Study of the integrable systems in mathematical physics and applied analysis

数学物理可积系统研究及应用分析

• 批准号: 15540219
• 负责人: OHMIYA Mayumi
• 金额: $ 1.98万
• 依托单位: Doshisha University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540219/
• 关键词: Darboux transformation; Iso-monodromy; Iso-spectral; Sine-Gordon equation; NMR quantum computer; Grobner basis; Wavelet analysis; Singular value decomposition; 変調不安定性; 楕円ソリトン; Lame方程式のスペクトル; Darboux変換; Groebner基底; Wavelet解析; 代数幾何的ポテンシャル; 楕円

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

我们阐明了由2^<nd> lame方程的双darboux转换获得的darboux-lame方程的异构粒子属性。使用它，我们成功地表征了Heun类型的微分方程，其单型可以通过覆盖地图将其转换为复杂投影线上的微分方程，从而可以通过覆盖地图进行计算。此外，我们开发了一种新的算法来求解微分方程。此外，我们将Darboux转化的光谱退化条件扩展到了非光谱案例。 Furthermore, we investigated the asymmetric 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.同时，我们使用Hirota的差异计划对这种不稳定现象进行了数字研究，并表明该方案即使对于不稳定现象也相当有用。 NMR量子计算中量子密度矩阵的重建。此外，我们研究了5个量子计算的GHz状态。另一方面，我们研究了Grobner的应用。特别是，我们实施了计算机代数系统“ ASIR”的Grobner基础的计算。我们研究了小波分析在非线性波运动的数值分析中的应用，并验证了贝尔金方法的某些效率。另一方面，我们将热声冷却系统的数学模型构建为非线性现象。此外，我们研究了应用小波分析的图像压缩的适当的奇异值分解算法。较少的

项目成果

Isospectral property of double Darboux transformation,

双达布变换的等谱性质，

• 期刊: Osaka Journal of Mathematics (in printing)
• 作者: 大宮眞弓;浦久保正美;Mayumi Ohmiya;Mayumi Ohmiya;Mayumi Ohmiya他2名;Mayumi Ohmiya他1名;Mayumi Ohmiya;Mayumi Ohmiya
• 通讯作者: Mayumi Ohmiya

International conference on Differential, Difference Equations and Their Application

微分、差分方程及其应用国际会议

• 发表时间: 2004
• 作者: 大宮眞弓;浦久保正美;Mayumi Ohmiya;Mayumi Ohmiya;Mayumi Ohmiya他2名;Mayumi Ohmiya他1名;Mayumi Ohmiya;Mayumi Ohmiya;Yoshihide Watanabe;Mayumi Ohmiya(分担執筆)(Ed. P.D.Siafarikas);Mayumi Ohmiya (Ed.P.D.Siafarikas)
• 通讯作者: Mayumi Ohmiya (Ed.P.D.Siafarikas)

Darboux-Lae equation and isomonodromic Deformation,

Darboux-Lae 方程和等单向变形，

• 发表时间: 2004
• 期刊: Abstract and Applied Analysis Vol.2004
• 作者: Mayumi Ohmiya 他2名;Mayumi Ohmiya;山口貴史 他1名;Mayumi Ohmiya;Mayumi Ohmiya;大宮眞弓 他2名;Mayumi Ohmiya;Mayumi Ohmiya;Mayumi Ohmiya;Mayumi Ohmiya;渡邊芳英他3名;Mayumi Ohmiya
• 通讯作者: Mayumi Ohmiya

Mayumi Ohmiya: "Darboux-Lame equation and isomonodromic deformation"Abstract and applied analysis. 印刷中.

Mayumi Ohmiya：“Darboux-Lame 方程和等单向变形”摘要和应用分析已出版。

Benjamin-Feir type instability of Sine-Gordon equation and spectrum of Lam'e equation II

Sine-Gordon 方程的 Benjamin-Feir 型不稳定性和 Lame 方程 II 的谱

• 发表时间: 2006
• 期刊: 研究集会報告「非線形波動および非線形力学系の現象と数理」(九州大学応用力学研究所) 電子版(未定)
• 作者: Mayumi Ohmiya 他2名