发表机构
Osaka Central Advanced Mathematical Institute; Osaka Metropolitan University（大阪中央高等数学研究所; 大阪公立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究将Fu和Nijhoff的KP层级直接线性化方案与佐藤格拉斯曼流形几何结合，扩展至双分量KP层级，揭示了柯西矩阵方法的几何解释及多分量推广的关联结构。
AI 中文摘要
Fu和Nijhoff针对KP层级及其约化的直接线性化方案，采用了含二次非线性项的无限矩阵U的演化方程组。U的部分矩阵元可被识别为佐藤格拉斯曼流形顶层胞腔的仿射坐标w_ij，这些矩阵元的演化方程与几何意义下代表KP层级的w_ij演化方程完全一致。通过向U的演化方程引入负流，该几何解释可扩展至U的其他矩阵元，扩展后的方程组与双分量KP层级本质等价。KP层级的柯西矩阵方法可在该几何视角下得到解释，Fu和Nijhoff非线性系统的多分量推广与AKNS、ASDYM层级相关，其中多分量佐藤格拉斯曼流形作为相关几何结构出现。
英文摘要
Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.
Comments62pages, no figure; (v2) many typos are corrected, e.g. definition of Omega, Sylvester equation in p.12, definition of S_k in p.38, eqs. (7), (32), (39), (48), (49), (70), (80), last equation in p.52, fourth equation of p.58, etc
