Study on Formal Integrability of Systems of Evolution Equations

演化方程组的形式可积性研究

基本信息

批准号：
13640140
负责人：
WATANABE Yoshihide
金额：
$ 1.15万
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-13640140/
关键词：
evolution equations formal integrability strong symmetries strong conservation laws computer algebra Darboux-Lame equation discrete Painleve equation solvable chaos 可積分系保存則対称性 strong symmetry stronng conservation law formal sym

项目摘要

・ We enumerate systems of evolution equations formally completely integrable in the sense of Sokolov and Shabat, that is, equations admitting strong symmetries and strong conservation laws. The calculation is performed with the computer algebra system REDUCE and under the two assumptions : (1) Time evolution is described by polynomials of dependent variables and their derivatives with respect to the spatial variable x, and such polynomials are homogeneous with respect to a suitable weight for the derivation. (2) Time evolution is linear with constant coefficients with respect to the highest order spatial derivative. We enumerate systems of 4-th order equations and get the list for the weight 1.・ The Gelfand-Dickey transformation transforms differential polynomials into symmetric polynomials, then the problem of finding the kernel of the Euler operator is reduced to the problem of finding invariants of the finite group called the Euler group, We have implemented the algorithm by Sturmfels which calculates generating invariants of finite groups to the computer algebra system Asir.・ We have calculated the monodromy group for the second order Darboux-Lame equation and established the method for calculating the position of singularities for such potentials.・ A birational realization of Affine-Weyl group A^<(1)>_<m-1> × A^<(1)>_<n-1> is given, and in terms of this representation, some discrete integrable systems are constructed. We also derived a hierarchy of discrete dynamical systems which admit the above affine-Weyl group as the symmetry group and proved that the hierarchy is obtained from the q-KP hierarchy by reduction.・ It is shown that two solvable chaotic systems, the arithmetic-harmonic mean (AHM) algorithm and the Ulam-von Neuman (UvN) map, admit determinantal solutions.

・我们列举了索科洛夫和沙巴特意义上的形式上完全可积的进化方程组，即承认强对称性和强守恒定律的方程组。计算是使用计算机代数系统 REDUCE 并在以下两个假设下进行的：（1）时间。演化由因变量及其关于空间变量 x 的导数的多项式描述，并且此类多项式对于推导的适当权重是齐次的 (2) 时间。演化对于最高阶空间导数是线性的，常数系数我们枚举四阶方程组并得到权重 1 的列表。· Gelfand-Dickey 变换将微分多项式变换为对称多项式，然后解决以下问题：寻找欧拉算子的核被简化为寻找称为欧拉群的有限群的不变量的问题，我们已经通过 Sturmfels 实现了该算法，该算法计算有限群的生成不变量到计算机代数系统Asir。・计算了二阶Darboux-Lame方程的单调群，并建立了计算此类势的奇点位置的方法。・Affine-Weyl群A^<(1的双有理实现)>_<m-1> × A^<(1)>_<n-1> 给出，并根据该表示，构造了一些离散可积系统，我们还推导了离散动力系统的层次结构。承认上述仿射-Weyl群作为对称群，并证明该层次是由q-KP层次通过约简得到的。・证明了两个可解的混沌系统，算术调和平均（AHM）算法和Ulam-冯诺依曼 (UvN) 映射，承认行列式解。