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, $V=1/2 \phi_3^2$. 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 &gt; 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 &gt; 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