The zero curvature representation for two dimensional integrable models is generalized to spacetimes of dimension d + 1 by the introduction of a d-form connection. The new generalized zero curvature conditions can be used to represent the equations of motion of some relativistic invariant field theories of physical interest in 2 + 1 dimensions (BF theories, Chern-Simons, 2 + 1 gravity and the CP 1 model) and 3 + 1 dimensions (self-dual Yang-Mills theory and the Bogomolny equations). Our approach leads to new methods of constructing conserved currents and solutions. In a submodel of the 2+1 dimensional CP 1 model, we explicitly construct an infinite number of previously unknown nontrivial conserved currents. For each positive integer spin representation of sl(2) we construct 2j +1 conserved currents leading to 2j + 1 Lorentz scalar charges.

We use the deformed sine-Gordon models recently presented by Bazeia et al to discuss possible definitions of quasi-integrability. We present one such definition and use it to calculate an infinite number of quasi-conserved quantities through a modification of the usual techniques of integrable field theories. Performing an expansion around the sine-Gordon theory we are able to evaluate the charges and the anomalies of their conservation laws in a perturbative power series in a small parameter which describes the "closeness" to the integrable sine-Gordon model. Our results indicate that in the case of the two-soliton scattering the charges are conserved asymptotically, i.e. their values are the same in the distant past and future, when the solitons are well separated. We back up our results with numerical simulations which also demonstrate the existence of long lived breather-like and wobble-like states in these models.Comment: 40 pages, 47 ps figure

We investigate higher grading integrable generalizations of the affine Toda systems, where the flat connections defining the models take values in eigensubspaces of an integral gradation of an affine Kac-Moody algebra, with grades varying from l to −l (l > 1). The corresponding target space possesses nontrivial vacua and soliton configurations, which can be interpreted as particles of the theory, on the same footing as those associated to fundamental fields. The models can also be formulated by a Hamiltonian reduction procedure from the so called two-loop WZNW models. We construct the general solution and show the classes corresponding to the solitons. Some of the particles and solitons become massive when the conformal symmetry is spontaneously broken by a mechanism with an intriguing topological character and leading to a very simple mass formula. The massive fields associated to non zero grade generators obey field equations of the Dirac type and may be regarded as matter fields. A special class of models is remarkable. These theories possess a U (1) Noether current which, after a special gauge fixing of the conformal symmetry, is proportional to a topological current. This leads to the confinement of the matter field inside the solitons, which can be regarded as a one dimensional bag model for QCD. These models are also relevent to the study of electron self-localization in (quasi)-one-dimensional electron-phonon systems.

We consider modifications of the nonlinear Schrödinger model (NLS) to look at the recently introduced concept of quasi-integrability. We show that such models possess an infinite number of quasi-conserved charges which present intriguing properties in relation to very specific space-time parity transformations. For the case of two-soliton solutions where the fields are eigenstates of this parity, those charges are asymptotically conserved in the scattering process of the solitons. Even though the charges vary in time their values in the far past and the far future are the same. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. Our findings may have important consequences on the applications of these models in several areas of non-linear science. We make a detailed numerical study of the modified NLS potential of the form V ∼ | ψ | 2 2+ε , with ε being a perturbation parameter. We perform numerical simulations of the scattering of solitons for this model and find a good agreement with the results predicted by the analytical considerations. Our paper shows that the quasi-integrability concepts recently proposed in the context of modifications of the sine-Gordon model remain valid for perturbations of the NLS model. arXiv:1206.5808v1 [hep-th]