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

通过对2^<和> Lame方程进行二重Darboux变换，得到了Darboux-Lame方程的等同构性质。利用这一点，我们成功地刻画了一元可精确计算的Heun型微分方程，并将其用覆盖映射变换成复射影线上的微分方程。此外，利用经典的阿佩尔引理，我们提出了一种求解微分方程的新算法。此外，我们将达布变换的谱退化条件推广到非谱情况。在此基础上，我们进一步研究了椭圆型多孤子解的渐近性质，并发现了椭圆型函数的新加法公式。另一方面，完成了强色散非线性系统的调制不稳定性研究，确定了正弦-戈多方程非同斜解波数的调制不稳定性区。同时，我们利用Hirota差分格式对这种不稳定现象进行了数值研究，并证明了这种格式即使对于不稳定现象也是相当有用的。一方面证明了双达布变换的等谱性质，并在一些具体情况下阐明了等单峰性质与等谱性质的等价性。此外，我们还研究了核磁共振量子计算中量子比特密度矩阵的重构公式。此外，我们还研究了5量子位核磁共振量子计算的GHZ态。另一方面，我们研究了Grobner基的应用。特别地，我们在计算机代数系统“Asir”中实现了环向理想的Grobner基的计算。研究了小波分析在非线性波动数值分析中的应用，验证了Beylkin方法的有效性。另一方面，我们建立了热声冷却系统作为非线性现象的数学模型。在此基础上，研究了应用小波分析进行图像压缩的奇异值分解算法。少

项目成果

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

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)

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

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