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Mathematics in ultradiscrete integrable systems

Research Project

Project/Area Number 12440046
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionThe University of Tokyo

Principal Investigator

TOKIHIRO Tetsuji  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (10163966)

Co-Investigator(Kenkyū-buntansha) KUNIBA Atsuo  The University of Tokyo, Graduate School of Arts and Sciences, Associate Professor, 大学院・総合文化研究科, 助教授 (70211886)
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)
SATSUMA Junkichi  The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70093242)
HIKAMI Kazuo  The University of Tokyo, Graduate School of Sciences, Assistant, 大学院・理学系研究科, 助手 (60262151)
TAKEBE Takashi  Ochanomizu Woman University, Faculty of Sciences, Associate Professor, 理学部, 助教授 (60240727)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥13,600,000 (Direct Cost: ¥13,600,000)
Fiscal Year 2003: ¥2,400,000 (Direct Cost: ¥2,400,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2001: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2000: ¥5,400,000 (Direct Cost: ¥5,400,000)
Keywordsultradiscrete systems / integrable systems / cellular automaton / integrable lattice models / crystal theory / tropical / 可解格子模型 / 箱玉系 / 超散系
Research Abstract

1.We have determined asymptotic behaviour of the fundamental cycle of periodic box-ball systems (PBBSs) of type (A_1^<(1)>)in the limit of the system size N, N → ∞. Due to the integrability of the PBBS, their orbits are located in a small region of the phase space the volume of which is proportional to exp[N]. We have proved that the maximum fundamental cycle is of order of exp[N ^<1/2>], but that almost all the fundamental cycle is less than exp[(log N)^2].
2.We constructed the geometric crystal, which was proposed by Berenstein and Kazhdan, for the affine Lie algebra Using the realization by matrices with spectral parameters, the birational transformation (tropical R matrix), which intertwines the tensor product of the geometric crystals, is obtained. We also proved its uniqueness.
The tropical R matrix is comutable with the geometric Kashiwara operators and satisfies the Yang-Baxter relation.
It does not preserve the inverse operation to addition formulae and produce piecewise linear equations for ultradiscrete systems.
3.We have constructed the combinatorial R matrices for B_n^<(1)>, D_n^<(1)>, A_<2n>^<(2)> and D_<n+1>^<(2)>. The soliton scattering in the lattices with Boltzmann weight given by these R matrices are expressed as the action by Wyle group. We also constructed an analogue to the inverse scattering methods.

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (34 results)

All Other

All Publications (34 results)

  • [Publications] T.Kimijima, T.Tokihiro: "Initial value problem of discrete periodic Toda equation and its ultradiscretization"Inverse Problems. 18. 1705-1732 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Yura, T.Tokihiro: "On a periodic soliton cellular automaton"J.Phys.A Math.Gen.. 35. 3787-3801 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] G.Hatayama, A.Kuniba, T.Takagi: "Scattering rules in soliton cellular automata associated with crystal bases"Contemporary Mathematics. 297. 151-182 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Kuniba, M.Okado: "Bethe equation at q=0, Mobius inversion formula, and weight multiplicities : II. X_n case"J.Alg.. 251. 577-618 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Takebe: "A note on modified KP hierarchy and its (yet another) dispersionless limit."Lett.Math.Phys.. 59. 157-172 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Kimijima, T.Tokihiro: "Initial value problem of discrete periodic Toda equation and its ultradiscretization"Inverse Problems. 18. 1705-1732 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] F.Yura, T.Tokihiro: "On a periodic soliton cellular automaton"J.Phys.A : Math.Gen.. 35. 3787-3801 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] G.Hatayama, A.Kuniba, T.Takagi: "Scattering rules in soliton cellular automata associated with crystal bases"Contemporary Mathematics. 297. 151-182 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] A.Kuniba, M.Okado: "Bethe equation at q=0, Mobius inversion formula, and weight multiplicities : II.X_n case"J.Alg.. 251. 577-618 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] T.Takebe: "A note on modified KP hierarchy and its (yet another) dispersionless limit."Lett.Math.Phys.. 59. 157-172 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] D.Yoshihara: "Fundamental cycle of a periodic box-ball system"J.Phys.A : Math.Gen.. 36. 99-121 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Hikami: "Torus knot and minimal model"Phys.Lett.. B575. 3033-3048 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Isojima: "Spider-web solutions of the coupled KP equation"J.Phys.A : Math.Gen.. 36. 9533-9552 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] A.Kuniba: "Geometric crystal and tropical R for D_n^<(1)>"Int.Math.Research Notices. 48. 2565-2620 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] J.Mada: "Asymptotoic behavior of fundamental cycle of periodic box-ball systems"J.Phys.A : Math.Gen.. 36. 7251-7268 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Takebe: "Trigonometric degeneration and Orbifold Wess-Zumino-Witten model I"J.Modern Phys.A. (in press).

    • Related Report
      2003 Annual Research Report
  • [Publications] T.Kimijima: "Initial value problem of discrete periodic Toda equation and its ultradiscretization"Inverse Problems. 18. 1705-1732 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] F.Yura: "On a periodic soliton cellular automaton"J.Phys. A : Math. Gen.. 35. 3787-3801 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] G.Hatayama: "Scattering rules in soliton cellular associated with crystal bases"Contemporary Mathematics. 297. 151-182 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Kuniba: "Bethe equation at q=0, Mobius inversion formula, and weight multiplicities : II. X_n case"J.Alg.. 251. 577-618 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Takebe: "A note on modified KP hierarchy and its (yet another) dispersionless limit"Lett.Math.Phys.. 59. 157-172 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.D.Lorenzo: "Quasi-classical descendeants of disordered vertex models with boundaries"Nucl.Phys. B. 644. 409-432 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Nobe: "From cellular automaton to difference equation : a general transformation method which preserves time evolution patterns"J. Phys. A : Math. Gen. 34. L371-L379 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Nagai: "Conserved quantities of box ball system"Glasgow Math. J.. 43A. 91-97 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] G.Hatayama: "Factorization of combinatorial $R$ matrices and associated cellular automata"J. Stat. Phys.. 102. 843-863 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] G.Hatayama: "Simple Algorithm for Factorized Dynamics of ${\mathfrakg}_n$-Automaton"J. Phys. A : Math. Gen. 34. 10697-10705 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Hikami: "Symmetry of cellular automaton"J. Phys. A : Math. Gen. 34. 2467-2467 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Hikami: "The Baxter equation for quantum discrete Boussinesq equation"Nucl. Phys.. B604. 580-602 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] A.Nobe: "Stable difference equations associated with elementary cellular automata"Japan Journal of Applied Mathematics. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] G.Kuroki: "Wess-Zumino-Witten model on elliptic curves at the critical level"Journal of Physics A. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Kuniba: "Bethe equation at $q=0$, M\"obius inversion formula, and weight multiplicities : I.$sl(2)$ case"Prog.in Math.. 191. 185-216 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] G.Hatayama: "The $A^<^>{(1)}_M$ automata related to crystals of symmetric tensors"J.Math.Phys.. 42. 274-308 (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Hikami: "Supersymmetric extension of the integrable box-ball system"J.Phys.A : Math.Gen.. 33. 4081-4094 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Hikami: "Supersymmetric Polychronakos spin chain.Motifs, distribution functions, and characters"Nucl.Phys.. B566. 511-528 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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