2002 Fiscal Year Final Research Report Summary
Study on Formal Integrability of Systems of Evolution Equations
| Project/Area Number |
13640140
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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 | DOSHISHA UNIVERSITY |
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Principal Investigator |
WATANABE Yoshihide Faculty of Engineering, Professor, 工学部, 教授 (50127742)
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| Co-Investigator(Kenkyū-buntansha) |
KONDO Koichi Kyushu Univ. Faculty of Math., Lecturer, 工学部, 専任講師 (30314397)
KAJIWARA Kenji Kyushu Univ. Faculty of Math., Associate Professor, 大学院・数理学研究院, 助教授 (40268115)
OHMIYA Mayumi Faculty of Engineering, Professor, 工学部, 教授 (50035698)
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| Project Period (FY) |
2001 – 2002
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| Keywords | evolution equations / formal integrability / strong symmetries / strong conservation laws / computer algebra / Darboux-Lame equation / discrete Painleve equation / solvable chaos |
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| Research Abstract |
・ We enumerate systems of evolution equations formally completely integrable in the sense of Sokolov and Shabat, that is, equations admitting strong symmetries and strong conservation laws. The calculation is performed with the computer algebra system REDUCE and under the two assumptions : (1) Time evolution is described by polynomials of dependent variables and their derivatives with respect to the spatial variable x, and such polynomials are homogeneous with respect to a suitable weight for the derivation. (2) Time evolution is linear with constant coefficients with respect to the highest order spatial derivative. We enumerate systems of 4-th order equations and get the list for the weight 1.
・ The Gelfand-Dickey transformation transforms differential polynomials into symmetric polynomials, then the problem of finding the kernel of the Euler operator is reduced to the problem of finding invariants of the finite group called the Euler group, We have implemented the algorithm by Sturmfels which calculates generating invariants of finite groups to the computer algebra system Asir.
・ We have calculated the monodromy group for the second order Darboux-Lame equation and established the method for calculating the position of singularities for such potentials.
・ A birational realization of Affine-Weyl group A^<(1)>_<m-1> × A^<(1)>_<n-1> is given, and in terms of this representation, some discrete integrable systems are constructed. We also derived a hierarchy of discrete dynamical systems which admit the above affine-Weyl group as the symmetry group and proved that the hierarchy is obtained from the q-KP hierarchy by reduction.
・ It is shown that two solvable chaotic systems, the arithmetic-harmonic mean (AHM) algorithm and the Ulam-von Neuman (UvN) map, admit determinantal solutions.
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Research Products
(14 results)
