田哲
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: 休暇村 支笏湖
  • Year and Date: 2011-08-05
• [Presentation] D型高階q-Painlev'e系について (2011)
  • Author(s): 増田哲
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: 休暇村支笏湖
  • Year and Date: 2011-08-05
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2011)
  • Author(s): Tetsu Masuda
  • Organizer: Infinite Analysis 11 -- Frontier of Integrability -
  • Place of Presentation: University of Tokyo, Japan
  • Year and Date: 2011-07-27
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2011)
  • Author(s): Tetsu Masuda
  • Organizer: Infinite Analysis 11-Frontier of Integrability-
  • Place of Presentation: University of Tokyo
  • Year and Date: 2011-07-27
• [Presentation] 離散冪函数の明示公式 (2011)
  • Author(s): 増田哲
  • Organizer: 日本数学会・無限可積分系セッション
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-23
• [Presentation] 離散冪函数の明示公式 (2011)
  • Author(s): 増田哲
  • Organizer: 超幾何方程式研究会
  • Place of Presentation: 神戸大学
  • Year and Date: 2011-01-06
• [Presentation] 離散羃函数の明示公式 (2011)
  • Author(s): 増田哲
  • Organizer: 超幾何方程式研究会
  • Place of Presentation: 神戸大学理学部
  • Year and Date: 2011-01-06
• [Presentation] 離散羃函数の明示公式 (2010)
  • Author(s): 増田哲
  • Organizer: 九州可積分系セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2010-11-12
• [Presentation] 離散羃函数の明示公式 (2010)
  • Author(s): 増田哲
  • Organizer: 非線形波動研究の新たな展開-現象とモデル化-
  • Place of Presentation: 九州大学応用力学研究所
  • Year and Date: 2010-10-29
• [Presentation] 離散羃函数の明示公式 (2010)
  • Author(s): 増田哲
  • Organizer: 非線形波動研究の新たな展開
  • Place of Presentation: 九州大学応用力学研究所
  • Year and Date: 2010-10-29
• [Presentation] 離散冪函数と Painleve VI 方程式の超幾何函数解 (2010)
  • Author(s): 増田哲
  • Organizer: 函数方程式論ワークショップ
  • Place of Presentation: 東京大学大学院数理科学研究科
  • Year and Date: 2010-07-26
• [Presentation] Painleve 系の対称性と特殊解 (2010)
  • Author(s): 増田哲
  • Organizer: デジタル解析学セミナー
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-01-27
• [Presentation] Painleve系の対称性と特殊解 (2010)
  • Author(s): 増田哲
  • Organizer: テジタル解析学セミナー
  • Place of Presentation: 早稲田大学
  • Year and Date: 2010-01-27
• [Presentation] The anti-self-dual Yang-Mills equation and the third Painleve equation (2010)
  • Author(s): Tetsu Masuda
  • Organizer: China-Japan Joint Workshop on Integrable Systems on the occasion of Prof. Hirota's 77th Birthday
  • Place of Presentation: Shaoxing Yonghe Manor Resort Hotel, Shaoxing
  • Year and Date: 2010-01-09
• [Presentation] The anti-self-dual Yang-Mills equation and the third Painleve equation (2010)
  • Author(s): Tetsu Masuda
  • Organizer: China-Japan Joint Workshop on Integrable Systems on the occasion of Prof. Hirota's 77th Birthday
  • Place of Presentation: Shaoxing Yonghe Manor Resort Hotel
  • Year and Date: 2010-01-09
• [Presentation] ASDYM 方程式と Ernst 方程式,ASDYM 方程式と Painleve III 方程式 (2009)
  • Author(s): 増田哲
  • Organizer: 自己双対 Yang-Mills 方程式の数理と可積分構造
  • Place of Presentation: 名古屋大学
  • Year and Date: 2009-11-08
• [Presentation] ASDYM方程式とErnst方程式, ASDYM方程式とPainleve III方程式 (2009)
  • Author(s): 増田哲
  • Organizer: -自己双対Yang-Mills方程式の数理と可積分構造
  • Place of Presentation: 名古屋大学
  • Year and Date: 2009-11-08
• [Presentation] 梅村多項式の判別式について (2009)
  • Author(s): 増田哲
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: かんぽの宿 鳥羽
  • Year and Date: 2009-08-02
• [Presentation] 梅村多項式の判別式について (2009)
  • Author(s): 増田哲
  • Organizer: 函数方程式論サマーセミナー
  • Place of Presentation: かんぽの宿鳥羽
  • Year and Date: 2009-08-02
• [Presentation] Discriminants of Umemura polynomials associated with P_V (2009)
  • Author(s): 増田哲
  • Organizer: Workshop on Accessory Parameters
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2009-06-21