Geometric methods in discrete integrablesystems

• Project/Area Number: 21340036
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Global analysis
• Research Institution: Kobe University
• Principal Investigator: YAMADA Yasuhiko 神戸大学, 大学院・理学研究科, 教授 (00202383)
• Co-Investigator(Kenkyū-buntansha): NOUMI Masatohi 神戸大学, 自然科学系先端融合研究環重点研究部, 教授 (80164672) ; OHTA Yasuhiro 神戸大学, 大学院・理学研究科, 教授 (10213745) ; SAITO Masahiko 神戸大学, 大学院・理学研究科, 教授 (80183044)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 離散可積分系 / ラックス表示 / 幾何学 / 特殊解 / パデ近似 / 共形場理論 / パンルヴェ方程式 / パデ補間 / 量子パンルヴェ方程式 / 特殊関数解 / 可積分系 / 楕円差分 / 超幾何函数 / 行列式表示 / q差分 / Lax形式 / AGT予想 / Fuji-Suzuki-Tsuda方程式 / ラックス形式 / 代数曲線

Research Abstract

We studied mainly the discrete Painleve equations. The Lax formalism for various cases including the elliptic difference Painleve equation of type E_8 was constructed, and its geometric characterization was clarified. We constructed the corresponding Pade interpolation problem and derive simple expressions for the Painleve equation, Lax form and special solutions. Relations among the Gauge theory, conformal field theory and quantum Painleve equations were also studied.

Research Products

• [Journal Article] Symmetries of quantum Lax equations for the Painleve equations (2013)
  • Author(s): Hajime Nagoya, Yasuhiko Yamada
  • Journal Title: Ann. Henri Poincare Volume: Online First Issue: 2 Pages: 313-344
  • DOI: 10.1007/s00023-013-0237-9
  • Peer Reviewed
• [Journal Article] Pade interpolation for elliptic Painleve equation (2013)
  • Author(s): M.Noumi, S.Tsujimoto and Y.Yamada
  • Journal Title: Springer Proceedings in Mathematics and Statistics Volume: 40 Pages: 463-482
  • DOI: 10.1007/978-1-4471-4863-0_18
  • ISBN: 9781447148623, 9781447148630
  • Peer Reviewed
• [Journal Article] Pad'e interpolation for elliptic Painlev'e equation (2013)
  • Author(s): M.Noumi, S.Tsujimoto and Y.Yamada
  • Journal Title: Springer Proceedings in Mathematics and Statistics Volume: 40 Pages: 463-482
  • Peer Reviewed
• [Journal Article] General High-Order Rogue Waves and Their Dynamics in the Nonlinear Schrodinger Equation (2012)
  • Author(s): Y.Ohta and J.Yang
  • Journal Title: Proc. R. Soc. Volume: A468 Issue: 2142 Pages: 1716-1740
  • DOI: 10.1098/rspa.2011.0640
  • Peer Reviewed
• [Journal Article] On the tau-Functions of the Reduced Ostrovsky Equation and the A_2^(2) Two-Dimensional Toda System (2012)
  • Author(s): B.-F.Feng, K.Maruno and Y.Ohta
  • Journal Title: J. Phys. A : Math. Theor. Volume: 45 Issue: 35 Pages: 355203-355203
  • DOI: 10.1088/1751-8113/45/35/355203
  • Peer Reviewed
• [Journal Article] ocalization with a surface operator, Irregular conformal blocks and Open topological string (2012)
  • Author(s): H.Awata, H.Fuji, H.Kanno, M.Manabe and Y.Yamada
  • Journal Title: Adv. Theor. Math. Phys. Volume: 16 Pages: 725-804
  • Peer Reviewed
• [Journal Article] Localization with a surface operator, Irregular conformal blocks and Open topological string (2012)
  • Author(s): H.Awata, H.Fuji, H.Kanno, M.Manabe and Y.Yamada
  • Journal Title: Adv. Theor. Math. Phys. Volume: 16 Pages: 725-804
  • Peer Reviewed
• [Journal Article] Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^(1) (2011)
  • Author(s): Y.Yamada
  • Journal Title: nt. Math. Res. Notices Volume: 17 Pages: 3823-3838
  • Peer Reviewed
• [Journal Article] Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves (2011)
  • Author(s): B.-F.Feng, J.Inoguchi, K.Kajiwara, K.Maruno and Y.Ohta
  • Journal Title: J. Phys. A : Math. Theor. Volume: 44 Issue: 39 Pages: 19-19
  • DOI: 10.1088/1751-8113/44/39/395201
  • Peer Reviewed
• [Journal Article] Discrete integrable systems and hodograph transformations arising from discrete plane curves (2011)
  • Author(s): B.-F.Feng, J.Inoguchi, K.Kajiwara, K.Maruno, Y.Ohta
  • Journal Title: J.Phys.A : Math.Theor. Volume: 44 Pages: 19-19
  • Peer Reviewed
• [Journal Article] A quantum isomonodromy equation and its application to N=2 SU(N)gauge theories (2011)
  • Author(s): Y.Yamada
  • Journal Title: J.Phys.A, Math.Theor. Volume: (Online版)
  • Peer Reviewed
• [Journal Article] Five-dimensional AGT conjecture and the deformed Virasoro algebra (2010)
  • Author(s): H.Awata and Y.Yamada
  • Journal Title: JHEP Volume: 01 Issue: 1
  • DOI: 10.1007/jhep01(2010)125
  • Peer Reviewed
• [Journal Article] Five-dimensional AGT relation and the deformed β-ensemble (2010)
  • Author(s): H.Awata and Y.Yamada
  • Journal Title: Prog. Theor. Phys. Volume: 124 Pages: 227-262
  • NAID: 10026986583
  • Peer Reviewed
• [Journal Article] Five-dimensional AGT relation and the deformed β-ensemble (2010)
  • Author(s): H.Awata Y.Yamada
  • Journal Title: Prog.Theor.Phys. Volume: 124 Pages: 227-262
  • NAID: 10026986583
  • Peer Reviewed
• [Journal Article] Lax formalism for q-Painleve equations with affine Weyl group symmetry of type E^{(1)}_n (2010)
  • Author(s): Y.Yamada
  • Journal Title: Int.Math.Res.Notices Volume: Online版
  • Peer Reviewed
• [Journal Article] Five-dimensional AGT conjecture and the deformed Virasoro algebra (2010)
  • Author(s): H.Awata, Y.Yamada
  • Journal Title: JHEP 01 Pages: 1-11
  • Peer Reviewed
• [Journal Article] Generalized energies and integrable Dn cellular automaton (2010)
  • Author(s): A.Kuniba, R.Sakamoto, Y.Yamada
  • Journal Title: Proceedings "Infinite Analysis 09 : New Trends in Quantum Integrable Systems"
  • Peer Reviewed
• [Journal Article] A Lax formalism for the elliptic difference Painleve equation (2009)
  • Author(s): Y.Yamada
  • Journal Title: SIGMA Volume: 5 Pages: 15-15
  • DOI: 10.3842/sigma.2009.042
  • Peer Reviewed
• [Journal Article] Generalized energies and integrable Dncellular automaton (2009)
  • Author(s): A.Kuniba, R.Sakamoto, and Y.Yamada
  • Journal Title: Infinite Analysis 09: New Trends in Quantum Integrable Systems Pages: 221-242
  • Peer Reviewed
• [Journal Article] Moduli spaces for linear differential equations and the Painleve equations (2009)
  • Author(s): M. van der Put and M.-H. Saito
  • Journal Title: Ann. Inst. Fourier (Grenoble) Volume: 59 Pages: 2611-2667
  • Peer Reviewed
• [Journal Article] A Lax formalism for the elliptic difference Painleve equation (2009)
  • Author(s): Y.Yamada
  • Journal Title: SIGMA 5 Pages: 15-15
  • Peer Reviewed
• [Presentation] Lax formalism for discrete Painleve equations (2013)
  • Author(s): Y.Yamada
  • Organizer: 8th IMACS International Conference on Nonlinear Evolution Equations (招待講演)
  • Place of Presentation: Georgia 大学.
  • Year and Date: 2013-03-25
• [Presentation] Symmetry of quantum Lax equations for the Painleve equations (2012)
  • Author(s): H.Nagoya and Y.Yamada
  • Organizer: 日本数学会無限可積分系セッション一般講演
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-18
• [Presentation] An interpolation problem related with q-E^(1)_8 Painleve equation (2011)
  • Author(s): Y.Yamada
  • Organizer: Infinit Analysis 11(招待講演)
  • Place of Presentation: 東大数理
  • Year and Date: 2011-07-28
• [Presentation] An interpolation problem related with q-E^{(1)}_8 Painleve equation (2011)
  • Author(s): Y.Yamada
  • Organizer: Infinite Analysis 11
  • Place of Presentation: 東京大学・数理科学研究科(東京都)(招待講演)
  • Year and Date: 2011-07-28
• [Presentation] 量子 Fuji-Suzuki-Tsuda 方程式とインスタントン分配関数 (2011)
  • Author(s): 山田泰彦
  • Organizer: 日本数学会無限可積分系セッション一般講演
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-22
• [Presentation] 量子Fuji-Suzuki-Tsuda方程式とインスタントン分配関数 (2011)
  • Author(s): 山田泰彦
  • 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(東京都)
  • Year and Date: 2011-03-22
• [Presentation] Quantum isomonodromy deformation and N=2 gauge theory (2011)
  • Author(s): 山田泰彦
  • Organizer: 物理セミナー(招待講演)
  • Place of Presentation: 静岡大理
  • Year and Date: 2011-02-04
• [Presentation] Quantum isomonodromy deformation and N=2 gauge theory (2011)
  • Author(s): 山田泰彦
  • Organizer: 物理セミナー
  • Place of Presentation: 静岡大学理学部(静岡県)(招待購演)
  • Year and Date: 2011-02-04
• [Presentation] Quantum isomonodromy deformation and N=2 gauge theory (2011)
  • Author(s): 山田泰彦
  • Organizer: 弦理論セミナー(招待講演)
  • Place of Presentation: 名大多元数理
  • Year and Date: 2011-01-24
• [Presentation] Quantum isomonodromy deformation and N=2 gauge theory (2011)
  • Author(s): 山田泰彦
  • Organizer: 弦理論セミナー
  • Place of Presentation: 名古屋大学(愛知県)(招待講演)
  • Year and Date: 2011-01-24
• [Presentation] AGT 予想が切り開いた新たな数学の世界 (2010)
  • Author(s): 山田泰彦
  • Organizer: 「量子可積分系の新展開」研究会(招待講演)
  • Place of Presentation: 静岡県・富士教育研修所
  • Year and Date: 2010-12-19
• [Presentation] AGT予想が切り開いた新たな数学の世界 (2010)
  • Author(s): 山田泰彦
  • Organizer: 「量子可積分系の新展開」研究会
  • Place of Presentation: (静岡県)富士教育研修所(招待講演)
  • Year and Date: 2010-12-19
• [Presentation] 共形場理論, モノドロミー保存変形と AGT 予想 (2010)
  • Author(s): 山田泰彦
  • Organizer: 表現論セミナー(招待講演)
  • Place of Presentation: 京大数理解析研究所
  • Year and Date: 2010-12-10
• [Presentation] 共形場理論,モノドロミー保存変形とAGT予想 (2010)
  • Author(s): 山田泰彦
  • Organizer: 表現論セミナー
  • Place of Presentation: 京都大学数理解析研究所(京都府)(招待講演)
  • Year and Date: 2010-12-10
• [Presentation] モノドロミー保存変形と N=2ゲージ理論 (2010)
  • Author(s): 山田泰彦
  • Organizer: 第4回日露ワーキングセミナー(招待講演)
  • Place of Presentation: 阪市大梅田文化交流センター.
  • Year and Date: 2010-11-30
• [Presentation] モノドロミー保存変形とN=2ゲージ理論 (2010)
  • Author(s): 山田泰彦
  • Organizer: 第4回日露ワーキングセミナー
  • Place of Presentation: 大阪市立大学梅田文化交流センター(大阪府)(招待講演)
  • Year and Date: 2010-11-30
• [Presentation] A quantum isomonodromy equation and its application to gauge theories (2010)
  • Author(s): 山田泰彦
  • Organizer: 神戸可積分系セミナー
  • Place of Presentation: 神戸大理
  • Year and Date: 2010-11-17
• [Presentation] A quantum isomonodromy equation and its application to gauge theories (2010)
  • Author(s): 山田泰彦
  • Organizer: 神戸可積分系セミナー
  • Place of Presentation: 神戸大学(兵庫県)
  • Year and Date: 2010-11-17
• [Presentation] N=2 ゲージ理論とモノドロミー保存変形 (2010)
  • Author(s): 山田泰彦
  • Organizer: 「重力・幾何・素粒子」研究会(招待講演)
  • Place of Presentation: 大阪市大理
  • Year and Date: 2010-09-29
• [Presentation] N=2ゲージ理論とモノドロミー保存変形 (2010)
  • Author(s): 山田泰彦
  • Organizer: 「重力・幾何・素粒子」研究会
  • Place of Presentation: 大阪市立大学(大阪府)(招待講演)
  • Year and Date: 2010-09-29
• [Presentation] E_n 型 q-Painleve 方程式の Lax 形式 (2010)
  • Author(s): 山田泰彦
  • Organizer: 日本数学会無限可積分系セッション一般講演
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] E_n型q-Painleve方程式のLax形式 (2010)
  • Author(s): 山田泰彦
  • Organizer: 日本数学会無限可積分系セッション
  • Place of Presentation: 名古屋大学(愛知県)
  • Year and Date: 2010-09-24
• [Presentation] CFT,モノドロミー保存変形, Nekrasov 関数 (2010)
  • Author(s): 山田泰彦
  • Organizer: BC 系と AGT 予想の周辺」研究会(招待講演)
  • Place of Presentation: 東大数理
  • Year and Date: 2010-09-13
• [Presentation] CFT,モノドロミー保存変形,Nekrasov関数 (2010)
  • Author(s): 山田泰彦
  • Organizer: 「BC系とAGT予想の周辺」研究会
  • Place of Presentation: 東京大学(東京都)(招待講演)
  • Year and Date: 2010-09-13
• [Presentation] q-Virasoro 代数とインスタントン分配関数 (2010)
  • Author(s): 粟田英資・山田泰彦
  • Organizer: 日本数学会無限可積分系セッション一般講演
  • Place of Presentation: 慶応大学
  • Year and Date: 2010-03-27
• [Presentation] q-Virasoro 代数とインスタントン分配関数 (2010)
  • Author(s): 粟田英資・山田泰彦
  • Organizer: 日本数学会(無限可積分系セッション・一般講演)
  • Place of Presentation: 慶應義塾大学(東京都)
  • Year and Date: 2010-03-27
• [Presentation] 離散 Painleve 方程式の Lax 形式 (2009)
  • Author(s): 山田泰彦
  • Organizer: 可積分系数理とその応用
  • Place of Presentation: はこだて未来大学
  • Year and Date: 2009-08-12
• [Presentation] 離散 Painleve 方程式の Lax 形式 (2009)
  • Author(s): 山田泰彦
  • Organizer: 可積分系数理とその応用
  • Place of Presentation: はこだて未来大学(北海道)
  • Year and Date: 2009-08-12
• [Presentation] Lax formalism for the elliptic difference Painleve equation (2009)
  • Author(s): Y.Yamada
  • Organizer: Geometric aspects of discrete and ultra - discrete integrable systems
  • Place of Presentation: Glasgow 大学
  • Year and Date: 2009-04-03
• [Presentation] Lax formalism for the elliptic difference Painleve equation (2009)
  • Author(s): 山田泰彦
  • Organizer: Geometric aspects of discrete and ultra-discrete in tegrable systems
  • Place of Presentation: Glasgow 大学(英国)
  • Year and Date: 2009-04-03
• [Presentation] Lax formalism for discrete Painlev'e equations
  • Author(s): Y.Yamada
  • Organizer: 8th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena
  • Place of Presentation: Univ. Georgia
  • Invited
• [Presentation] Symmetry of quantum Lax equations for the Painlev'e equations
  • Author(s): H.Nagoya, Y.Yamada
  • Organizer: 日本数学会2012年度秋季総合分科会無限可積分系セッション
  • Place of Presentation: 九州大学
• [Remarks] 山田泰彦
  • URL: http://www.math.kobe-u.ac.jp/HOME/yamaday/yamada.html