两个可积系统及其约化

数学物理学报 ›› 2014, Vol. 34 ›› Issue (5): 1304-1312.

两个可积系统及其约化

郭秀荣¹, 胡美燕²

1.山东科技大学基础课部山东泰安 271000;
2.山东科技大学信息系山东泰安 271000)

收稿日期:2013-04-15 修回日期:2014-07-08 出版日期:2014-10-25 发布日期:2014-10-25
基金资助:
国家自然科学基金(11371361)、山东省自然科学基金(ZR2012AQ011)、山东省高等学校科技计划项目(J14LI58)和山东科技大学群星计划项目(QX2013143)资助

Two Integrable Systems and Reductions

GUO Xiu-Rong¹, HU Mei-Yan²

1.Department of Basic Courses, Shandong University of Science and Technology, Shandong Taian 271000;
2.Department of Information Engineering, Shandong University of Science and Technology, Shandong Taian 271000

Received:2013-04-15 Revised:2014-07-08 Online:2014-10-25 Published:2014-10-25
Supported by:
国家自然科学基金(11371361)、山东省自然科学基金(ZR2012AQ011)、山东省高等学校科技计划项目(J14LI58)和山东科技大学群星计划项目(QX2013143)资助

摘要/Abstract

摘要：

该文引入了一个李代数,然后定义了其相应的两个圈代数,利用圈代数构造了两个等谱问题, 其相容性条件导出了两个可积动力系统.通过约化这样的系统,得到了某些有趣的非线性方程, 如Burgers方程、组合KdV-MKdV方程和Kuramoto-Sivashinsky方程以及KdV方程的一种推广形式. 最后,利用贝尔多项式讨论了广义KdV方程的可积性质,包括双线性形式、Lax对、贝克隆变换和无穷守恒律等.

关键词: 可积系统, 贝克隆变换, 贝尔多项式

Abstract:

A Lie algebra is presented whose two corresponding loop algebras are followed to obtained, which are used to introduce two kinds of isospectral problems whose compatibility leads to two integrable dynamics. Some interesting nonlinear evolution equations are produced by reduction of the two dynamics, including the well-known Burgers equation, a generalized combined KdV-MKdV equation and the Kuramoto-Sivashinsky equation and a generalized form of the KdV equation. Then with the help of Bell polynomials we investigate some integrable properties of the generalized form of the KdV equation, such as bilinear representation, Lax pair, B"{a}cklund transformation, infinite conservation laws, and so on.

Key words: Integrable system, B\"{a}cklund transformation, Bell polynomials

中图分类号:

35Q51

郭秀荣, 胡美燕. 两个可积系统及其约化[J]. 数学物理学报, 2014, 34(5): 1304-1312.

GUO Xiu-Rong, HU Mei-Yan. Two Integrable Systems and Reductions[J]. Acta mathematica scientia,Series A, 2014, 34(5): 1304-1312.

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