论文标题
基质孤子的热带限制和纠缠杨巴克斯特地图
Tropical limit of matrix solitons and entwining Yang-Baxter maps
论文作者
论文摘要
我们考虑一个基质重构问题，即，对于Yang-baxter映射，它是矩阵KP方程中“纯” 2-Soliton解的偏振图。使用LAX矩阵及其逆，相关的重构问题决定了另一个映射，这不是Yang-Baxter方程的解决方案，而是满足Yang-Baxter方程的混合版本以及Yang-Baxter地图。在最近的工作中，此类地图被称为“纠缠Yang-Baxter地图”。实际上，从基质KP方程的纯2-Soliton溶液中获得的极化图，并且已经用于矩阵KDV还原，通常不是Yang-baxter映射，但由两个地图或其倒置中的一个描述。我们阐明了为什么Yang-Baxter方程较弱的版本会通过在“热带极限”中探索纯的3-Soliton解决方案，其中3-索顿相互作用分解为2-Soliton的相互作用。在这里，通过二维Toda晶格方程的矩阵概括产生的纯孤子溶液详细阐述了纯孤子溶液，在此中，我们遇到了与KP情况相同的Yang-Baxter地图，表明了一种通用性。
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.
