We consider a matrix refactorization problem, i.e., a "Lax representation", for the Yang-Baxter map that originated as the map of polarizations from the "pure" 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang-Baxter equation, but satisfies a mixed version of the Yang-Baxter equation together with the Yang-Baxter map. Such maps have been called "entwining Yang-Baxter maps" in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is NOT in general a Yang-Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang-Baxter equation holds, by exploring the pure 3-soliton solution in the "tropical limit", where the 3-soliton interaction decomposes into 2-soliton interactions. Here this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang-Baxter maps as in the KP case, indicating a kind of universality.

中文翻译：

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矩阵孤子的热带极限和缠绕的杨-巴克斯特图

我们考虑一个矩阵重构问题，即“松散表示”，对于 Yang-Baxter 映射，它起源于矩阵 KP 方程的“纯”2-孤子解的极化映射。使用 Lax 矩阵及其逆矩阵，相关的重构问题确定另一个映射，该映射不是 Yang-Baxter 方程的解，但满足 Yang-Baxter 方程与 Yang-Baxter 映射的混合版本。这种地图在最近的工作中被称为“缠绕杨-巴克斯特地图”。事实上，从矩阵 KP 方程的纯 2-孤子解获得的极化图，并且已经用于矩阵 KdV 约简，通常不是杨-巴克斯特图，但它由两个图之一描述或他们的逆。我们澄清了为什么杨-巴克斯特方程的较弱版本成立，通过探索“热带极限”中的纯 3-孤子解，其中 3-孤子相互作用分解为 2-孤子相互作用。这里详细阐述了通过二元 Darboux 变换生成的二维 Toda 晶格方程的矩阵推广的纯孤子解，在那里我们遇到了与 KP 情况相同的缠绕 Yang-Baxter 映射，表明了一种普遍性。