Research on the spectrum and the monodromy related to the algebro-geometric potentials

与代数几何势相关的谱和单峰性研究

• 批准号: 13640195
• 负责人: OHMIYA Mayumi
• 金额: $ 0.32万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640195/
• 关键词: Algebro-geometric potential; Darboux transformation; Spectral preserving deformation; Monodromy preserving deformation; Fibering; cyclic action; Computer algebra system; Invariant ring; 代数幾何的楕円ポテンシャル; Calogero系; 楕円ソリトン; 非線形波動; 超幾何方程式; ソリトン

When the n-th Lame potential (n≧2), which is the most typical algebro-geometric potential, satisfies a kind of the degenerate condition, the Darboux-Lame equations are defined, and studied from the both of the viewpoints of isospectral deformation and isomonodromic deformation. The resulting equations are shown to be isospectral by verifying that those potentials solve the higher order KdV equation.On the one hand, those equations are shown to be isomonodromic if and only if the deformation parameter is restricted to be on some specific algebraic curve. It is also shown that the 2nd Darboux-Lame equation is transformed to the equation of Heun type with the four regular singular points by some 3-fold covering map from the underlying elliptic curve to the protective line. In addition, if the deformation parameter coincides with zero, this Heun type equation degenerates to Gauss' hypergeometric equation with the three regular singular points. By vitue of this fact and its isomonodromic property, the monodromy group associated with this Heun type equation can be concretely computed.Moreover, a kind of fibering condition of the 3-manifolds over the circle is clarified to study the fibering structure of the Jacobi variety corresponding to the spectral curve.To calculate concretely the monodromy representation, it is necessary to clarify the structure of the invariant ring associated with the finete group. An algorithm to compute the generator of the invariant ring is explored and inplemented with the computer algebra system Asir.

当最典型的代数几何势--n阶Lame势（n ≠ 2）满足一种退化条件时，定义了Darboux-Lame方程，并从等谱形变和等单道形变两个角度研究了Darboux-Lame方程.通过证明这些势解高阶KdV方程，证明了所得到的方程是等谱的;一方面，证明了这些方程是等单道的当且仅当变形参数被限制在某一特定的代数曲线上;证明了第二类Darboux-Lame方程可以通过从椭圆曲线到保护线的3重覆盖映射转化为具有4个正则奇点的Heun型方程.另外，当变形参数为零时，Heun型方程退化为具有三个正则奇点的Gauss超几何方程。利用这一事实和它的等单值性，可以具体地计算与这类Heun型方程相联系的单值群，并阐明了圆上三维流形的一种重叠条件，以研究谱曲线对应的Jacobi簇的重叠结构，具体地计算单值表示，有必要阐明与Finete群相关联的不变环的结构。研究了一种计算不变环生成元的算法，并在计算机代数系统Asir上实现。

项目成果

渡邊芳英, 鍋島勇: "Asirによる有限群の不変式環の生成元の計算"数式処理. 8巻4号. 25-26 (2002)

Yoshihide Watanabe、Isamu Nabeshima：“使用 Asir 计算有限群不变环的生成元”，第 8 卷，第 4 期。25-26 (2002)。

M. Ohmiya, M. Urakubo: "Darboux-Lame equations and iso-monodromic deformation"Proceeding of the conference on the nonlinear wave phenomena-theory and application. 209-214 (2002)

M. Ohmiya，M. Urakubo：“Darboux-Lame 方程和等单向变形”非线性波现象理论与应用会议论文集。

Mayumi Ohmiya, Masami Urakubo: "Darboux-Lame equations and iso-monodromic deformation"研究集会「非線形波動現象の理論と応用」報告集. (印刷中).

Mayumi Ohmiya、Masami Urakubo：“Darboux-Lame方程和等单向变形”研究会议“非线性波现象的理论与应用”报告集（正在出版）。

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。

Y. Watanabe, I. Nabeshima: "Computation of the generator of the invariant ring associated with the finite group by Asir"Surikaisekikenkyuusho-koukyuuroku. Vol. 1295. 17-28 (2002)

Y. Watanabe，I. Nabeshima：“Asir 与有限群相关的不变环生成元的计算”Surikaisekikenkyuusho-koukyuuroku。