【作者】 吴丽华；

【导师】 耿献国；

【作者基本信息】 郑州大学 ， 基础数学， 2012， 博士

【摘要】 本文主要分为如下两个部分：其一,借助于Lenard递推序列及零曲率方程,推导出与偶的3×3超矩阵谱问题相联系的新的超KdV方程族和超KN方程族,并建立了它们的广义双Hamilton结构和无穷守恒律；其二,基于三角曲线理论及代数几何知识,构造了三族与3×3矩阵谱问题相联系的孤子方程的拟周期解.第二章中,依据超李代数,将经典的2×2矩阵谱问题扩充为偶的3×3超矩阵谱问题.利用相容性条件,即零曲率方程,导出了新的超KdV方程族及超KN方程族.应用超迹公式讨论了这两族超孤子方程的Hamilton结构,并建立了超KdV方程及超KN方程的无穷守恒律.我们知道,孤子方程的拟周期解不仅揭示了解的内部结构,描述了非线性现象的拟周期行为,或孤子方程的可积性特征,而且可以利用它约化出多孤子解,椭圆函数解及其它形式的解.因此,研究孤子方程的拟周期解就变得十分重要.本文第三章到第五章,分别讨论了三族与3×3矩阵谱问题相联系的孤子方程的拟周期解.通过方程族的Lax矩阵的特征多项式,定义了一条三角曲线κg,并将其紧化为亏格为9的三叶Riemann面.在此Riemann面上引入适当的Baker-Akhiezer函数,亚纯函数及椭圆变量,从而将孤子方程分解为可解的Dubrovin-type常微分方程系统.然后,在Abel映射下流被拉直.进一步,根据亚纯函数及B aker-Akhiezer函数零点和奇点的性质,利用第二类和第三类Abel微分.Riemann定理及Riemann-Roch定理得到了它们的Riemann theta函数表示.最后,结合亚纯函数或(及)Baker-Akhiezer函数的Riemann theta函数表示和它们的渐近性质,便给出了孤子方程族的拟周期解.更多还原

【Abstract】 The thesis can be mainly divided into two parts. First, with the help of Lenard recursion equations and the zero-curvature equation, we derive a new super KdV hierarchy and super KN hierarchy related to two even3×3matrix spectral problems. Moreover, generalized bi-Hamiltonian structures and infinite conservation laws of these two hierarchies are established; On the other hand, based on the theory of trigonal curve and the knowledge of algebraic geometry, we construct the quasi-periodic solutions of three hierarchies of soliton equations associated with three3×3matrix spectral problems.In chapter two, by extending the corresponding classical2×2spectral problems to the even3×3super matrix spectral problems according to super Lie algebra, we propose a new super KdV hierarchy and super KN hierarchy resorting to the compatible condition i.e. zero-curvature equation. Applying the super trace iden-tity, we discuss the generalized bi-Hamlitonian structures of the two super soliton hierarchies. Furthermore, we establish the infinite sequence of conserved quantities of the super KdV equation and super KN equation.As we all know, quasi-periodic solutions of soliton equations not only reveal in-herent structure mechanism of solutions and describe the quasi-periodic behavior of nonlinear phenomenon or characteristic for the integrability of soliton equations, but also can be reduced to find multi-soliton solutions, elliptic function solutions, and others. Therefore, the research on the quasi-periodic solutions of soliton equations is of greatest importance. From chapter three to five, we discuss the quasi-periodic solutions of three hierarchies of soliton equations associated with three different3×3matrix spectral problems, respectively. With the aid of the characteristic polynomial of Lax matrix for soliton hiearachy, we define a trigonal curve κg, and then its com- pactification becomes a three-sheeted Riemann surface of arithmetic genus g. We introduce the appropriate Baker-Akhiezer function, meromorphic function and ellip-tic variables on the three-sheeted Riemann surface, from which soliton equations are decomposed into the system of solvable Dubrovin-type ordinary differential equa-tions. Then, under the Abel map, the flows of soliton hierarchy are straightened. Furthermore, in accordance with the properties of the zeros and singularities of the meromorphic function and Baker-Akhiezer function, we get their Riemann theta function representations by means of the second and third Abel differentials, Rie-mann theorem and Riemann-Roch theorem. Combing the Riemann theta function representations of the meromorphic function or(and) the Baker-Akhiezer function with their asymptotic properties, we finally obtain the quasi-periodic solutions of soliton equations.更多还原