2005 Fiscal Year Final Research Report Summary
Application of the discrete integrable systems on the semi-infinite lattice to the system of the bi-orthogonal polynomials
| Project/Area Number |
15540119
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
TSUJIMOTO Satoshi KYOTO UNIVERSITY, Graduate School of Informatics, Lecturer, 情報学研究科, 講師 (60287977)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Yoshimasa KYOTO UNIVERSITY, Graduate School of Informatics, Professor, 情報学研究科, 教授 (50172458)
NAGAI Atsushi Nippon University, College of Industrial Technology, Lecturer, 生産工学部, 講師 (90304039)
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| Project Period (FY) |
2003 – 2005
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| Keywords | soliton / discrete system / integrable system / orthogonal polynomial / identity of determinant / bi-orthogonal polynomial / Lotka-Volterra equation / molecule solution |
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| Research Abstract |
The purpose of this research is to construct the theory of the spectral transformations of the bi-orthogonal polynomials and the continued fractions from the point of the view of the discrete integrable systems on the semi-infinite lattice. We mainly study the relationship between the bi-orthogonal polynomials which have the three-term relations and the Toda-type discrete integrable systems. We give main results:
・The R1 type and the R2 type discrete integrable systems associated with the generalized eigen value problems
We study the bilinear equations of the R1 type and the R2 type discrete integrable systems. Then we clarify the relationship with the Toda type discrete systems including the nonautonomous discrete Toda equation and the relativistic discrete Toda equation, and derive their Backlund transformations.
・The discrete integrable system associated with the rational interpolation functions
We derive a discrete integrable system associated with Frobenius-Stickelberger's pioneering research on the rational interpolation functions and Thiele's theory on the Pade approximations. Moreover we give the bilinear equations of this integrable system and show that these bilinear equations are derived from the discrete KP equation and the two-dimensional discrete Toda equation. A generalised Lotka-Volterra equation and a generalized epsilon algorithm are presented.
・The nonautonomous discrete Toda equatio
From the research on the bilinear equation of the R1 chain and the R2 chain, we have new results on the nonautonomous discrete Toda equation. The Darboux transformations of the nonautonomous discrete Toda equation are derived from the Darboux transformation of the discrete KP equation and the two-dimensional discrete Toda equation.
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