摘要
本论文研究了三个全离散可积模型和一个(2+1)维导数Schwarzian KdV方程.主要内容可分为三部分：
其一,讨论Adler-Bobenko-Suris列表中的方程H1(离散势KdV方程)和特殊H3模型(离散势MKdV方程).给出了它们的Lax对,经非线性化得到一些可积辛映射,运用这些映射和同一个Liouville可积平台上之离散相流的可换性,求得H1模型和特殊H3模型的有限亏格解,并得到离散KdV方程(由Nijhoff给出)之特解的theta函数表达式.
其二,通过一个离散谱问题的非线性化推导出由Veselov给出的离散Neumann模型,基于装配了有限亏格位势的Lax矩阵和Darboux矩阵的交换关系,借助于Baker-Akhiezer-Kriechever函数求出方程的一个特解.
其三,构造出两个(1+1)维导数Schwarzian KdV方程的零曲率表示,二者相容的结果即为一个(2+1)维导数Schwarzian KdV方程.由Lax对的非线性化和Hamilton相流的拉直,求得方程的有限参数特解和Abel-Jacobi解.
This thesis investigates three fully discrete integrable models and a (2+1)-dimensional derivative Schwarzian KdV equation. It consists chiefly of three parts:
Firstly, the H1model and the special H3model in the Adler-Bobenko-Suris list, i.e. the lattice potential KdV equation and the lattice potential MKdV equation are discussed. New Lax pairs of them are given, by which integrable symplectic maps are constructed through a non-linearization procedure. Resorting to these maps and the permutability of the discrete phase flows sharing the same Liouville platform, finite genus solutions of the H1model as well as the special H3model are calculated. Besides, a special solution expressed with theta function of the lattice KdV equation with Nijhoff's discretization is also arrived.
Secondly, the Veselov's discrete Neumann system is derived through non-linearization of a discrete spectral problem. Based on the commutation relation between the Lax matrix and the Darboux matrix with finite genus potentials, a special solution is obtained with the help of the Baker-Akhiezer-Kriechever function.
Finally, the zero-curvature expressions of two (1+1)-dimensional derivative Schwarzian KdV equations are constructed. According to there compatibility, a (2+1)-dimensional derivative Schwarzian KdV equation is got.Through non-linearization of the Lax pairs and straightening out of the Hamilton phase flows, the special solutions with finite pa-rameters and the Abel-Jacobi solutions are calculated.
引文
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