# A connection between the classical r-matrix formalism and covariant Hamiltonian field theory

## TL;DR

This paper establishes a covariant Poisson bracket framework for integrable field theories, revealing the classical r-matrix structure's covariant nature across spacetime variables, demonstrated through key models like sine-Gordon and NLS.

## Contribution

It introduces the classical r-matrix within a covariant Poisson bracket for the Lax connection, unifying integrability structures with spacetime covariance.

## Key findings

- Classical r-matrix appears in covariant Poisson brackets, indicating integrability.
- Demonstrated with sine-Gordon, NLS, and mKdV models.
- Shows the covariant r-matrix structure contrasts with traditional single-time Hamiltonian formalism.

## Abstract

We bring together aspects of covariant Hamiltonian field theory and of classical integrable field theories in $1+1$ dimensions. Specifically, our main result is to obtain for the first time the classical $r$-matrix structure within a covariant Poisson bracket for the Lax connection, or Lax one form. This exhibits a certain covariant nature of the classical $r$-matrix with respect to the underlying spacetime variables. The main result is established by means of several prototypical examples of integrable field theories, all equipped with a Zakharov-Shabat type Lax pair. Full details are presented for: $a)$ the sine-Gordon model which provides a relativistic example associated to a classical $r$-matrix of trigonometric type; $b)$ the nonlinear Schr\"odinger equation and the (complex) modified Korteweg-de Vries equation which provide two non-relativistic examples associated to the same classical $r$-matrix of rational type, characteristic of the AKNS hierarchy. The appearance of the $r$-matrix in a covariant Poisson bracket is a signature of the integrability of the field theory in a way that puts the independent variables on equal footing. This is in sharp contrast with the single-time Hamiltonian evolution context usually associated to the $r$-matrix formalism.