与四阶矩阵特征问题相关的约束流与完全可积系统  

The Constraint Flows and Compietely Integrable System for the Fourth-order Matrix Eigenvalue Problem

作　　者：赵亚纯 王淑红[2] 袁书娟[3] 

机构地区：[1]南京市第十四中学数学组,江苏南京210031 [2]内蒙古民族大学数计学院,内蒙古通辽028043 [3]河北工业大学理学院,天津300130

出　　处：《石家庄铁道学院学报》2007年第2期30-36,共7页Journal of Shijiazhuang Railway Institute

摘　　要：主要讨论与四阶矩阵特征值问题相联系的孤子方程及其Lax上,利用位势函数与特征函数之间的Bargmann约束,将四阶特征值问题及相应的伴随特征值问题非线性化,获得新的有限维Hamilton系统,并应用r-矩阵理论证明了新的有限维Hamilton系统在Liouville意义下的完全可积性。最后借助于在Liouville意义下完全可积Hamilton系统的对合解得到孤子方程族解的对合表示。In this paper, we discuss the 4th-order matrix eigenvalue problem ： φx = Mφ ,the soliton equation and its Lax pairs with 4th-order matrix eigenvalue problem, then based on Bargmann constraint between the potential and the eigenfunction , a new finite-dimensional Hamiltonian system is obtained by nonlinearzation of the eigenvalue problem and its adjointone. Between the Lax pairs of nonlinearzation and r -matrix, according to the r-matrix theory, it is proved that the new finite-dimensional Hamiltonian system is completely integrable in the Liouville sense. Then based on the involutive solution of completely integrable Hamihonian system in the Liouville sense, involution representation of the solution for the evolution equations are generated.

关 键 词：R-矩阵 辛结构 POISSON括号 完全可积性 

分 类 号：O193[理学—数学]