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

Understand

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.