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

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 -matrix structure within a covariant Poisson bracket for the Lax connection, or Lax one form. This exhibits a certain covariant nature of the classical -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 Schrödinger 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.