Solitons and Almost-Intertwining Matrices

TL;DR

This paper introduces almost-intertwining matrices and provides a simple formula for KP hierarchy tau-functions, connecting soliton solutions to eigenvalue dynamics and integrable particle systems.

Contribution

It defines almost-intertwining matrices and links them to KP tau-functions, soliton solutions, and eigenvalue dynamics, revealing new connections in integrable systems.

Findings

Tau-functions expressed via triples of matrices

Includes soliton and rational solutions of KP hierarchy

Eigenvalue dynamics relate to Ruijsenaars-Schneider system

Abstract

We define the set of almost-intertwining matrices to be all triples(X,Y,Z) of n x n matrices for which XZ=YX+T for some rank one matrix T. A surprisingly simple formula is given for tau-functions of the KP hierarchy in terms of such triples. The tau-functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of the matrix X are considered, leading in special cases to the Ruijsenaars-Schneider particle system.