A multisymplectic approach to defects in integrable classical field theory

TL;DR

This paper develops a multisymplectic formalism for integrable classical field theories with defects, introducing a second Poisson bracket for space and demonstrating how it simplifies the analysis of defects and Liouville integrability.

Contribution

It introduces a novel multisymplectic approach with dual Poisson brackets, enabling a new interpretation of defects as canonical transformations and proving integrability.

Findings

The new Poisson bracket simplifies defect analysis.

Defects can be interpreted as canonical transformations.

Liouville integrability is proven in the presence of defects.

Abstract

We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1+1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schr\"odinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen B\"acklund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.