【作者】 杨潇；

【导师】 曹策问；

【作者基本信息】 郑州大学 ， 基础数学， 2007， 博士

【摘要】 本文对两个连续谱问题(正、负KN谱问题)及一个离散谱问题进行了研究。主要讨论这三个谱问题产生的一批孤子方程的可积分解，得到了KN等谱族产生的dNS，dmKdV，2＋1维mKP，及含离散变量的1＋1维导数Toda，2＋1维导数Toda方程的参数解，给出了2＋1维mKP，2＋1维导数Toda方程的有限亏格解。本文采用Lax对非线性化方法，从基本恒等式出发，将特征值问题非线性化，得到Bargmann映射，辛映射及Moser矩阵。用幂级数方法，得到守恒积分。在此基础上将流拉直，证明守恒积分的对合性及独立性，得到系统的可积性。同时，将所得非线性发展方程分解，利用Abel-Jacobi坐标得到方程的有限参数解，最后利用Abel反演，求出2＋1维方程的有限亏格解。更多还原

【Abstract】 Two KN spectral problems (positive , negative) and a discrete spectral problem are investigated in this thesis. Our discussion is mainly focus on the integrable decomposition of the soliton equations derived from the spectral problems. Soliton equations of the KN isospectral hierarchy, such as dNS, dmKdV, and 2+1mKP are obtained. Meanwhile, equations with a discrete variable are presented, including 1+1 dToda, 2+1 dToda. Furthermore, all equations are given finite parameter solutions, and finite genus solutions of two 2+1 dimensional equations are gotten.Based on the fundamental identity, the eigenvalue problem is nonlinearized through the Lax pair nonlinearization technique, Bargmann mapping, symplectic mapping and Moser matrix are obtained. Under the help of power series, conserved integrables are obtained, then continuous and discrete flows are straightened, the involutivity and independence of conserved integrables are proved, thus the nonlinear systems we get are integrable. And the nonlinear evolutionary equations are decomposed, their Abel-Jacobi solutions are obtained. In the end, finite genus solutions of 2+1 dimensional soliton equations are given.更多还原