On the symmetric formulation of the Painleve IV equation

TL;DR

This paper introduces a symmetric framework for the Painleve IV equation that connects Hamiltonian formalism with the symmetric form, utilizing pseudo-differential Lax formalism and affine Weyl group symmetries.

Contribution

It presents a novel approach linking Hamiltonian formalism and symmetric Painleve IV, employing Darboux-Backlund transformations and affine Weyl group actions.

Findings

Symmetries of Painleve IV are characterized in a new symmetric framework.

Darboux-Backlund transformations are interpreted as maps between Hamiltonian solutions.

Affine Weyl group actions are realized through 'square-root' transformations.

Abstract

Symmetries and solutions of the Painleve IV equation are presented in an alternative framework which provides the bridge between the Hamiltonian formalism and the symmetric Painleve IV equation. This approach originates from a method developed in the setting of pseudo-differential Lax formalism describing AKNS hierarchy with the Darboux-Backlund and Miura transformations. In the Hamiltonian formalism the Darboux-Backlund transformations are introduced as maps between solutions of the Hamilton equations corresponding to two allowed values of Hamiltonian's discrete parameter. The action of the generators of the extended affine Weyl group of the $A_2$ root system is realized in terms of three "square-roots" of such Darboux-Backlund transformations defined on a multiplet of solutions of the Hamilton equations.