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Synthetic Research on nonlinear complete integrable systems and combinatorics

Research Project

Project/Area Number 09304013
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field 解析学
Research InstitutionThe University of Tokyo

Principal Investigator

OKAMOTO Kazuo  the University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (40011720)

Co-Investigator(Kenkyū-buntansha) SATSUMA Junkichi  The Univ. of Tokyo, Graduate School of Math. Sci., Prof., 大学院・数理科学研究科, 教授 (70093242)
KATSURA Toshiyuki  The Univ. of Tokyo, Graduate School of Math. Sci., Prof., 大学院・数理科学研究科, 教授 (40108444)
KOHNO Toshitake  the Univ. of Tokyo, Graduate School of Math. Sci., Prof., 大学院・数理科学研究科, 教授 (80144111)
TAKANO Kyo-ichi  Kobe Univ., Faculty of Sci. Department of Math., Prof., 理学部, 教授 (10011678)
MATSUO Atsushi  The Univ. of Tokyo, Graduate School of Math. Sci., Assoc. Prof., 大学院・数理科学研究科, 助教授 (20238968)
山本 昌宏  東京大学, 大学院・数理科学研究科, 助教授 (50182647)
小木曽 啓示  東京大学, 大学院・数理科学研究科, 助教授 (40224133)
河東 泰之  東京大学, 大学院・数理科学研究科, 助教授 (90214684)
川又 雄二郎  東京大学, 大学院・数理科学研究科, 教授 (90126037)
Project Period (FY) 1997 – 2000
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥24,700,000 (Direct Cost: ¥24,700,000)
Fiscal Year 2000: ¥6,600,000 (Direct Cost: ¥6,600,000)
Fiscal Year 1999: ¥6,100,000 (Direct Cost: ¥6,100,000)
Fiscal Year 1998: ¥5,500,000 (Direct Cost: ¥5,500,000)
Fiscal Year 1997: ¥6,500,000 (Direct Cost: ¥6,500,000)
KeywordsIntegrable system / Nonliear CIS / Painleve equations / Garnier systems / Bilinear Forms / Affine Weyl groups / Backlund Trqnsformation / symmetries / 双称形型式 / 対称性 / 非線完全積分可能系 / 双線形形式 / 完全積分可能素 / 組み合わせ論 / 初期値空間 / 多変数特殊関数 / 代数解 / 楕円曲線 / パンルヴェ階層 / 非線形完全積分可能系 / 代数幾何学符号 / シューア関数 / KP方程式 / 離散化と超離散化 / ソリトンセルオートマトン
Research Abstract

The present project has been supported by Grant-in-Aid for Scientific Research from 1997 to 2000. The aim of our pursuite is double: studies on nonlinear completely integrable systems from viewpoints of combinatorics, and various approaches to theory of combinatorics in terms of completely integrable systems. In particular, our main subjects of this research project are listed as follows:
(a) theoretical investigation on nonlinear completely integrable systems (CIS, in short) ,
(b) application of the theory to various domains in mathematical sciences,
(c) symmmeries of completely integrable systems,
The Painleve equations are surely one of the most important examples of nonlinear inteegrable systems. The head investigator of the project has published an article on the Painleve equations, cited at the top of references of this reprt; the former half of this paper is devoted to an survey of results on the Painleve equations and recent results on the Garnier sytems are given in the latter hal … More f. Corresponding to each of the subjects mentioned above, we make a summary of results, obtained during promotion of the present project.
(a) A geometrical interpretation is given to the space of initial conditions, which had been constructed by head investigator for the Painleve equations. By the use of this viewpoint, a geometrical characterization and classification are established for integrable systems, not only of continuous type but also of discrete one.
(b) A new structure of hierarchy is discovered for the Garnier systems, which are known as extension of the Painleve equations to several variable cases. We persuade that the former admits the general solution of the latter, as special solitions.
(c) Another structure of hierarchy for the Painleve equations has been established mathematically; in this case, extendes equations have the same group of symmetries as that of the Painleve equations. We can insist without hesitation that our present investigation on nonlinear completely integrable systems is giving fruitful results to theories and applications of the subjects, and we convince ourselves of development of researches on this domain.
The investigators of this research project have continued their studies on integrable systems and announced their own results obtained during four years, 1997-2000, in various occasins. They have published some of results in journals. Less

Report

(5 results)
  • 2001 Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (32 results)

All Other

All Publications (32 results)

  • [Publications] OKAMOTO Kazuo: "The Hamiltonicus associated with Painleve equations""The Painleve propertyione century later" CRM Series in Math. Phys. 735-787 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] OKAMOTO Kazuo: "Special polynomials associated with the rational solutions and the Hirota bilinear relations of the 2^<nd> & 4^<th> Painleve equations"Nagoya Math. J.. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] KOHNO Toshitake: "Vassilier invariants of braids and iterated integrals"Adv. Stud. Pure. Math.. 27. 157-168 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] KATSURA Toshiyuki: "On a stratification of the moduli of K3 surfaces"J. Eur. Math. Soc.. 2. 259-290 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] SATSUMA Junkichi: "Nonautonomous discrete integrable systems"Chaos. Solitons. Fractals. 11. 121-135 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] MATSUO Atsushi: "The automorphism group of the Haring code vertex operator"J. Algebra. 228. 204-226 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] OKAMOTO Kazuo: "The Hamiltnians associated with Painleve equations"CRM Series in Math. Phys, "The Painleve Property: One Century later". 735-737 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] OKAMOTO Kazuo: "Special Polynomials associated with the rational slutions and Hirota bilinear relation of 2^<nd> and 4^<th> Panleve equations"Nagoya Math. J.. 159. 179-200 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] KOHNO Toshitake: "Vassiliev invariants of braids and iterated integrals"Adv. Stud. Pure Math.. 27. 157-168 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] KATSURA Toshiyuki: "On a stratification of the moduli of K3 surfaces"J. Eur. Math. Soc.. 2. 259-290 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] SATSUMA Junkichi: "Nonautonomous discrete integrable systems"Chaos, Solitons, Fractals. 11. 121-135 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] MATSUO Atsusi: "The automorphism group of the Hamming code vertex operator"J. Algebra. 228. 204-226 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] TAKANO Kyoichi: "Defining manifolds for the Painleve equations""Toward the exact WKB analysys of differential equations, linear or non-linear" Kyoto. 261-269 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] OKAMOTO Kazuo: "The Hamiltonians associated with the Painleve equations""The Painleve property i one century later"CRM Series in Math.Phys.. 735-787 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] OKAMOTO Kazuo: "Special polynomials associated with the rational solutions and the Hirota bilinear relations of 2^<nd> & 4^<th> Painleve equations"Nagoya Math.J.. 159. 179-200 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] KOHNO Toshitake: "Vassiliev invariants of braids and iterated integrals"Adv.Stud.Pure Math.. 27. 157-168 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] KATSURA Toshiyuki: "On a stratification of the moduli of K3 surfaces"J.Eur.Math.Soc.. 2. 259-290 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] SATSUMA Junkichi: "Nonautonomous discrete integrable systems"Chaos Solitons Fractals. 11. 121-135 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] MATSUO Atsushi: "The antomorphism group of the Hamming code vertex operator"J.Algebra. 228. 204-226 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] OKAMOTO Kazuo: "The Hamiltoniaus associated to the Painleve equations"The Painleve Property : one century later, Springer. 735-788 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] KOHNO Toshitake: "Vassiliev invariants of braids and iterated integrals"Advanced Studies in pure Math.. (発表予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] KATSURA Toshiyuki: "On the distribution of linear codes around the [7,4,3]-Hamming code"Max-Planck-Institut fiir Math. Preprint Series. 3. 1-8 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] YAMAMOTO Masahiro: "Uniqueness and stability in multidimensional hyperbolic inverse problems"J. Math. Pures. Appl.. 78. 65-98 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] TAKANO Kyo-ichi (共著): "On some Hamiltonian structures of Painleve systems, II"J. Math. Soc. JAPAN. 51. 843-866 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] MATSUO Atsushi (共著): "The automorphism group of the Hamming code Vertex algebra"Jaursal of Algebra. (発表予定). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.OKAMOTO et al.: "Special Polynonisals associated with the Painlcve equations." Proceedings on “Integrable systems and Algebraic Geometry". 349-372 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Kazuo OKAMOTO: "The Hamiltonians associated with the Painleve equations" Proc.of the Summer School at Cargise. (発表予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Kazuo OKAMOTO: "Polynoial Hamiltomian structure for the A_3-system" Kumamoto J.Math.10. 31-43 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yujiro KAWAMATA: "On the cone of divisors of Calabi-Yau fibre spaces" Internat.J.Math. 8. 665-687 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Toshitaka KOHNO: "Monodrony of conformal blocks," Proc.Appl.Math.Warkshops KAiST. 179-186 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] Keiji OGUiSO: "On Vorontsov's theorem on K3 surfaces with nonsymplectic group actions" Proc.AMS. (発表予定). (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Yasuyuki KAWAHiGASHI: "Orbifold subfactors from Hecke algebras II" Comm.Math.Phys.(発表予定). (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1997-04-01   Modified: 2016-04-21  

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