On Dirichlet, Poncelet and Abel problems
On Dirichlet, Poncelet and Abel problems
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关于 Dirichlet、Poncelet 和 Abel 问题
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发表时间：
2009
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通讯作者：
A. Zhedanov
中科院分区：
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作者：
V. Burskii;A. Zhedanov
给出了天才1的椭圆曲线理论应用于偏微分方程组、几何、代数、分析和物理等问题之间的联系。讨论了任意双二次代数曲线上弦方程的Dirichlet问题和其他边值问题解的唯一性。证明了解的非唯一性当且仅当与该曲线对应的两个二次曲线的Poncelet问题有周期轨道。类似地，证明了来自不同数学领域的其他问题与之等价。其中有代数Pell-Abel方程的可解性问题和推广了著名的三角矩问题的矩问题的不确定性问题。上述问题的可解性判据可以用$hetainBbbQ$形式给出，其中数$heta=m/n$与具体问题数据相联系。在Pad‘e插值理论中，我们还证明了上述问题与现代数学物理问题如Toda链的椭圆解、经典Heisenberg$xy$链的静态解、椭圆网格上的双正交有理函数等问题的密切关系。
We offer connections between %Theory of elliptic curve of genius 1 applies to some problems of PDE, geometry, algebra, analysis and physics. The uniqueness of the solution of the Dirichlet problem and some another boundary value problems for the string equation inside of an arbitrary biquadratic algebraic curve is considered. It is shown that the solution is non-unique if and only if corresponding the Poncelet problem for two conics %associated with the curve has a periodic trajectory. Similarly some other problems %from different %fields of mathematics are proved to be equivalent to it. Among them there are the solvability problem of the algebraic Pell-Abel equation and an indeterminacy problem of a moment problem that generalizes well-known trigonometrical moment problem. Solvability criterions of above problems can be presented in form $ hetainBbb Q$ where the number $ heta=m/n$ is connected with the concrete problem data. We demonstrate also an intimate relation of the above mentioned problems with such problems of the modern mathematical physics as elliptic solutions of the Toda chain, static solutions of the classical Heisenberg $XY$-chain and biorthogonal rational functions on elliptic grids in the theory of the Pad'e interpolation.