2000 Fiscal Year Final Research Report Summary
Eigenvolues of random matrices and soliton equations
| Project/Area Number |
11440044
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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 |
Basic analysis
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
SHIOTA Takahiro Kyoto Univ, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20243008)
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| Project Period (FY) |
1999 – 2000
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| Keywords | matrix integral / soliton equations / KP hierarchy / Toda lattice / tau function / Virasoro symmetry |
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| Research Abstract |
Together with M.Adler and P.van Moerbeke, we studied an integrable system called Pfaff lattice (obtained by imposing certain symmetry condition to the 2-dimensional Toda lattice and restricting the time evolutions to the locus s=-t ; this is a subsystem of the D'_∞-hierarchy of Jimbo and Miwa, and is closely related to the coupled KP hierarchy of Hirota and Kakei), and constructed its tau function, bilinear identity, Fay identities, and a quasiperiodic solution. Combined with the work of Adler, Horozov, van Moerbeke on the Lax representation of the Pfaff lattice, etc., this work shows the general theory of the Pfaff lattice. Symmetric and symplectic matrix integrals give tau functions of the Pfaff lattice. We also studied the Virasoro conditions satisfied by those special solutions.
Together with E.Horozov, we proved that the spectral curve of a rank-1 commutative ring of ordinary differential operators, whose coefficients are rational functions of the independent variable, is a rational curve with only cusps. This refines a similar result of G.Wilson in which the bispectrality of the ring was assumed.
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