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

我们阐明了通过对 2^<nd> Lame 方程进行双重 Darboux 变换得到的 Darboux-Lame 方程的等单性，利用这一点，我们成功地刻画了 Heun 型微分方程，通过将其变换为此外，应用经典的阿佩尔引理，我们开发了一种新的求解微分方程的算法，此外，我们还扩展了微分方程的求解方法。进一步，我们应用该结果研究了椭圆多孤子解的渐近行为，并发现了椭圆函数的新加法公式。研究了强色散非线性系统的调制不稳定性，同时确定了Sine-Gordo n方程非同宿解的波数调制不稳定区域。这次，我们利用广田差分格式对此类不稳定现象进行了数值研究，结果表明该格式对于不稳定现象也相当有用。一方面，我们证明了双达布变换的等谱性质，并且阐明了等单谱性质和等谱性质在某些特定情况下的等价性。此外，我们还研究了核磁共振量子计算中量子位密度矩阵的重构公式。另一方面，我们研究了Grobner基的应用，特别是在计算机代数系统“Asir”中实现了Grobner基的计算。一方面，我们将小波分析应用于非线性波运动的数值分析，验证了Beylkin方法的一定有效性；另一方面，我们构建了热声冷却系统的非线性现象的数学模型，并研究了适当的奇异性。应用小波分析的图像压缩的值分解算法。

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。

Appell's lemma and Λ-operator as a tool of solving ODEs

阿佩尔引理和 Λ 算子作为求解 ODE 的工具

• 发表时间: 2005
• 期刊: The WSEAS transactions on Systems Vol.4
• 作者: Mayumi Ohmiya 他2名;Mayumi Ohmiya;山口貴史 他1名;Mayumi Ohmiya;Mayumi Ohmiya
• 通讯作者: Mayumi Ohmiya