We consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to $\mathbb{Z}$-graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the $\mathbb{Z}$-graded graph.

中文翻译：

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哈密​​顿系统，Toda格，孤子，加权Z梯度图上的Lax对

我们考虑离散一维非线性方程，并提出将其提升为 $\mathbb{ž}$分级图。我们确定了允许人们将一维解决方案提升到图上的解决方案的条件。特别是，我们证明了分形分形图中静态势的孤子的存在。我们还表明，即使对于拓扑有趣图的简单示例，相应的非平凡Lax对和关联的ary变换也不会提升到图上的Lax对。$\mathbb{ž}$渐变图。