| Co-Investigator(Kenkyū-buntansha) |
KAJIWARA Kenji Doshisha University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (40268115)
OHMIYA Mayumi Doshisha University, Faculty of Engineering, Professor, 工学部, 教授 (50035698)
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| Budget Amount *help |
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥800,000 (Direct Cost: ¥800,000)
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| Research Abstract |
・The Hamiltonian formalism for semi-discrete evolution equations is introduced by using the Schouten bracket defined in the ring with the shift operator. It is hoped that integrable evolution equations admit bi-Hamiltonian structure and it is proved that some of the semi-discrete evolution equations including the Toda-Lattice equation admit bi-Hamiltonian structure. However, there may be some equations which are integrable in some sense but do not seem to admit bi-Hamiltonian structure.
・The discrete Euler operator defined in terms of the difference operator is derived from the discrete variational problem and properties of which is examined in detail. Especially, the relation between such Euler operator and the usual discrete Euler operator defined in terms of the shift operator is clarified and by using these relations, the kernel and the image of this new type of discrete Euler operator is determined. Further, if one uses such discrete Euler operator, instead of the usual Euler opera
… More
tor, in the construction of Hamiltonian formalism for semi-discrete evolution equations, it is proved that it has some advantages over the usual formalism.
・A Hamilton's canonical equation with finite degrees of freedom, which is equivalent to the stationary KdV equation is derived from the Deift-Trubowitz type algebraic trace formula. This equation is proved to be completely integrable in Liouville's sense, and explicitly involves the information on the spectrum for algebro-geometric potentials which is constructed from the solution of the equation. Further, it is shown that, under certain degeneracy condition, the equation is reduced to the Neumann system, which describes the motion of a harmonic oscillator restricted to the sphere. Also, the relation between the equation and the Dubrovin-Novikov system is clarified.
・Study on the symmetry and the τ functions of the continuous and discrete Painleve equations is accomplished. Especially, starting with the determinant formulae for the classical solution of the Painleve equation, it is proved that such determinant flormulae are also valid for transcendental solutions of the Painleve equation. In order to investigate the solutions of the discrete equations, it is shown to be effective to use singularity confinement test and in this view point, symmetry of the solution of the discrete Painleve equation is exploited. Less
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