79 research outputs found
Recent mathematical developments in the Skyrme model
In this review we present a pedagogical introduction to recent, more
mathematical developments in the Skyrme model. Our aim is to render these
advances accessible to mainstream nuclear and particle physicists. We start
with the static sector and elaborate on geometrical aspects of the definition
of the model. Then we review the instanton method which yields an analytical
approximation to the minimum energy configuration in any sector of fixed baryon
number, as well as an approximation to the surfaces which join together all the
low energy critical points. We present some explicit results for B=2. We then
describe the work done on the multibaryon minima using rational maps, on the
topology of the configuration space and the possible implications of Morse
theory. Next we turn to recent work on the dynamics of Skyrmions. We focus
exclusively on the low energy interaction, specifically the gradient flow
method put forward by Manton. We illustrate the method with some expository toy
models. We end this review with a presentation of our own work on the
semi-classical quantization of nucleon states and low energy nucleon-nucleon
scattering.Comment: 129 pages, about 30 figures, original manuscript of published Physics
Report
Baby Skyrmion Strings
We provide analytical and numerical evidence of the existence of classically
stable, string-like configurations in a 2+1 dimensional analog of the Skyrme
model. The model contains a conserved topological charge usually called the
baryon number. Our strings are non-topological solitons which have a constant
baryon number per unit length. The energy per length containing one baryon is,
however, less than the energy of an isolated baryon (radially symmetric ``baby
Skyrmion") in a region of the parameter space, which suggests a degree of
stability for our configurations. In a limiting case, our configuration
saturates a Bogomolnyi-type bound and is degenerate in energy per baryon with
the baby Skyrmion. In another limiting case, the energies are still degenerate
but do not saturate the corresponding Bogomolnyi-type bound. Nonetheless, we
expect the string to be stable here. Both limiting cases are solvable
analytically.Comment: Latex, (revtex), with one figure in a separate postscript file, 12
page
Solitons in a Baby-Skyrme model with invariance under area preserving diffeomorphisms
We study the properties of soliton solutions in an analog of the Skyrme model
in 2+1 dimensions whose Lagrangian contains the Skyrme term and the mass term,
but no usual kinetic term. The model admits a symmetry under area preserving
diffeomorphisms. We solve the dynamical equations of motion analytically for
the case of spinning isolated baryon type solitons. We take fully into account
the induced deformation of the spinning Skyrmions and the consequent
modification of its moment of inertia to give an analytical example of related
numerical behaviour found by Piette et al.. We solve the equations of motion
also for the case of an infinite, open string, and a closed annular string. In
each case, the solitons are of finite extent, so called "compactons", being
exactly the vacuum outside a compact region. We end with indications on the
scattering of baby-Skyrmions, as well as some considerations as the properties
of solitons on a curved space.Comment: 30 pages, 5 figures, revtex, major modifications, conclusions
modifie
Low Energy Skyrmion-Skyrmion Scattering
We study the scattering of Skyrmions at low energy and large separation using
the method proposed by Manton of truncation to a finite number of degrees
freedom. We calculate the induced metric on the manifold of the union of
gradient flow curves, which for large separation, to first non-trivial order is
parametrized by the variables of the product ansatz. (presented at the Lake
Louise Winter Institute, 1994)Comment: 6 page
On the Strong Coupling Limit of the Faddeev-Hopf Model
The variational calculus for the Faddeev-Hopf model on a general Riemannian
domain, with general Kaehler target space, is studied in the strong coupling
limit. In this limit, the model has key similarities with pure Yang-Mills
theory, namely conformal invariance in dimension 4 and an infinite dimensional
symmetry group. The first and second variation formulae are calculated and
several examples of stable solutions are obtained. In particular, it is proved
that all immersive solutions are stable. Topological lower energy bounds are
found in dimensions 2 and 4. An explicit description of the spectral behaviour
of the Hopf map S^3 -> S^2 is given, and a conjecture of Ward concerning the
stability of this map in the full Faddeev-Hopf model is proved.Comment: 21 pages, 0 figure
The kinetic energy and and the geometric structure in the sector of the Skyrme model: A study using the Atiyah-Manton Ansatz
We study the construction of the collective-coordinate manifold in the baryon
number two sector of the Skyrme model. To that end we use techniques of
adiabatic large amplitude collective motion, which treat potential and kinetic
energy on an equal footing. In this paper the starting point is the Ansatz
proposed by Atiyah and Manton (Phys.~Lett.~{\bf 438B}, 222 (1989)), which
allows a study of the dynamics using a finite and small number of variables.
From these variables we choose a subset of collective ones. We then study the
behavior of inertial parameters along parts of the collective manifold, and
study the dynamical parts of the interaction.Comment: FAU-T3-94/1, 42 pages. 21 postscript figures can be included in the
text using epsf.sty. Postscript file of complete manuscript avalailabe as
ftp://theorie3.physik.uni-erlangen.de/pub/publications/NRWAMSk.ps.g
The Effect of low Momentum Quantum Fluctuations on a Coherent Field Structure
In the present work the evolution of a coherent field structure of the
Sine-Gordon equation under quantum fluctuations is studied. The basic equations
are derived from the coherent state approximation to the functional
Schr\"odinger equation for the field. These equations are solved asymptotically
and numerically for three physical situations. The first is the study of the
nonlinear mechanism responsible for the quantum stability of the soliton in the
presence of low momentum fluctuations. The second considers the scattering of a
wave by the Soliton. Finally the third problem considered is the collision of
Solitons and the stability of a breather.
It is shown that the complete integrability of the Sine-Gordon equation
precludes fusion and splitting processes in this simplified model.
The approximate results obtained are non-perturbative in nature, and are
valid for the full nonlinear interaction in the limit of low momentum
fluctuations. It is also found that these approximate results are in good
agreement with full numerical solutions of the governing equations. This
suggests that a similar approach could be used for the baby Skyrme model, which
is not completely integrable. In this case the higher space dimensionality and
the internal degrees of freedom which prevent the integrability will be
responsable for fusion and splitting processes. This work provides a starting
point in the numerical solution of the full quantum problem of the interaction
of the field with a fluctuation.Comment: 15 pages, 9 (ps) figures, Revtex file. Some discussion expanded but
conclusions unchanged. Final version to appear in PR
Stable spinning optical solitons in three dimensions
We introduce spatiotemporal spinning solitons (vortex tori) of the
three-dimensional nonlinear Schrodinger equation with focusing cubic and
defocusing quintic nonlinearities. The first ever found completely stable
spatiotemporal vortex solitons are demonstrated. A general conclusion is that
stable spinning solitons are possible as a result of competition between
focusing and defocusing nonlinearities.Comment: 4 pages, 6 figures, accepted to Phys. Rev. Let
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