Multisymplectic approach to integrable defects in the sine-Gordon model

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 Bäcklund 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.

arXiv comment23 pages, some typos corrected and references updated, journal version

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• A connection between the classical r-matrix formalism and covariant Hamiltonian field theory
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• Integrable crosscaps in classical sigma models
• A Dual Construction of the Isotropic Landau-Lifshitz Model
• Matrix representations of multidimensional integral and ergodic operators
• New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations