4 research outputs found
Static Hopfions in the extended Skyrme-Faddeev model
We construct static soliton solutions with non-zero Hopf topological charges
to a theory which is an extension of the Skyrme-Faddeev model by the addition
of a further quartic term in derivatives. We use an axially symmetric ansatz
based on toroidal coordinates, and solve the resulting two coupled non-linear
partial differential equations in two variables by a successive over-relaxation
(SOR) method. We construct numerical solutions with Hopf charge up to four, and
calculate their analytical behavior in some limiting cases. The solutions
present an interesting behavior under the changes of a special combination of
the coupling constants of the quartic terms. Their energies and sizes tend to
zero as that combination approaches a particular special value. We calculate
the equivalent of the Vakulenko and Kapitanskii energy bound for the theory and
find that it vanishes at that same special value of the coupling constants. In
addition, the model presents an integrable sector with an infinite number of
local conserved currents which apparently are not related to symmetries of the
action. In the intersection of those two special sectors the theory possesses
exact vortex solutions (static and time dependent) which were constructed in a
previous paper by one of the authors. It is believed that such model describes
some aspects of the low energy limit of the pure SU(2) Yang-Mills theory, and
our results may be important in identifying important structures in that strong
coupling regime.Comment: 22 pages, 42 figures, minor correction
Nonlinear sigma models solvable by the Aratyn-Ferreira-Zimerman ansatz
Nonlinear sigma models compatible with the aratyn-Ferreira-Zimerman ansatz
are discussed, the latter ansatz automatically leading to configurations with
definite values of the Hopf index. These models are allowed to involve a weight
factor which is a function of one of the toroidal coordinates. Depending on the
choice of the weight factor, the field equation takes various forms. In one
model with a special weight factor, the field equation turns out to be the
fifth Painleve equation. This model suggests the existence of a knot soliton
strictly confined in a finite spatial volume. Some other interesting cases are
also discussed.Comment: 19 pages. Two column style is changed to preprint styl