The Q1 lattice equation, a member in the Adler–Bobenko–Suris list of 3D consistent lattices, is investigated. By using the multidimensional consistency, a novel Lax pair for Q1 equation is given, which can be nonlinearized to produce integrable symplectic maps. Consequently, a Riemann theta function expression for the discrete potential is derived with the help of the Baker–Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps.

中文翻译：

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通过非线性可积辛映射对 Q1 晶格方程进行代数几何积分

研究了 Q1 晶格方程，它是 3D 一致晶格的 Adler-Bobenko-Suris 列表中的成员。通过使用多维一致性，给出了 Q1 方程的一个新的 Lax 对，它可以被非线性化以产生可积辛映射。因此，离散势的黎曼 theta 函数表达式是在 Baker-Akhiezer 函数的帮助下导出的。该表达式基于从可积辛映射迭代生成的离散相流的交换性，导致 Q1 晶格方程的代数几何积分。