【作者】 孙业朋；

【导师】 徐西祥；

【作者基本信息】 山东科技大学 ， 运筹学与控制论， 2003， 硕士

【摘要】 本文分别构造了具有2个位势和3个位势的等谱特征值问题。从等谱问题出发，利用屠格式导出了著名的广义Burgers方程族和一类新的MKdV-NLS方程族，及一族离散的非线性演化方程，且证明了它们均是Liouville可积的广义Hamilton方程族。其中MKdV-NLS方程族还具有双Hamilton结构。同时，应用非线性化技巧，证明了在Bargmann约束下，MKdV-NLS方程族的Lax对可被非线性化为两个有限维Liouville完全可积系。又通过构造Loop代数，分别得到了广义Burgers方程的可积耦合、MKdV-NLS方程族的可积耦合及Dirac方程族的一类扩展可积模型。最后利用Darboux变换方法，通过构造不同的Darboux矩阵，分别得到了混合的非线性Schrdinger方程的N-波Darboux变换，WKI方程的Darboux变换和一族新的离散孤子方程的Darboux变换及精确解。更多还原

【Abstract】 In this thesis, isospectral eigenvalue problems, which contain two and three potentials respectively, are established. Starting from isospectral problems, Tu’s scheme is applied to generate a well-known generalized Burgers equation hierarchy, a class of new MKdV-NLS equation hierarchy and a family of discrete nonlinear evolution equations. They are shown to be Liouville integrable Hamiltonian systems. Moreover, MKdV-NLS hierarchy have Binary Hamiltonian structures. Then the nonlinearization procedure is applied to the eigenvalue problem of MKdV-NLS hierarchy. Under Bargmann constraint, it is shown that Lax pairs are nonlinearized to be two finite-dimensional Liouville completely integrable system. ConstuctingLoop algebra G leads to integrable couplings of the generalized Burgers hierarchy, integrable couplings of the MKdV-NLS hierarchy and a class of expandingintegrable model of Dirac hierarchy. Finally, by using Darboux transformation method, various of Darboux matrices are obtained. Darboux transformations of the mixed nonlinear schr dinger equation, WKI equation and a family of new discrete soliton equations are constructed. Furthermore, their exact solutions are derived.更多还原