一种扩张AKNS可积方程族的方法  

An approach for enlarging integrable hierarchy of AKNS hierarchy

作　　者：冯莉莉[1] 于发军[1] 

机构地区：[1]沈阳师范大学数学与系统科学学院,沈阳110034

出　　处：《沈阳师范大学学报（自然科学版）》2016年第3期329-332,共4页Journal of Shenyang Normal University:Natural Science Edition

基　　金：辽宁省科技厅自然科学基金资助项目(2015020029)

摘　　要：目前人们从反对称矩阵李代数的角度出发,基本都是围绕着2×2Lax对进行研究,而对4×4Lax对的讨论的还比较少。可积耦合系统是当代非线性学科的一个重要研究内容,可积Hamiltonian系统理论在各个学科都有着深远的意义,利用它能推导出许多有意义的非线性演化方程。巧妙利用6个基元获得新的loop代数,将2×2AKNS方程族的Lax对扩张成4×4AKNS方程族的Lax对,进而获得其可积耦合系统。首先,构建一个4×4的反对称李代数。然后,利用伴随零曲率方程获得递推算子L,选定合适的初始值带入递推方程中,得到一个新的可积耦合方程族和广义的AKNS方程。最后,应用迹恒等式和屠格式,成功地建立了相应可积耦合方程族的Hamiltonian结构。In this paper, we start from a antisymmetric matrix Lie algebra problem. Because most of works focus on the Lax pairs of 2 × 2 Lax pairs, there is less work to search for the Lax pairs of 4 × 4 Lax pairs. Integrable coupling system is an interesting content in nonlinear science. The Hamiltonian structure of integrable coupling hierarchy has an important meaning in other studies, which can derive many nonlinear soliton equations. We obtain a new loop algebra by using six elements of matrix Lie algebra and enlarge AKNS hierarchy with 2× 2 Lax pairs to AKNS hierarchy with 4 × 4 Lax pairs, which can get its integrable coupling AKNS system. We construct a 4 ×4 Lax pairs with antisymmetric matrix Lie algebra. By zero -- curvature representation, a recurrence operator L is presented, then we get a new integrable coupling equation hierarchy and find a generalized AKNS equation hierarchy. At last, its Hamiltonian structure is obtained through the trace identity and Tu scheme.

关 键 词：AKNS方程族 可积系统 李代数 HAMILTONIAN结构 

分 类 号：O175.2[理学—数学]