Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.

中文翻译：

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长波-短波型方程的层次：解的准周期行为及其表示

基于 Lenard 递归关系和零曲率方程，我们推导出与具有三个势的 3 × 3 矩阵谱问题相关的长波-短波类型方程的层次结构。借助于Lax矩阵的特征多项式，定义了一条三角曲线，在该曲线上引入了Baker-Akhiezer函数和两个亚纯函数。分析亚纯函数的一些性质，包括无限点的渐近展开，我们得到了 Baker-Akhiezer 函数的本质奇点和因数。利用代数曲线理论，最终根据黎曼 theta 函数推导出整个层次结构的准周期解。