37 research outputs found
Conservation laws in Skyrme-type models
The zero curvature representation of Zakharov and Shabat has been generalized
recently to higher dimensions and has been used to construct non-linear field
theories which either are integrable or contain integrable submodels. The
Skyrme model, for instance, contains an integrable subsector with infinitely
many conserved currents, and the simplest Skyrmion with baryon number one
belongs to this subsector. Here we use a related method, based on the geometry
of target space, to construct a whole class of theories which are either
integrable or contain integrable subsectors (where integrability means the
existence of infinitely many conservation laws). These models have
three-dimensional target space, like the Skyrme model, and their infinitely
many conserved currents turn out to be Noether currents of the
volume-preserving diffeomorphisms on target space. Specifically for the Skyrme
model, we find both a weak and a strong integrability condition, where the
conserved currents form a subset of the algebra of volume-preserving
diffeomorphisms in both cases, but this subset is a subalgebra only for the
weak integrable submodel.Comment: Latex file, 22 pages. Two (insignificant) errors in Eqs. 104-106
correcte
Soliton stability in some knot soliton models
We study the issue of stability of static soliton-like solutions in some non-linear field theories which allow for knotted field configurations. Concretely, we investigate the AFZ model, based on a Lagrangian quartic in first derivatives with infinitely many conserved currents, for which infinitely many soliton solutions are known analytically. For this model we find that sectors with different (integer) topological charge (Hopf index) are not separated by an infinite energy barrier. Further, if variations which change the topological charge are allowed, then the static solutions are not even critical points of the energy functional. We also explain why soliton solutions can exist at all, in spite of these facts. In addition, we briefly discuss the Nicole model, which is based on a sigma-model type Lagrangian. For the Nicole model we find that different topological sectors are separated by an infinite energy barrier
Investigation of the Nicole model
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
Integrable subsystem of Yang--Mills dilaton theory
With the help of the Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2)
Yang-Mills field, we find an integrable subsystem of SU(2) Yang-Mills theory
coupled to the dilaton. Here integrability means the existence of infinitely
many symmetries and infinitely many conserved currents. Further, we construct
infinitely many static solutions of this integrable subsystem. These solutions
can be identified with certain limiting solutions of the full system, which
have been found previously in the context of numerical investigations of the
Yang-Mills dilaton theory. In addition, we derive a Bogomolny bound for the
integrable subsystem and show that our static solutions are, in fact, Bogomolny
solutions. This explains the linear growth of their energies with the
topological charge, which has been observed previously. Finally, we discuss
some generalisations.Comment: 25 pages, LaTex. Version 3: appendix added where the equivalence of
the field equations for the full model and the submodel is demonstrated;
references and some comments adde
A BPS Skyrme model and baryons at large Nc
Within the class of field theories with the field contents of the Skyrme
model, one submodel can be found which consists of the square of the baryon
current and a potential term only. For this submodel, a Bogomolny bound exists
and the static soliton solutions saturate this bound. Further, already on the
classical level, this BPS Skyrme model reproduces some features of the liquid
drop model of nuclei. Here, we investigate the model in more detail and,
besides, we perform the rigid rotor quantization of the simplest Skyrmion (the
nucleon). In addition, we discuss indications that the viability of the model
as a low energy effective field theory for QCD is further improved in the limit
of a large number of colors N_c.Comment: latex, 23 pages, 1 figure, a numerical error in section 3.2
corrected; matches published versio
Investigation of restricted baby Skyrme models
A restriction of the baby Skyrme model consisting of the quartic and
potential terms only is investigated in detail for a wide range of potentials.
Further, its properties are compared with those of the corresponding full baby
Skyrme models. We find that topological (charge) as well as geometrical
(nucleus/shell shape) features of baby skyrmions are captured already by the
soliton solutions of the restricted model. Further, we find a coincidence
between the compact or non-compact nature of solitons in the restricted model,
on the one hand, and the existence or non-existence of multi-skyrmions in the
full baby Skyrme model, on the other hand.Comment: latex, 18 pages, 2 figures; some typos correcte
Comment on ``Reduction of static field equation of Faddeev model to first order PDE'', arXiv:0707.2207
The authors of the article Phys. Lett. B 652 (2007) 384, (arXiv:0707.2207),
propose an interesting method to solve the Faddeev model by reducing it to a
set of first order PDEs. They first construct a vectorial quantity , depending on the original field and its first derivatives, in terms of which
the field equations reduce to a linear first order equation. Then they find
vectors and which identically obey this linear
first order equation. The last step consists in the identification of the with the original as a function of the original field.
Unfortunately, the derivation of this last step in the paper cited above
contains an error which invalidates most of its results
Compact shell solitons in K field theories
Some models providing shell-shaped static solutions with compact support
(compactons) in 3+1 and 4+1 dimensions are introduced, and the corresponding
exact solutions are calculated analytically. These solutions turn out to be
topological solitons, and may be classified as maps and suspended
Hopf maps, respectively. The Lagrangian of these models is given by a scalar
field with a non-standard kinetic term (K field) coupled to a pure Skyrme term
restricted to , rised to the appropriate power to avoid the Derrick
scaling argument. Further, the existence of infinitely many exact shell
solitons is explained using the generalized integrability approach. Finally,
similar models allowing for non-topological compactons of the ball type in 3+1
dimensions are briefly discussed.Comment: 10 pages, latex, 2 figures, change in title and introduction.
Discussion section, 2 figures and references adde
Integrability in Theories with Local U(1) Gauge Symmetry
Using a recently developed method, based on a generalization of the zero
curvature representation of Zakharov and Shabat, we study the integrability
structure in the Abelian Higgs model. It is shown that the model contains
integrable sectors, where integrability is understood as the existence of
infinitely many conserved currents. In particular, a gauge invariant
description of the weak and strong integrable sectors is provided. The
pertinent integrability conditions are given by a U(1) generalization of the
standard strong and weak constraints for models with two dimensional target
space. The Bogomolny sector is discussed, as well, and we find that each
Bogomolny configuration supports infinitely many conserved currents. Finally,
other models with U(1) gauge symmetry are investigated.Comment: corrected typos, version accepted in J. Phys.
New Integrable Sectors in Skyrme and 4-dimensional CP^n Model
The application of a weak integrability concept to the Skyrme and
models in 4 dimensions is investigated. A new integrable subsystem of the
Skyrme model, allowing also for non-holomorphic solutions, is derived. This
procedure can be applied to the massive Skyrme model, as well. Moreover, an
example of a family of chiral Lagrangians providing exact, finite energy
Skyrme-like solitons with arbitrary value of the topological charge, is given.
In the case of models a tower of integrable subsystems is obtained. In
particular, in (2+1) dimensions a one-to-one correspondence between the
standard integrable submodel and the BPS sector is proved. Additionally, it is
shown that weak integrable submodels allow also for non-BPS solutions.
Geometric as well as algebraic interpretations of the integrability conditions
are also given.Comment: 23 page