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

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.

arXiv comment38 pages

References in corpus (24)

• The Sine-Gordon Solitons as a N-Body Problem
• The boundary sine-Gordon theory: classical and semi-classical analysis
• Spectrum of Quantum Transfer Matrices via Classical Many-Body Systems
• Relativistic Classical Integrable Tops and Quantum R-matrices
• Random matrix ensembles associated with Lax matrices
• Poisson-Lie interpretation of trigonometric Ruijsenaars duality
• Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction
• On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models
• A class of Calogero type reductions of free motion on a simple Lie group
• Deformations of Calogero-Moser Systems
• High values of disorder-generated multifractals and logarithmically correlated processes
• Rational Top and its Classical R-matrix
• The hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars-Schneider-van Diejen models: Lax matrices and duality
• The master T-operator for the Gaudin model and the KP hierarchy
• Trigonometric version of quantum-classical duality in integrable systems
• Integrable random matrix ensembles
• Spin Calogero models associated with Riemannian symmetric spaces of negative curvature
• On the non-relativistic limit of the quantum sine-Gordon model with integrable boundary condition
• Trigonometric Sutherland systems and their Ruijsenaars duals from symplectic reduction
• Supersymmetric quantum spin chains and classical integrable systems
• Action-angle duality between the C(n)-type hyperbolic Sutherland and the rational Ruijsenaars-Schneider-van Diejen models
• Duality between the trigonometric BC(n) Sutherland system and a completed rational Ruijsenaars-Schneider-van Diejen system
• Scattering theory of the hyperbolic BC(n) Sutherland and the rational BC(n) Ruijsenaars--Schneider--van Diejen models
• On the classical r-matrix structure of the rational BC(n) Ruijsenaars-Schneider-van Diejen system

Cited by in corpus (9)

• Quantum Lax pairs via Dunkl and Cherednik operators
• On factorized Lax pairs for classical many-body integrable systems
• Trigonometric and elliptic Ruijsenaars-Schneider systems on the complex projective space
• Global description of action-angle duality for a Poisson-Lie deformation of the trigonometric ${\mathrm{BC}}_n$ Sutherland system
• Quantum-classical duality for Gaudin magnets with boundary
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• Quantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
• Dunkl and Cherednik operators
• Lax matrices for a 1-parameter subfamily of van Diejen--Toda chains