Higher order Painlevé equations and their symmetries via reductions of a class of integrable models

H Aratyn,J F Gomes,A H Zimerman

doi:10.1088/1751-8113/44/23/235202

Abstract

Higher order Painlevé equations and their symmetry transformations belonging to extended affine Weyl groups A(1)n are obtained through a self-similarity limit of a class of pseudo-differential Lax hierarchies with symmetry inherited from the underlying generalized Volterra lattice structure. In particular, an explicit example of the Painlevé V equation and its Bäcklund symmetry is obtained through a self-similarity limit of a generalized KdV hierarchy from Aratyn et al (1995 Int. J. Mod. Phys. A 10 2537).

Highlights

The aim of this work is to explore integrable origins of higher order Painleve equations and their extended affine Weyl symmetry groups. Towards this goal this paper investigates a self-similarity limit of a special class of pseudo-differential Lax hierarchies of the constrained KP hierarchy with symmetry structure defined by Backlund transformations induced by a discrete structure of the Volterra type lattice
This is illustrated for the special cases of M = 1, 2 for which the generators of the extended affine Weyl symmetry group are derived from the Backlund transformation of Section 2
It is well-known that symmetries of many continuum KP-type hierarchies are governed by discrete lattice-like structures

Summary

Introduction

The aim of this work is to explore integrable origins of higher order Painleve equations and their extended affine Weyl symmetry groups Towards this goal this paper investigates a self-similarity limit of a special class of pseudo-differential Lax hierarchies of the constrained KP hierarchy with symmetry structure defined by Backlund transformations induced by a discrete structure of the Volterra type lattice. This is illustrated for the special cases of M = 1, 2 for which the generators of the extended affine Weyl symmetry group are derived from the Backlund transformation of Section 2.

A “half-integer” lattice

Basic facts about the 2M -bose constrained KP hierarchy

General Construction

Outlook