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Tzitzeica solitons versus relativistic Calogero–Moser three-body clusters

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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 Omega of the relativistic systems with 2N particles and N antiparticles, there exists a 2N-dimensional Poincare-invariant submanifold Omega(P) 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 Omega(P). This correspondence leads to a picture of the soliton as a cluster of two particles and one antiparticle in their lowest internal energy state.

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