The Sine-Gordon Solitons as a N-Body Problem

TL;DR

This paper models N-soliton solutions of the sine-Gordon equation as an N-body problem, revealing a relativistic generalization of the Calogero model with a quadratic Poisson bracket and a dynamical r-matrix.

Contribution

It introduces a relativistic N-body model based on sine-Gordon solitons, extending the Calogero model with new Poisson bracket and r-matrix structures.

Findings

Quadratic Poisson bracket of the Lax matrix.

Dynamical r-matrix in the model.

Contrast with the linear Poisson bracket in Calogero model.

Abstract

We consider the N-soliton solutions in the sine-Gordon model as a N-body problem. This leads to a relativistic generalization of the Calogero model first introduced by Ruijsenaars. We show that the fundamental Poisson bracket of the Lax matrix is quadratic, and the $r$-matrix is a dynamical one. This is in contrast to the Calogero model where the fundamental Poisson bracket of the Lax matrix is linear.