基于三角曲线理论的高维可积系统的有限亏格解

As one of the most important branches of modern nonlinear science, integrable systems are able to describe nonlinear phenomenon and reveal its nature in various scientific and engineering fields. Among the research on them, searching for the exact analytic solutions and establishing a systematical approach have always been a significant one. Motivated by the complexity of multidimensional integrable systems associated with third-order matrix spectral problems and insufficient investigation on their algebro-geometric integration, we shall take the idea of decomposition and apply the theory of the trigonal curves defined by the characteristic polynomial of Lax matrices to construct finite genus solution of the multidimensional integrable systems with the help of some algebro-geometric tools, such as Baker-Akhiezer functions, meromorphic functions. And then we expect to develop a systematical and effective method. Based on these, we try to extend the algebro-geometric approach of the classical integrable systems to discuss the finite genus solutions for super integrable systems.

可积系统作为现代非线性科学的重要分支之一，可以描述许多科学和工程领域的非线性现象，进而揭示其本质。寻找精确解析解并建立系统的求解方法一直是可积系统的主要研究内容之一。针对与3阶矩阵谱问题相联系的高维可积系统本身的复杂性及其代数几何积分缺乏系统研究，本项目拟应用高维分解思想和Lax矩阵特征多项式定义的三角曲线理论，借助Baker-Akhiezer函数、亚纯函数等代数几何工具，构造与3阶矩阵谱问题相联系的高维可积系统的有限亏格解，并最终形成一套系统有效的方法。在此基础上，将经典可积系统的代数几何积分方法推广到超可积系统，探讨其有限亏格解。

随着科技的不断发展，越来越多的非线性可积模型出现在自然科学和工程领域。从应用的角度出发，最好是能给出这些非线性可积系统的精确解。本项目应用三角曲线理论和代数几何工具，研究了几个与3阶矩阵谱问题相联系的1+1维可积方程族的有限亏格解（代数几何解或拟周期解）。进一步，探讨了三角曲线理论在高维和超可积系统的应用,但由于高维分解及超可积系统谱曲线的复杂性，目前暂未得到理想的成果。此外，利用Darboux变换、Bäcklund变换及互反变换，讨论了一些经典可积方程和CH型方程的精确解，包括光滑孤子解、圈孤子解、怪波解等。