Direct linearization, Cauchy matrix and Sato Grassmannian
Abstract
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 wij of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of wij 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.
Related
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BibTeX
@misc{takasaki2026direct,
title = {{Direct linearization, Cauchy matrix and Sato Grassmannian}},
author = {Kanehisa Takasaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24538},
eprint = {2608.24538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24538},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Direct linearization, Cauchy matrix and Sato Grassmannian
AU - Kanehisa Takasaki
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2608.24538
UR - https://arxiv.org/abs/2608.24538
AB - 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.
ER -