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2001 Fiscal Year Final Research Report Summary

Complete Integrability and Structure of Solutions for Nonlinear Discrete Equations

Research Project

Project/Area Number 10440057
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

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)

Co-Investigator(Kenkyū-buntansha) YANO Koichi  Aoyama Gakuin University, College of Science and Engineering, Professor, 理工学部, 教授 (60114691)
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)
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)
OHTA Yasuhiro  Hiroshima University, Graduate School of Engineering, Assistant Professor, 大学院・工学研究科, 助手 (10213745)
Project Period (FY) 1998 – 2001
KeywordsSoliton / Integrable System / Difference Equation / Ultra-discretization / Singularity Confinement / Painleve Equation
Research Abstract

For the purpose of clarifying the structure of "singularity confinement", extending the concept to ultra-discrete system, and revealing algebraic and geometrical structure of discrete integrable systems, we have obtained the following results.
(1) We have shown the algebraic structure of solution method for nonautonomous discrete integrable systems and constructed solutions expressed by special functions for the systems. We have also proposed new nonlinear relations among the solutions.
(2) For some discrete Painleve equations, we have clarified their self-duality and the structure of nonlinear relations among the solutions.
(3) We have obtained an ultra-discrete version of the Toda molecule equation and the explicit expression of its solutions. We have also studied its algebraic structure and symmetry, and discussed a possibility of applying the system to the field of mathematical engineering.
(4) By considering a nonlinear integro-differential equation as a difference-differential equation, we have studied its integrable structure. We have also studied the structure of solutions for a discrete analogue of coupled nonlinear wave equations.
(5) We have presented a geometrical approach to the equation which satisfies the singularity confinement criterion but exhibits chaotic behaviour. We have also given a method to caluculate the algebraic entropy by using the theory of intersection numbers.
(6) We have studied geometrical structure of the discrete Painleve equations and shown the relation between the solution space of Toda lattice equation and the solutions of discrete Painleve equations.

  • Research Products

    (63 results)

All Other

All Publications (63 results)

  • [Publications] A.Nagai: "The Toda Molecule Equation and the ε-Algorithms"Mathematics of Computation. 67. 1565-1575 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Nagai: "Ultra-discrete Toda Molecule Equation"Phys. Lett. A. 244. 383-388 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Ramani: "Self-Duality and Schlesinger Chains for the Asymmetric d-PII and 8-PIII Equations"Comm. Math. Phys.. 192. 67-76 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara: "Determinant Structure of the Rational Solutions for the Painleve IV Equation"J. Phys. A : Math. Gen.. 31. 2431-2446 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] R.Willox: "Quadilaferal Lattices and Eigeufunction Potentials for N-Component ICP Hierarchies"Phys. Lett. A. 252. 163-172 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.M.Tamihmani: "Nonlinear Integro-differential Equations as Discrete Systems"Inverse Problem. 15. 787 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.J.Ablowits: "On Integrability and Chaos in Discrete systems"Chaos. Solitons and Fractals. 11. 159-169 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] J.J.C.Nimmo: "Applications of parboux Transformations to the Self-Dual Yang-Miles Equations"Theor. Mathe. Phys.. 122. 239-246 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Rawani: "Discrete Integrable Systems from Continuous Painleve Equations through Limiting Procedures"Nonlinearity. 13. 1073-1086 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Ramani: "Lineanzable Mappings and the Low-Growth Criterion"J. Phys. A : Math. Gen.. 33. L287-L292 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohta: "Determinant and Plattian Solutions for Discrete Soliton Equations"CRM Proceedings and Lecture Notes. 25. 339-345 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Nagai: "Conserved Quantities of Box and Boll System"Glasgow Math. J.. 43A. 91-97 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Tokihiro: "Proof of Solitonical Nature of Box and Ball Systems by means of Inverse Ultra-discretization"Inverse Problem. 15. 1639-1662 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.J.Ablowitz: "On Discretization of the Vector Nonlinear Schrodinger Equations"Phys. Lett. A. 253. 287-304 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohta: "Singularity Confinement and Algebraic Entropy : the Case of the Discrete Painleve Equation"Phys. Lett. A. 262. 152-157 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.M.Tamizhmani: "Integrability Criteria for Differential-Difference Systems : A Comparison of Singularity Confinement and Low-growth Reguirement"J. Phys. A : Math. Gen.. 32. 6679-6685 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] R.Willox: "Nonautonomous Discrete Integrable Systems"Chaos, Solitons and Fractals. 11. 121-135 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohta: "Plattian Solution for Coupled A'screte Nonlinear Shrodinger Equation"Chaos. Solitons and Fractals. 11. 91-95 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] G.Hatayama: "The A^<(1)>_M Automata Related to Crystals of Symmetric Tensors"J. Math. Phys.. 42. 274-308 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] F.W.Nijhoff: "On Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation"Stud. Appl. Math.. 106. 261-314 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Ramani: "A Geometrical Description of the Painleve VI and V Equations"Comm. Math. Phys.. 217. 315-329 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Ramani: "The Q Painleve V Equation and its Geometricol Description"J. Phys. A : Math. Gen.. 34. 2505-2513 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohta: "A Bilinear Approach to a Plottian Self-Dual Yang-Miles Equation"Goasgow Math. J.. 43A. 99-108 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Lafortune: "Blending Two Discrete Integrability Criteria : Singularity Confinement and Algebraic Entropy"CRM Proceedings and Lecture Notes. 29. 299-311 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Y.Ohta: "An Attine Wyle Group Approach to the Eight-Parameter Discrete Painleve Equation"J. Phys. A : Math. Gen.. 34. 10523-10532 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] S.Kakei: "A Differential-Difference System Related to Toroidal Lie Algebra"J. Phys. A : Math. Gen.. 34. 10585-10592 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kajiwara: "Determinant Formula for the Toda and Discrete Toda Equations"Funkcial. Ekvac.. 44. 291-307 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Takenawa: "A Geometrical Approach to Singularity Continenent and Algebraic Entropy"J. Phys. A : Math. Gen.. 34. L95-L102 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Takenawa: "Algebraic Entropy and the Space of Initial Values for Discrete Dynamical Systems"J. Phys. A : Math. Gen.. 34. 10533-10545 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Takenawa: "Discrete Dynamical Systems Associated with Root Systems of Indefinite Type"Commun. Math.. 224. 657-681 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] J.Satsuma: "Scattering"Academic Press. 9 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Nagai: "The Toda Molecule Equation and the ε-Algorithms"Mathematics of Computation. Vol. 67. 1565-1575 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Nagai: "Ultra-discrete Toda Molecule Equation"Phys. Letter. Vol. A244. 383-388 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Ramani: "Self-Duality and Schlesinger Chains for the Asymmetric d-PII and q-PIII Equations"Comm, Math. Phys.. Vol. 192. 67-76 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kajiwara: "Determinant Structure of the Rational Solutions for the Painleve IV Equation"J. Phys. A : Math. Gen.. Vol. 31, No.10. 2431-2446 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] B. Grammaticos: "Degeneration through Coalescence of the q-Painleve VI Equations"J. Phys. A : Math. Gen.. Vol. 31, No.15. 3545-3558 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R. Willox: "Quadrilateral Lattices and < Eigenfunction Potensials for N-component KP Hierarchies"Phys. Letter. Vol. A252. 163-172 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. M. Tamizhmani: "Nonlinear Integro-differential Equations as Discrete Systems"Inverse Problem. Vol. 15. 787 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Tokihiro: "Proof of Solitonical Nature of Box and Ball Systems by means of Inverse Ultra-discretization"Inverse Problem. Vol. 15. 1639-1662 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. J. Ablowitz: "On Discretizations of the Vector Nonlinear Sehrodinger Equation"Phys. Lett. A. Vol. 253, No. 5,6. 287-304 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Ohta: "Singularity Confinement and Algebraic Entropy : the Case of the Discrete Painleve Equations"Phys. Lett. A. Vol.262, No. 2,3. 152-157 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.M.Tamizhmani: "Integrability Criteria for Differential-Difference Systems : A Comparison of Singularity Confinement and Low-Growth Requirements"J. Phys. A : Math. Gen.. Vol. 32, No. 38. 6679-6685 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] R. Willox: "Nonautonomous Discrete Integrable Systems"Chaos, Solitons and Fractals. Vol. 11. 121-135 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Ohta: "Pfaffian Solution for Coupled Discrete Nonlinear Schrodinger Equation"Chaos, Solitons and Fractals. Vol. 11, No.1-3. 91-95 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M. J. Ablowitz: "On Integrability and Chaos in Discrete Systems"Chaos, Solitons and Fractals. Vol. 11, No.1-3. 159-169 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J J. C. Nimmo: "Applications of Darboux Transformations to the Self-Dual Yang-Mills Equations"Theor. math. Phys.. Vol. 122, No. 2. 239-246 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Ramani: "Discrete Integrable Systems from Continuous Painleve o Equations through Limiting Procedures"Nonlinearity. Vol. 13, No. 4. 1073-1086 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Ramani: "Linearizable Mappings and the Low-Growth Criterion"J. Phys. A : Math. Gen.. Vol. 33, No. 31. L287-L292 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Ohta: "Determinant and Pfaffian Solutions for Discrete Soliton Equations"CRM Proceedings and Lecture Notes. Vol. 25. 339-345 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Nagai: "Conserved Quantities of Box and Ball System"Glasgow Math. J.. Vol. 43A. 91-97 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] J.Satsuma: "Scattering in Cellular Automata"Scattering ed. by R. Pike and P. Sabatier ( Academic Press, London). 1754-1772 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] G.Hatayama: "The A (1) M Automata Related to Crystals of Symmetric Tensors"J. Math. Phys.. Vol. 42. 274-308 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] F. W. Nijhoff: "On Discrete Painleve Equations Associated with the Lattice KdV Systems and the Painleve VI Equation"Stud. Appl. Math.. Vol. 106, No. 3. 261-314 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Ramani: "A Geometrical Description of the Discrete Painleve VI and V Equations"Comm. Math. Phys.. Vol. 217, No. 2. 315-329 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A. Ramani: "The q-PainleveV Equation and Its Geometrical Description"J. Phys. A : Math. Gen.. Vol. 34, No. 11. 2505-2513 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Ohta: "A Bilinear Approach to a Pfaffian Self-Dual Yang-Mills Equation"Glasgow Math. J.. Vol. 43A. 99-108 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Lafortune: "Blending Two Discrete Integrability Criteria : Singularity Confinement and Algebraic Entropy"CRM Proceedings and Lecture Notes. Vol. 29. 299-311 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Y. Ohta: "An Affine Weyl Group Approach to the Eight-Parameter Discrete Painleve Equation"J. Phys. A : Math. Gen.. Vol.34, No. 48. 10523-10532 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] S. Kakei: "A Differential-Difference System Related to Toroidal Lie Algebra"J. Phys. A : Math. Gen.. Vol.34, No. 48. 10585-10592 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Kajiwara: "Determinant Formulas for the Toda and Discrete Toda Equations"Funkcial. Ekvac. Vol.44, No. 2. 291-307 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Takenawa: "A Geometrical Approach to Singularity Confinement and Algebraic Entropy"J. Phys. A : Math. Gen.. Vol. 34. L95-L102 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Takenawa: "Algebraic Entropy and the Space of Initial Values for Discrete Dynamical Systems"J. Phys. A : Math. Gen.. Vol. 34. 10533-10545 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T. Takenawa: "Discrete Dynamical Systems Associated with Root Systems of Indefinite Type"Commun. Math.. Vol. 224. 657-681 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2003-09-17  

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