From KP/UC hierarchies to Painleve equations

TL;DR

This paper explores the deep connection between Painleve equations and integrable hierarchies like KP and UC, revealing how specific reductions lead to Painleve equations and their properties.

Contribution

It demonstrates that homogeneity and periodicity reductions of KP and UC hierarchies produce Painleve equations, clarifying their Lax formalism, symmetries, and solutions.

Findings

Reduction of hierarchies yields Painleve equations.

Provides explicit descriptions of Lax formalism and symmetries.

Constructs algebraic solutions using Schur functions.

Abstract

We study the underlying relationship between Painleve equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painleve equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g., Lax formalism, Hirota bilinear relations for tau-functions, Weyl group symmetry, and algebraic solutions in terms of the character polynomials, i.e., the Schur function and the universal character.