代数曲线在与高阶矩阵谱问题相联系的可积系统中的应用

Based on the theory of algebraic curves, the project is devoted to study the algebro-geometric solutions of integrable systems associated with high order matrix spectral problems. On one hand, we discuss algebro-geometric solutions of integrable systems associated with different kinds of 3×3 discrete matrix spectral problems to find the new case where the associated Riemann surface has one infinite point. On the other hand, we construct algebro-geometric solutions of integrable systems associated with 4×4 matrix spectral problems by means of the theory of tetragonal curves. It mainly focuses on the following aspects: (1) Construct hierarchies of integrable equations associated with high order matrix spectral problems; (2) Introduce the algebraic curves through the characteristic polynomials of the Lax matrices and define the Baker-Akhiezer functions and appropriate meromorphic functions on the associated compact Riemann surfaces; (3) On the basis of divisors, asymptotic properties and Riemann-Roch theorem, Riemann theta function representations of meromorphic functions and Baker-Akhiezer functions are obtained, from which we can yield the algebro-geometric solutions to hierarchies of integrable equations. The research has important theoretical significance in finding new integrable systems, enriching and improving the application of algebraic curves in integrable systems.

本项目主要基于代数曲线理论研究与高阶矩阵谱问题相联系的可积系统的代数几何解。一方面，考虑与不同类型的3×3离散矩阵谱问题相联系的可积系统的代数几何解，研究三角曲线具有一个无穷远点的新情况。另一方面，利用四方曲线的理论构造与4×4矩阵谱问题相联系的可积系统的代数几何解。研究内容包括：(1)从高阶矩阵谱问题出发，构造可积方程族；(2)借助Lax矩阵的特征多项式引入代数曲线，在其紧化的Riemann面上定义Baker-Akhiezer函数和合适的亚纯函数；(3)通过分析其因子和渐近性质，利用Riemann-Roch定理构造亚纯函数和Baker-Akhiezer函数的Riemannθ函数表示，从而给出整个可积方程族的代数几何解。本项目对于寻找新的可积系统，丰富和完善代数曲线在可积系统中的应用具有重要的理论意义。

本项目主要对可积系统代数几何解的构造和孤子、呼吸子、怪波等局域波解的求解展开研究。基于代数曲线理论和达布变换方法，得到了如下的结果：1. 借助Lax矩阵的特征多项式引入代数曲线，在其紧化的Riemann面上定义Baker-Akhiezer函数和合适的亚纯函数，得到若干与高阶矩阵谱问题相联系的可积方程族的代数几何解，如Itoh-Narita-Bogoyavlensky晶格族、广义Merola-Ragnisco-Tu晶格族等；2. 利用广义达布变换方法和调制不稳定性分析，研究了若干非线性可积方程的怪波解、呼吸子解和孤子解，如三层耦合Maxwell-Bloch方程、(3+1)维KdV方程等。本项目对于寻找新的可积系统，丰富和完善代数曲线在可积系统中的应用具有重要的理论意义。

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