Generalized eikonal knots and new integrable dynamical systems

Journal of Mathematical Physics

Vázquez

et al. 2006

Self Cite

We study soliton solutions of the Nicole model - a non-linear four-dimensional field theory consisting of the CP^1 Lagrangian density to the non-integer power 3/2 - using an ansatz within toroidal coordinates, which is indicated by the conformal symmetry of the static equations of motion. We calculate the soliton energies numerically and find that they grow linearly with the topological charge (Hopf index). Further we prove this behaviour to hold exactly for the ansatz. On the other hand, for the full three-dimensional system without symmetry reduction we prove a sub-linear upper bound, analogously to the case of the Faddeev-Niemi model. It follows that symmetric solitons cannot be true minimizers of the energy for sufficiently large Hopf index, again in analogy to the Faddeev-Niemi model.Comment: Latex, 35 pages, 1 figur

Section: Infinitely Many Conserved Currentsmentioning

confidence: 99%

Investigation of the Nicole model

Journal of Mathematical Physics

Vázquez

et al. 2006

Self Cite

“…The first case provides the integrability condition for the integrable submodels of the Faddeev-Niemi, Nicole and Baby Skyrme model, as was discussed, e.g., in [1], [9]. The second case provides the generalized integrability conditions which were introduced by Wereszczyński in [10], as we shall discuss in the next section. Observe that the first condition, the complex eikonal equation, is model independent, whereas the second condition F b = 0 depends on the model, i.e., on the Lagrangian.…”

Section: Conserved Currentsmentioning

confidence: 99%

“…For details on the pertinent geometry and topology we refer, e.g., to [13], [14], [15]. Now we proceed in two steps, analogously to the calculation in [10]. Firstly, we insert this ansatz into the integrability condition Eq.…”

Section: Soliton Models Of Wereszczyńskimentioning

confidence: 99%

See 1 more Smart Citation

Generalized integrability conditions and target space geometry

2005

Physics Letters B

In some higher dimensional nonlinear field theories integrable subsectors with infinitely many conservation laws have been identified by imposing additional integrability conditions. Originally, the complex eikonal equation was chosen as integrability condition, but recently further generalizations have been proposed. Here we show how these new integrability conditions may be derived from the geometry of the target space and, more precisely, from the Noether currents related to a certain class of target space transformations.Comment: 14 pages, Late

“…Higher solitons have to be calculated numerically [7]. Some generalization of the Nicole model have been discussed in [8], [9], where again only one analytical soliton solution could be found in each model. Both the AFZ model and the Nicole model (and their generalizations) allow for static finite energy solutions because their kinetic term is chosen non-polynomial in order to have a scale invariant energy and avoid Derrick's theorem.…”

Section: Introductionmentioning

confidence: 99%

Pullback of the volume form, integrable models in higher dimensions and exotic textures

Klimas

Journal of Mathematical Physics

et al. 2009

Self Cite

On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification A procedure allowing for the construction of Lorentz invariant integrable models living in d + 1 dimensional space time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized zero curvature formulation and infinitely many conserved quantities. A close relation between the Lagrange density of the integrable models and the pullback of the pertinent volume form on target space is established. Moreover, we show that the conserved currents are Noether currents generated by the volume-preserving diffeomorphisms. Further, we show how such models may emerge via Abelian projection of some gauge theories. Then we apply this framework to the construction of integrable models with exotic textures. Particularly, we consider integrable models providing exact suspended Hopf maps, i.e., solitons with a nontrivial topological charge of 4 ͑S 3 ͒ХZ 2 . Finally, some families of integrable models with solitons of n ͑S n ͒ type are constructed. Infinitely many exact solutions with arbitrary value of the topological index are found. In addition, we demonstrate that they saturate a Bogomolny bound.