【作者】 徐秀丽；

【导师】 龚新波；

【作者基本信息】 山东科技大学 ， 应用数学， 2008， 硕士

【摘要】 本文主要研究了非线性演化方程族的生成以及非线性演化方程族的扩展可积模型。第一章概述了孤立子理论的产生和发展、研究概况及其研究意义。在第二章中,首先,运用（2+1）-维的零曲率方程得到了（2+1）-维的JM族。其次,根据已有的Lie代数,通过线性组合得到了一个6维的Lie代数,然后,构造出相应的loop代数,并由此设立一个广义谱问题,运用屠格式直接获得了（2+1）-维JM族的可积扩展模型。再次,在一个多分量loop代数的基础上,运用（2+1）-维的零曲率方程得到多分量（2+1）-维JM族。在第三章中,首先,构造一个新的矩阵等谱问题,由离散的屠格式得到非线性晶格方程,利用迹恒等式得到其Hamilton结构。其次,利用半直和的方法扩充已有的Lie代数得到新的Lie代数,在其基础上求得已知可积系的三个可积耦合。因此,得到了一种求可积耦合的简便方法。最后,建立了一个新的代数系统G_M并将其扩充到高维代数系统G_jM,进而构造出相应的loop代数（?）_jM,作为其应用导出了离散可积系的多分量可积系。在第四章中,借助于循环数,构造了一个新的loop代数,并由此得到具有双Hamilton结构的广义Toda族。更多还原

【Abstract】 This paper discusses the formulation of the nonlinear evolution hierarchy as well as their expanding integrable systems.In the first chapter,historical origin and some researches of soliton theory together with its research meaning are presented.In the second chapter,firstly based on an old Lie algebra,a six dimensional Lie algebra is constructed through the linear combination,then the corresponding loop algebra is obtained.From the loop algebra a generalized isospectral problems are designed.Using Tu scheme,an expanding integrable model of（2+1）-dimensional JM hierarchy is worked out.Secondly,on the base of a multi-component loop algebra,multi-component（2+1）-dimensional JM hierarchy is worked out by use of（2+1）-dimensional zero-curvature equation.In the third chapter,firstly,a discrete matrix spectral problem is introduced,a hierarchy of nonlinear lattice equation is obtained by Tu scheme,then its Hamilton structure is presented by use of the extended trace identity. Secondly,we obtain its three integrable coupling systems with the help of two semi-direct sum Lie algebras.Thirdly,by using a new algebraic system G_M,a multi-component discrete integrable hierarchy is obtained,and an new extended algebraic system G_jM of G_M are presented,from which the integrable couplings system of the multi-component lattice hierarchy are obtained.In the fourth chapter,with the help of the cycled numbers,a higher-dimensional loop algebra is constructed.By employing loop algebra A₃^*,a generalized Toda hierarchy is obtained with possessing 2-Hamilton structure,which is also reduced to the well-known Toda hierarchy.更多还原