Integrable Origins of Higher Order Painleve Equations

TL;DR

This paper explores the derivation of higher order Painleve equations from pseudo-differential Lax hierarchies, revealing their integrable structure and symmetry properties related to affine Weyl groups.

Contribution

It introduces a novel method to obtain higher order Painleve equations via self-similarity limits of Lax hierarchies with affine Weyl group symmetries.

Findings

Higher order Painleve equations are derived from Lax hierarchies.

The equations exhibit invariance under extended affine Weyl groups.

The approach links integrable systems with algebraic symmetry structures.

Abstract

Higher order Painleve equations invariant under extended affine Weyl groups $A^{(1)}_n$ are obtained through self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized Volterra lattice structure.