Grassmannians, Nonlinear Wave Equations and Generalized Schur Functions

TL;DR

This paper introduces generalized Schur functions linked to Grassmannians, providing a new approach to constructing solutions for the KP hierarchy of nonlinear PDEs, expanding tau-functions in terms of these functions.

Contribution

It generalizes Schur polynomials to new functions connected to Grassmannians, offering a novel method for solving KP equations.

Findings

Generalized Schur functions relate to Grassmannian geometry.

They enable expansion of tau-functions in KP hierarchy.

Coefficients satisfy Plücker relations.

Abstract

A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear partial differential equations. Specifically, just as the Schur polynomials are used to expand tau-functions as a sum, it is shown that it is natural to expand a quotient of tau-functions in terms of these generalized Schur functions. The coefficients in this expansion are found to be constrained by the Pl\"ucker relations of a grassmannian.