1,656 research outputs found
Topological discrete kinks
A spatially discrete version of the general kink-bearing nonlinear
Klein-Gordon model in (1+1) dimensions is constructed which preserves the
topological lower bound on kink energy. It is proved that, provided the lattice
spacing h is sufficiently small, there exist static kink solutions attaining
this lower bound centred anywhere relative to the spatial lattice. Hence there
is no Peierls-Nabarro barrier impeding the propagation of kinks in this
discrete system. An upper bound on h is derived and given a physical
interpretation in terms of the radiation of the system. The construction, which
works most naturally when the nonlinear Klein-Gordon model has a squared
polynomial interaction potential, is applied to a recently proposed continuum
model of polymer twistons. Numerical simulations are presented which
demonstrate that kink pinning is eliminated, and radiative kink deceleration
greatly reduced in comparison with the conventional discrete system. So even on
a very coarse lattice, kinks behave much as they do in the continuum. It is
argued, therefore, that the construction provides a natural means of
numerically simulating kink dynamics in nonlinear Klein-Gordon models of this
type. The construction is compared with the inverse method of Flach, Zolotaryuk
and Kladko. Using the latter method, alternative spatial discretizations of the
twiston and sine-Gordon models are obtained which are also free of the
Peierls-Nabarro barrier.Comment: 14 pages LaTeX, 7 postscript figure
Kinks in dipole chains
It is shown that the topological discrete sine-Gordon system introduced by
Speight and Ward models the dynamics of an infinite uniform chain of electric
dipoles constrained to rotate in a plane containing the chain. Such a chain
admits a novel type of static kink solution which may occupy any position
relative to the spatial lattice and experiences no Peierls-Nabarro barrier.
Consequently the dynamics of a single kink is highly continuum like, despite
the strongly discrete nature of the model. Static multikinks and kink-antikink
pairs are constructed, and it is shown that all such static solutions are
unstable. Exact propagating kinks are sought numerically using the
pseudo-spectral method, but it is found that none exist, except, perhaps, at
very low speed.Comment: Published version. 21 pages, 5 figures. Section 3 completely
re-written. Conclusions unchange
Magnetic bubble refraction and quasibreathers in inhomogeneous antiferromagnets
The dynamics of magnetic bubble solitons in a two-dimensional isotropic
antiferromagnetic spin lattice is studied, in the case where the exchange
integral J(x,y) is position dependent. In the near continuum regime, this
system is described by the relativistic O(3) sigma model on a spacetime with a
spatially inhomogeneous metric, determined by J. The geodesic approximation is
used to describe low energy soliton dynamics in this system: n-soliton motion
is approximated by geodesic motion in the moduli space of static n-solitons,
equipped with the L^2 metric. Explicit formulae for this metric for various
natural choices of J(x,y) are obtained. From these it is shown that single
soliton trajectories experience refraction, with 1/J analogous to the
refractive index, and that this refraction effect allows the construction of
simple bubble lenses and bubble guides. The case where J has a disk
inhomogeneity (taking the value J_1 outside a disk, and J_2<J_1 inside) is
considered in detail. It is argued that, for sufficiently large J_1/J_2 this
type of antiferromagnet supports approximate quasibreathers: two or more
coincident bubbles confined within the disk which spin internally while their
shape undergoes periodic oscillations with a generically incommensurate period.Comment: Conference proceedings paper for talk given at Nonlinear Physics
Theory and Experiment IV, Gallipoli, Italy, June 200
Easy plane baby skyrmions
The baby Skyrme model is studied with a novel choice of potential, . This "easy plane" potential vanishes at the equator of the target
two-sphere. Hence, in contrast to previously studied cases, the boundary value
of the field breaks the residual SO(2) internal symmetry of the model.
Consequently, even the unit charge skyrmion has only discrete symmetry and
consists of a bound state of two half lumps. A model of long-range
inter-skyrmion forces is developed wherein a unit skyrmion is pictured as a
single scalar dipole inducing a massless scalar field tangential to the vacuum
manifold. This model has the interesting feature that the two-skyrmion
interaction energy depends only on the average orientation of the dipoles
relative to the line joining them. Its qualitative predictions are confirmed by
numerical simulations. Global energy minimizers of charges B=1,...,14,18,32 are
found numerically. Up to charge B=6, the minimizers have 2B half lumps
positioned at the vertices of a regular 2B-gon. For charges B >= 7, rectangular
or distorted rectangular arrays of 2B half lumps are preferred, as close to
square as possible.Comment: v3: replaced with journal version, one new reference, one deleted
reference; 8 pages, 5 figures v2: fixed some typos and clarified the
relationship with condensed matter systems 8 pages, 5 figure
Kink dynamics in a novel discrete sine-Gordon system
A spatially-discrete sine-Gordon system with some novel features is
described. There is a topological or Bogomol'nyi lower bound on the energy of a
kink, and an explicit static kink which saturates this bound. There is no
Peierls potential barrier, and consequently the motion of a kink is simpler,
especially at low speeds. At higher speeds, it radiates and slows down.Comment: 10 pages, 7 figures, archivin
Quantum lump dynamics on the two-sphere
It is well known that the low-energy classical dynamics of solitons of
Bogomol'nyi type is well approximated by geodesic motion in M_n, the moduli
space of static n-solitons. There is an obvious quantization of this dynamics
wherein the wavefunction evolves according to the Hamiltonian H_0 equal to
(half) the Laplacian on M_n. Born-Oppenheimer reduction of analogous mechanical
systems suggests, however, that this simple Hamiltonian should receive
corrections including k, the scalar curvature of M_n, and C, the n-soliton
Casimir energy, which are usually difficult to compute, and whose effect on the
energy spectrum is unknown. This paper analyzes the spectra of H_0 and two
corrections to it suggested by work of Moss and Shiiki, namely H_1=H_0+k/4 and
H_2=H_1+C, in the simple but nontrivial case of a single CP^1 lump moving on
the two-sphere. Here M_1=TSO(3), a noncompact kaehler 6-manifold invariant
under an SO(3)xSO(3) action, whose geometry is well understood. The symmetry
gives rise to two conserved angular momenta, spin and isospin. A hidden
isometry of M_1 is found which implies that all three energy spectra are
symmetric under spin-isospin interchange. The Casimir energy is found exactly
on the zero section of TSO(3), and approximated numerically on the rest of M_1.
The lowest 19 eigenvalues of H_i are found for i=0,1,2, and their spin-isospin
and parity compared. The curvature corrections in H_1 lead to a qualitatively
unchanged low-level spectrum while the Casimir energy in H_2 leads to
significant changes. The scaling behaviour of the spectra under changes in the
radii of the domain and target spheres is analyzed, and it is found that the
disparity between the spectra of H_1 and H_2 is reduced when the target sphere
is made smaller.Comment: 35 pages, 3 figure
The p-weak gradient depends on p
Given \u3b1 > 0, we construct a weighted Lebesgue measure on Rnfor which the family of nonconstant curves has p-modulus zero for p 64 1 + \u3b1 but the weight is a Muckenhoupt Ap weight for p > 1 + \u3b1. In particular, the p-weak gradient is trivial for small p but nontrivial for large p. This answers an open question posed by several authors. We also give a full description of the p-weak gradient for any locally finite Borel measure on R
The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps
The most fruitful approach to studying low energy soliton dynamics in field
theories of Bogomol'nyi type is the geodesic approximation of Manton. In the
case of vortices and monopoles, Stuart has obtained rigorous estimates of the
errors in this approximation, and hence proved that it is valid in the low
speed regime. His method employs energy estimates which rely on a key
coercivity property of the Hessian of the energy functional of the theory under
consideration. In this paper we prove an analogous coercivity property for the
Hessian of the energy functional of a general sigma model with compact K\"ahler
domain and target. We go on to prove a continuity property for our result, and
show that, for the CP^1 model on S^2, the Hessian fails to be globally coercive
in the degree 1 sector. We present numerical evidence which suggests that the
Hessian is globally coercive in a certain equivariance class of the degree n
sector for n>1. We also prove that, within the geodesic approximation, a single
CP^1 lump moving on S^2 does not generically travel on a great circle.Comment: 29 pages, 1 figure; typos corrected, references added, expanded
discussion of the main function spac
Kink Dynamics in a Topological Phi^4 Lattice
It was recently proposed a novel discretization for nonlinear Klein-Gordon
field theories in which the resulting lattice preserves the topological
(Bogomol'nyi) lower bound on the kink energy and, as a consequence, has no
Peierls-Nabarro barrier even for large spatial discretizations (h~1.0). It was
then suggested that these ``topological discrete systems'' are a natural choice
for the numerical study of continuum kink dynamics. Giving particular emphasis
to the phi^4 theory, we numerically investigate kink-antikink scattering and
breather formation in these topological lattices. Our results indicate that,
even though these systems are quite accurate for studying free kinks in coarse
lattices, for legitimate dynamical kink problems the accuracy is rather
restricted to fine lattices (h~0.1). We suggest that this fact is related to
the breaking of the Bogomol'nyi bound during the kink-antikink interaction,
where the field profile loses its static property as required by the
Bogomol'nyi argument. We conclude, therefore, that these lattices are not
suitable for the study of more general kink dynamics, since a standard
discretization is simpler and has effectively the same accuracy for such
resolutions.Comment: RevTeX, 4 pages, 4 figures; Revised version, accepted to Physical
Review E (Brief Reports
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