A multisymplectic approach to defects in integrable classical field theory

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 Schrödinger (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 Bäcklund 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.

17 pages, final version with titles for the references

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• Darboux-Backlund transformations, dressing & impurities in multi-component NLS
• Type-II super-Backlund transformation and integrable defects for the N=1 super sinh-Gordon model
• A connection between the classical r-matrix formalism and covariant Hamiltonian field theory
• On the origin of dual Lax pairs and their $r$-matrix structure
• Multiform description of the AKNS hierarchy and classical r-matrix
• Hamiltonian multiform description of an integrable hierarchy
• Defects in the supersymmetric mKdV hierarchy via Backlund transformations
• Jumps and twists in affine Toda field theories
• Integrability of generalised type II defects in affine Toda field theory
• Integrable crosscaps in classical sigma models
• A Dual Construction of the Isotropic Landau-Lifshitz Model
• An algebraic approach to discrete time integrability
• The Ablowitz-Ladik system on a graph
• On the nonlinear Schrödinger equation with a time-dependent boundary condition
• Time-dependent defects in integrable soliton equations