Lax representation of the hyperbolic van Diejen dynamics with two coupling parameters

TL;DR

This paper develops a Lax pair for the hyperbolic van Diejen system with two coupling parameters, enabling explicit solutions and analysis of scattering, and proposes a candidate for a three-parameter extension.

Contribution

It constructs a Lax representation for the two-parameter hyperbolic van Diejen system and explores its integrability and scattering properties, also suggesting a Lax matrix for the three-parameter case.

Findings

Lax pair constructed for the two-parameter system

Dynamics solvable via projection method

First integrals match van Diejen's Hamiltonians

Abstract

In this paper, we construct a Lax pair for the classical hyperbolic van Diejen system with two independent coupling parameters. Built upon this construction, we show that the dynamics can be solved by a projection method, which in turn allows us to initiate the study of the scattering properties. As a consequence, we prove the equivalence between the first integrals provided by the eigenvalues of the Lax matrix and the family of van Diejen's commuting Hamiltonians. Also, at the end of the paper, we propose a candidate for the Lax matrix of the hyperbolic van Diejen system with three independent coupling constants.