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2006 Fiscal Year Final Research Report Summary

Mathematical theory of dynamical systems of Painleve type

Research Project

Project/Area Number 15340057
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKYUSHU UNIVERCITY

Principal Investigator

KAJIWARA Kenji  Kyushu University, Faculty of Mathematics, Associate Professor, 大学院数理学研究院, 助教授 (40268115)

Co-Investigator(Kenkyū-buntansha) NOUMI Masatoshi  Kobe University, Department of Mathematics, Professor, 大学院自然科学研究科, 教授 (80164672)
YAMADA Yasuhiko  Kobe University, Department of Mathematics, Professor, 理学部, 教授 (00202383)
IWASAKI Katsunori  Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (00176538)
Project Period (FY) 2003 – 2006
KeywordsPainleve equations / discrete Painleve equations / τ functions / hypergeometric functions / elliptic curve / affine Weyl group / integrable systems / chaotic systems
Research Abstract

1. We have extended the theory of symmetric form for q-Painleve IV equation with (A_2+A_1)^<(1)> affine Weyl group symmetry to formulate the q-KP hierarchy. By the similarity reduction we have constructed the hierarchy of discrete systems with (A_<m-1>+A_1)^<(1)> affine Weyl group symmetry, and further, that with (A_<m-1>+A_<n-1>)^<(1)> affine Weyl group symmetry.
2. We have presented a formulation of the elliptic Painleve equation and its generalizations, the former of which is located on the top of all the Painleve systems of second order. Namely, we have formulated the time evolution and the Backlund transformations as Cremona transformations on the configuration space of generic points in the complex projective space, and given their realization as birational transformations parametrized by the theta functions on the level of the τ functions. We have also formulated the time evolution as the addition formula on the moving pencil of plane cubic curves, and clarified the geometric mea … More ning of the Hamilitonians of the Painleve differential equations.
3. Applying the above formulation, we have constructed the simplest hypergeometric solutions to the elliptic Painleve equation and all the q-Painleve equations, and completed the list of coalscent diagram of hypergeometric functions, starting from the elliptic hypergeometric function _<10>E_9 to the q-Airy function.
4. Combining the algebro-geometric formulation of the Painleve VI equation and the ergodic theory of birational mapping on the algebraic surface by using the Riemann-Hilbert correspondence, we have shown that the nonlinear monodromy of the Painleve VI equation is chaotic along almost all the loops.
5. We have shown that the entries of Hankel determinant formula for the solutions of the Painleve differential equations arise as coefficients of asymptotic expansion of the ratio of solutions to the auxiliary linear problem, and that this phenomenon originates from the structure of the KP hierarchy.
6. Applying the above results we have discussed some new extentions or new solutions to the discrete and ultradiscrete Toda equation. Less

  • Research Products

    (52 results)

All 2007 2006 2005 2004 2003 Other

All Journal Article (50 results) Book (2 results)

  • [Journal Article] An ergodic study of Painleve VI2007

    • Author(s)
      K.Iwasaki et al.
    • Journal Title

      Mathematische Annalen (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Dynamics of the sixth Painleve equation2007

    • Author(s)
      M.Inaba et al.
    • Journal Title

      Seminaires et Congres (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Point configurations, Cremona transformations and the elliptic difference Painleve equation2007

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Seminaires et Congres (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equation of type A_4^<(1)>2007

    • Author(s)
      T.Hamamoto et al.
    • Journal Title

      Journal of Physics A : Mathematical and Theor et al. (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Chaos in the sixth Painleve equation2007

    • Author(s)
      K.Iwasaki et al.
    • Journal Title

      RIMS Kokyuroku Bessatsu (In press)

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An ergodic study of Painleve VI2007

    • Author(s)
      K.Iwasaki et al.
    • Journal Title

      Mathematische Annalen (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dynamics of the sixth Painleve equation2007

    • Author(s)
      M.Inaba et al.
    • Journal Title

      Seminaires et Congres (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Point configurations, Cremona transformations and the elliptic difference Painleve equation2007

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Seminaires et Congres (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equation of type A_4^<(1)>2007

    • Author(s)
      T.Hamamoto et al.
    • Journal Title

      Journal of Physics A : Mathematical and Theoretical (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Chaos in the sixth Painleve equation2007

    • Author(s)
      K.Iwasaki et al.
    • Journal Title

      RIMS Kokyuroku Bessatsu (in press)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection2006

    • Author(s)
      A.Kuniba et al.
    • Journal Title

      Nuclear Physics B 744

      Pages: 299-327

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equations of type VI, Part I2006

    • Author(s)
      M.Inaba et al.
    • Journal Title

      Publication of Research Institute for Mathematical Sciences 42

      Pages: 987-1089

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painleve equations of type VI, Part II2006

    • Author(s)
      M.Inaba et al.
    • Journal Title

      Advanced Studies in Pure Mathematics 45

      Pages: 387-432

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Generating function associated with the Hankel determinant formula for the solutions of the Painleve IV equation2006

    • Author(s)
      N.Joshi et al.
    • Journal Title

      Funkcialaj Ekvacioj 49

      Pages: 451-468

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equation of type (A_1+A_')^<(1)>2006

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      International Mathematics Research Notices 2006

      Pages: Article ID 84619

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] PainleveおよびToda方程式の解のHankel determinant formulaと補助線形問題2006

    • Author(s)
      梶原 健司
    • Journal Title

      九州大学応用力学研究所研究集会報告 17ME-S2

      Pages: Article No. 41

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equation of type (A_1+A_1')^<(1)>2006

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      International Mathematics Research Notices 2006, Article ID

      Pages: 84619

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hankel determinant formulas for the solutions of the Painleve and Toda equations, and their auxiliary linear problems (in Japanese)2006

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Reports on RIAM Symposium, Kyushu University 17ME-S2, Article No.41

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Cubic pencils and Painleve Hamiltonians2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Funkcialaj Ekvacioj 48

      Pages: 147-160

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Construction of hypergeometric solutions to the q-Painleve equations2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      International Mathematics Research Notices 2005

      Pages: 1441-1463

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Generating function related to the Okamoto polynomials for the Painleve IV equation2005

    • Author(s)
      H.Goto et al.
    • Journal Title

      Bulletin of the Australian Mathematical Society 71

      Pages: 517-526

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Soliton solutions for the non-autonomous discrete-time Toda lattice equation2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Journal of Physics A : Mathematical and General 38

      Pages: 6363-6370

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Box-ball system with reflecting end2005

    • Author(s)
      A.Kuniba et al.
    • Journal Title

      Journal of Nonlinear Mathematical Physics 12

      Pages: 475-507

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] q-Painleve IIおよびV方程式の超幾何解2005

    • Author(s)
      濱本 太郎 他
    • Journal Title

      九州大学応用力学研究所研究集会報告 16ME-S1

      Pages: 260-266

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] q-Painleve方程式の超幾何解2005

    • Author(s)
      梶原 健司 他
    • Journal Title

      京都大学数理解析研究所講究録 1422

      Pages: 77-98

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A geometric description of the elliptic Painleve equation2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Rokko Lectures in Mathematics 18

      Pages: 43-48

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Soliton solutions for the non-autonomous discrete-time Toda lattice equation2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Journal of Physics A : Mathematical and General. 38

      Pages: 6363-6370

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve II and IV equations2005

    • Author(s)
      T.Hamamoto et al.
    • Journal Title

      Reports of RIAM Symposium, Kyushu University 16ME-S1

      Pages: 260-266

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equations2005

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      RIMS Kokyuroku, Kyoto University 1422

      Pages: 77-98

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Backlund transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence2004

    • Author(s)
      M.Inaba et al.
    • Journal Title

      International Mathematics Research Notices 2004

      Pages: 1-30

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions2004

    • Author(s)
      M.Noumi et al.
    • Journal Title

      Advanced Studies in Pure Mathematics 40

      Pages: 371-442

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Generating function associated with the determinant formula for solutions of the Painleve II equation2004

    • Author(s)
      N.Joshi et al.
    • Journal Title

      Asterisque 274

      Pages: 67-78

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Symmetries in Painleve equations2004

    • Author(s)
      M.Noumi et al.
    • Journal Title

      Sugaku Expositions 17

      Pages: 203-218

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Hypergeometric solutions to the q-Painleve equations2004

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      International Mathematics Research Notices 2004

      Pages: 2497-2521

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Cremona変換と楕円差分Painleve方程式2004

    • Author(s)
      梶原 健司 他
    • Journal Title

      京都大学数理解析研究講究録 1400

      Pages: 197-263

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Okamoto多項式のHankel行列式表示に付随する母函数2004

    • Author(s)
      後藤弘道他
    • Journal Title

      九州大学応用力学研究所研究集会報告 15ME-S3

      Pages: 220-226

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Generating function associated with the Hankel determinant formula for the Okamoto polynomials (in Japanese)2004

    • Author(s)
      H.Goto et al.
    • Journal Title

      Reports of RIAM Symposium, Kyushu University 15ME-S3

      Pages: 220-226

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On a q-Painleve III equation. I : derivation, symmetry and Riccati type solutions2003

    • Author(s)
      K.Kajiwara
    • Journal Title

      Journal of Nonlinear Mathematical Physics 10

      Pages: 86-102

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] q-Painleve systems arising from q-KP hierarchy2003

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Letters in Mathematical Physics 62

      Pages: 259-268

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On a q-Painleve III equation. II : rational solutions2003

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Journal of Nonlinear Mathematical Physics 10

      Pages: 282-303

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] _<10>E_9 solution to the elliptic Painleve equation2003

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      Journal of Physics A : Mathematical and General 36

      Pages: L263-L272

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Isolated singularity, Witten's Laplacian, and duality for twisted de Rham cohomology2003

    • Author(s)
      K.Iwasaki
    • Journal Title

      Communications in Partial Differential Equations 28

      Pages: 61-82

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Cohomology groups for recurrence relations and contiguity relations of hypergeometric systems2003

    • Author(s)
      K.Iwasaki
    • Journal Title

      Journal of the Mathematical Society of Japan 55

      Pages: 289-321

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] An area-preserving action of the modular group on cubic surfaces and the Painleve VI equation2003

    • Author(s)
      K.Iwasaki
    • Journal Title

      Communications in Mathematical Physics 242

      Pages: 185-219

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Geometric crystal and tropical R for D_n^<(1)>2003

    • Author(s)
      A.Kuniba et al.
    • Journal Title

      International Mathematics Research Notices 2003

      Pages: 2565-2620

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Multiple elliptic hypergeometric series. An approach from the Cauchy determinant2003

    • Author(s)
      Y.Kajihara et al.
    • Journal Title

      Indagtiones Mathematicae 14

      Pages: 395-421

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] 離散パンルヴェ方程式の理論 : 現状と展望2003

    • Author(s)
      梶原 健司
    • Journal Title

      九州大学応用力学研究所研究集会報告 14ME-S7

      Pages: 144-154

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Witten's Laplacian and duality for twisted de Rham cohomology2003

    • Author(s)
      K.Iwasaki
    • Journal Title

      Communications in Partial Differential Equations 28

      Pages: 61-82

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Theory of discrete Painleve equations : state of art and perspectives (in Japanese)2003

    • Author(s)
      K.Kajiwara
    • Journal Title

      Reports of RIAM Symposium, Kyushu University 14ME-S7

      Pages: 144-154

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Cremona transformations and the elliptic difference Painleve equation (in Japanese)

    • Author(s)
      K.Kajiwara et al.
    • Journal Title

      RIMS Kokyuroku, Kyoto University 1400

      Pages: 197-263

    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] 共形場理論入門2005

    • Author(s)
      山田 泰彦
    • Total Pages
      265
    • Publisher
      培風館
    • Description
      「研究成果報告書概要(和文)」より
  • [Book] Painleve equations through symmetry2004

    • Author(s)
      M.Noumi
    • Total Pages
      156
    • Publisher
      American Mathematical Society
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2008-05-27  

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