2005 Fiscal Year Final Research Report Summary
Study of the integrable systems in mathematical physics and applied analysis
| Project/Area Number |
15540219
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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 |
Global analysis
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| Research Institution | Doshisha University |
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Principal Investigator |
OHMIYA Mayumi Doshisha University, Faculty of Engineering, Professor, 工学部, 教授 (50035698)
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| Co-Investigator(Kenkyū-buntansha) |
WATANABE Yoshihide Doshisha University, Faculty of Engineering, Professor, 工学部, 教授 (50127742)
KONDO Koichi Doshisha University, Faculty of Engineering, Lecturer, 工学部, 専任講師 (30314397)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Darboux transformation / Iso-monodromy / Iso-spectral / Sine-Gordon equation / NMR quantum computer / Grobner basis / Wavelet analysis / Singular value decomposition |
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| Research Abstract |
We clarified the isomonodromic property of the Darboux-Lame equation obtained by the double Darboux transformation for the 2^<nd> Lame equation. Using this, we succeeded to characterize the differential equations of Heun type whose monodromy can be exactly calculable by transforming it to the differential equation on the complex projective line by a covering map.
Moreover, applying the classical Appell's lemma, we developed a new algorithm of solving the differential equations. In addition, we extended the spectral degenerate condition of Darboux transformation to the non-spectral case. Furthermore, we investigated the asymptotic behavior of the elliptic multi-soliton solutions applying this result, and discovered the new addition formula of the elliptic function.
On the other hand, we accomplished the study of the modulation instability of the strongly dispersive nonlinear system, and decided the modulation instability zone of the wave numbers of the nonhomoclinic solution to Sine-Gordo
… More
n equation. At the same time, we carried out the numerical study of such unstable phenomena using Hirota's difference scheme, and showed that this scheme was fairly useful even for the unstable phenomena.
On the one hand, we proved the iso-spectral property of the double Darboux transformation, and clarified the equivalence of the iso-monodromic property and the iso-spectral property in some specific case.
Moreover, we studied the formula for the reconstruction of qubit density matrix in NMR quantum computing. Moreover, we studied the GHZ state of 5 qubits NMR quantum computing.
On the other hand, we studied the application of Grobner basis. In particular, we implemented the computation of Grobner basis of the toric ideal to the computer algebra system "Asir".
We studied the application of the wavelet analysis to the numerical analysis of the nonlinear wave motion, and verified certain efficiency of Beylkin's method. On the other hand, we constructed the mathematical model of the heat acoustic cooling system as nonlinear phenomena. Moreover, we studied the appropriate singular value decomposition algorithm for the image compression applying the wavelet analysis. Less
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