The Heisenberg hierarchy and its Hamiltonian structure are derived respectively by virtue of the zero-curvature equation and the trace identity. With the help of the Lax matrix, we introduce an algebraic curve $K_n$ of arithmetic genus n, from which we define meromorphic function ϕ and straighten out all of the flows associated with the Heisenberg hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit theta function representations of solutions for the whole Heisenberg hierarchy as a result of the asymptotic properties of ϕ.

中文翻译：

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海森堡层次结构的代数几何构造

借助零曲率方程和迹线身份分别推导了海森堡层次结构及其哈密顿结构。在Lax矩阵的帮助下，我们引入了代数曲线${ķ}_{ñ}$算术属n的元素，我们定义亚纯函数ϕ，并理顺与所有在Abel–Jacobi坐标下的Heisenberg层次结构相关的流。最后，我们实现了对整个海森堡层次结构的渐进性质的结果解决方案的明确的θ函数表示φ。