Multisymplectic approach to integrable defects in the sine-Gordon model

TL;DR

This paper applies multisymplectic systems theory to the sine-Gordon model with defects, introducing a second Poisson bracket to unify and improve the Hamiltonian description of the model.

Contribution

It introduces a second Poisson bracket for the sine-Gordon model, enabling a unified Hamiltonian framework for integrable defects and resolving issues with Poisson brackets at the defect point.

Findings

Unified Hamiltonian description with a new Poisson bracket.

Resolved Poisson bracket evaluation issues at the defect.

Reinterpreted defect conditions via canonical transformations.

Abstract

Ideas from the theory of multisymplectic systems, introduced recently in integrable systems by the author and Kundu to discuss Liouville integrability in classical field theories with a defect, are applied to the sine-Gordon model. The key ingredient is the introduction of a second Poisson bracket in the theory that allows for a Hamiltonian description of the model that is completely equivalent to the standard one, in the absence of a defect. In the presence of a defect described by frozen B\"acklund transformations, our approach based on the new bracket unifies the various tools used so far to attack the problem. It also gets rid of the known issues related to the evaluation of the Poisson brackets of the defect matrix which involve fields at coinciding space point (the location of the defect). The original Lagrangian approach also finds a nice reinterpretation in terms of the canonical transformation representing the defect conditions.