Study on Integrability of Evolution Equations with Computer

演化方程的计算机可积性研究

基本信息

批准号：
11640140
负责人：
WATANABE Yoshihide
金额：
$ 1.09万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

・The Hamiltonian formalism for semi-discrete evolution equations is introduced by using the Schouten bracket defined in the ring with the shift operator. It is hoped that integrable evolution equations admit bi-Hamiltonian structure and it is proved that some of the semi-discrete evolution equations including the Toda-Lattice equation admit bi-Hamiltonian structure. However, there may be some equations which are integrable in some sense but do not seem to admit bi-Hamiltonian structure.・The discrete Euler operator defined in terms of the difference operator is derived from the discrete variational problem and properties of which is examined in detail. Especially, the relation between such Euler operator and the usual discrete Euler operator defined in terms of the shift operator is clarified and by using these relations, the kernel and the image of this new type of discrete Euler operator is determined. Further, if one uses such discrete Euler operator, instead of the usual Euler opera … More tor, in the construction of Hamiltonian formalism for semi-discrete evolution equations, it is proved that it has some advantages over the usual formalism.・A Hamilton's canonical equation with finite degrees of freedom, which is equivalent to the stationary KdV equation is derived from the Deift-Trubowitz type algebraic trace formula. This equation is proved to be completely integrable in Liouville's sense, and explicitly involves the information on the spectrum for algebro-geometric potentials which is constructed from the solution of the equation. Further, it is shown that, under certain degeneracy condition, the equation is reduced to the Neumann system, which describes the motion of a harmonic oscillator restricted to the sphere. Also, the relation between the equation and the Dubrovin-Novikov system is clarified.・Study on the symmetry and the τ functions of the continuous and discrete Painleve equations is accomplished. Especially, starting with the determinant formulae for the classical solution of the Painleve equation, it is proved that such determinant flormulae are also valid for transcendental solutions of the Painleve equation. In order to investigate the solutions of the discrete equations, it is shown to be effective to use singularity confinement test and in this view point, symmetry of the solution of the discrete Painleve equation is exploited. Less

・通过使用与移位操作员在环中定义的schouten括号来介绍半混凝土演化方程的汉密尔顿格式。希望综合进化方程录入双 - 哈米尔顿结构，并证明某些半差异演化方程在内，包括toda-lattice方程式录入双 - 哈米尔顿结构。但是，可能有一些方程在某种意义上是可以集成的，但似乎不允许双 - 哈米尔顿结构。・离散的Euler操作员根据差异操作员的定义是从离散的变异问题和属性详细检查的属性中得出的。尤其是，阐明了根据移位操作员定义的这种Euler操作员与通常的离散EULER操作员之间的关系，并通过使用这些关系，确定了这种新型离散Euler操作员的内核和图像。此外，如果有人使用这样的离散的欧拉运算符，而不是通常的欧拉运算符……更多的tor，在构建哈密顿式格式的半差异进化方程时，证明它具有与通常的汉密尔顿的常规方程式相比，它具有一定的优势。 Deift-Trubowitz型代数痕量公式。事实证明，该方程在liouville的意义上是完全可以集成的，并且明确涉及从方程解决方案构建的代数几何势的频谱中的信息。此外，据表明，在某些退化条件下，方程还原为Neumann系统，该系统描述了仅限于球体的谐波振荡器的运动。同样，方程式和杜布罗罗文 - 诺维科夫系统之间的关系也得到了阐明。・研究对称性和连续和离散的painleve方程的τ函数。尤其是，从帕克莱夫方程的经典解决方案的确定公式开始，证明这种确定的燃叶也适用于帕克莱夫方程的先验解决方案。为了研究离散方程的溶液，使用奇异性限制测试被证明是有效的，在此观点中，利用离散点painleve方程的解的对称性。