三角曲线与孤子族的代数几何解
Trigonal curves and algebro-geometric solutions to soliton hierarchies
摘要：
利用孤子族的Lax矩阵的线性组合，通过特征方程引入三角曲线，确定了定义为Baker-Akhiezer函数比的亚纯函数的零点和极点的Dubrovin型方程.在与拉克斯对相关的Abel-Jacobi坐标下，我们理顺了孤子族中的所有流，并通过观察Baker-Akhiezer函数的渐近行为，得到了孤子族的Riemann θ函数的代数几何解.我们分析了四分量AKNS孤子族的方式，它导致三角曲线适用于任意孤子族的代数几何解决方案的建设的一般理论。
英文摘要：
Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
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