2003 Fiscal Year Final Research Report Summary
Mathematics in ultradiscrete integrable systems
| Project/Area Number |
12440046
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | The University of Tokyo |
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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)
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| 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)
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| Project Period (FY) |
2000 – 2003
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| Keywords | ultradiscrete systems / integrable systems / cellular automaton / integrable lattice models / crystal theory / tropical |
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| 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.
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