Linear $r$-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

TL;DR

This paper develops a linear r-matrix algebra framework for multi-particle systems with polynomial potentials separable in parabolic coordinates, utilizing Lax representations and dynamical r-matrices.

Contribution

It introduces a novel r-matrix algebra with variable-dependent r-matrices for systems separable in parabolic coordinates, advancing integrable systems theory.

Findings

Constructed a hierarchy of Lax representations for the systems.

Derived the associated dynamical r-matrix algebra.

Discussed the dynamical Yang-Baxter equation in this context.

Abstract

We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of $2\times 2$ matrices for the whole hierarchy, we construct the associated linear $r$-matrix algebra with the $r$-matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed.