A multisymplectic approach to defects in integrable classical field theory
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.
KeywordsIntegrable Field Theories Integrable Hierarchies
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- N.R. Jungwirth et al., A single-molecule approach to ZnO defect studies: single photons and single defects, arXiv:1402.1773.
- F. Nabarro, Theory of crystaline dislocation, Clarendon Press, Oxford U.K. (1967).Google Scholar
- T. Lubensky et al., Topological defects and interactions in Nematic emulsions, cond-mat/9707133.
- D. Vollhardt and C. Wolfe, The phases of Helium 3, Taylor & Francis (1990).Google Scholar
- A. Doikou, A note on GLN type-I integrable defects, J. Stat. Mech. (2014) P02002.Google Scholar
- T. De Donder, Théorie invariante du calcul des variations, Gauthier-Villars, Paris France (1935).Google Scholar
- E.K. Sklyanin, On complete integrability of the Landau-Lifshitz equation, LOMI E-79-3 (1980).
- E. Sklyanin, Bäcklund transformations and Baxters Q-operator, nlin/0009009.