Tzitzeica solitons versus relativistic Calogero–Moser three-body clusters

Abstract

We establish a connection between the hyperbolic relativistic Calogero–Moser systems and a class of soliton solutions to the Tzitzeica equation (also called the Dodd–Bullough–Zhiber–Shabat–Mikhailov equation). In the 6N-dimensional phase space Ω of the relativistic systems with 2N particles and N antiparticles, there exists a 2N-dimensional Poincaré-invariant submanifold ΩP corresponding to N free particles and N bound particle-antiparticle pairs in their ground state. The Tzitzeica N-soliton tau functions under consideration are real valued and obtained via the dual Lax matrix evaluated in points of ΩP. This correspondence leads to a picture of the soliton as a cluster of two particles and one antiparticle in their lowest internal energy state.