Faces of Relativistic Toda Chain

TL;DR

This paper reveals that the relativistic Toda chain is a reduction of the 2D Toda hierarchy, establishing connections to gauge equivalence with the discrete AKNS hierarchy and applications in biorthogonal polynomials.

Contribution

It demonstrates the reduction of the relativistic Toda chain to the 2D Toda hierarchy and explores its integrable properties and applications in polynomial systems.

Findings

RTC is a special reduction of 2DTL

RTC is gauge equivalent to discrete AKNS hierarchy

Applications to biorthogonal polynomial systems

Abstract

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of: Lax representation, orthogonal polynomial systems, and $\tau$-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed.