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Study on Construction and Classification of Nonautonomous Nonlinear Integrable Systems Based on Symmetry of Bilinear Form

基于双线性形式对称性的非自治非线性可积系统的构造与分类研究

基本信息

批准号：
13640118
负责人：
OHTA Yasuhiro
金额：
$ 1.6万
依托单位：
HHIROSHIMA 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-13640118/
关键词：
Integrable System Bilinear Form Painleve Equation Backlund Transformation Toda Lattice 非線形可積文系

项目摘要

1. We found several new nonautonomous nonlinear integrable second order ordinary difference equations which have forward-backward asymmetry. By using suitable dependent variable transformations, those forward-backward asymmetric integrable mappings are transformed into a class of discrete Painleve equations. Moreover we generalized those systems to even-odd asymmetric forms.2. We revealed the deep relation between the Laurent series expansion around a singular point for continuous Painleve equations and the small parameter in singularity confinement for discrete Painleve equations. By using the common feature for those two small parameters, we formulated a method to find an infinitesimal symmetry and applied it for simple examples.3. It was shown that by applying some dependent variable transformations for linearizable mappings we can create nonautonomous nonlinear integrable second order ordinary difference equations which have positive algebraic entropy. This remarkable example give a new aspect for the discrete integrability criteria.4. We studies the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painleve equation to an homographic mapping. We showed that the degree grows like n^3. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we presented a method for the construction of integrable Nth-order mapping with degree growth n^N.

1.我们发现了几个新的具有前后不对称性的非自治非线性可积二阶常微分方程解。通过适当的因变量变换，将这些前向后向非对称可积映射转化为一类离散的Painleve方程。此外，我们将这些系统推广到奇偶不对称形式。揭示了连续Painleve方程在奇点附近的Laurent级数展开式与离散Painleve方程奇性约束中的小参数之间的深层联系。利用这两个小参数的共同特征，我们给出了一种求无穷小对称性的方法，并将其应用于简单的例子。证明了通过对可线性化映射进行一些依赖变量变换，我们可以得到具有正的代数熵的非自治的非线性可积二阶常差分方程组。这一显著的例子为离散可积判据提供了一个新的方面。研究了一类由离散Painleve方程与单应映射耦合而成的三阶可积映射的初始条件迭代的次数增长性。在满足奇点约束条件的特殊情况下，我们发现了一个较慢的二次增长。最后给出了n^N次增长的N阶可积映射的构造方法。