The Heisenberg hierarchy and its Hamiltonian structure are obtained respectively by use of the zero curvature equation and the trace identity. With the help of the Lax matrix we introduce an algebraic curve \({\mathcal {K}}_{n}\) of arithmetic genus n, from which we define meromorphic function \(\phi \) and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we get the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of \(\phi \).

中文翻译：

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海森堡层次结构的准周期解

利用零曲率方程和迹线身份分别获得海森堡层次结构和哈密顿结构。借助于Lax矩阵，我们引入了算术类n的代数曲线\（{\ mathcal {K}} _ {n} \），从中定义了亚纯函数\（\ phi \）并理顺了所有在Abel–Jacobi坐标下与Heisenberg层次结构关联的流。最后，由于\（\ phi \）的渐近性质，我们获得了整个Heisenberg层次结构的解决方案的显式theta函数表示。