The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting. The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

中文翻译：

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具有意外渐近行为的二维 Toda 晶格和类粒子解的解公式

本文的第一个主要目的是根据三个独立的算子参数推导出标量二维 Toda 格子的显式求解公式，改进 [31] 中的工作。这是通过研究 2d-Toda 格的非对易版本，将其孤子解推广到非对易设置来实现的。应用程序部分的目的是展示从矩阵数据获得的解决方案族表现出丰富多样的渐近行为。第一个指标是，在文献中广泛研究的网络结构（参见 [4] 和其中的参考资料）是一个子家族。然后构造另外三类解（具有越来越不寻常的行为），并推导出它们的渐近线。