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

Solitons on tori and soliton crystals

Necessary conditions for a soliton on a torus $M=\R^m/\Lambda$ to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be $L^2$ orthogonal to \ee, the space of parallel symmetric bilinear forms on $TM$, and, further, a certain symmetric bilinear form on \ee, called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice $\Lambda$, there is a baby Skyrme model which supports a soliton crystal of periodicity $\Lambda$. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies in particular that the celebrated Skyrme crystal of Castillejo {\it et al.}, and Kugler and Shtrikman, passes both tests.Comment: 24 pages, revised version to be published. Added an existence proof for baby Skyrmions of arbitrary degree on a general two-torus for a model with general potential. Otherwise, minor improvement

Lump dynamics in the CP^1 model on the torus

The topology and geometry of the moduli space, M_2, of degree 2 static solutions of the CP^1 model on a torus (spacetime T^2 x R) are studied. It is proved that M_2 is homeomorphic to the left coset space G/G_0 where G is a certain eight-dimensional noncompact Lie group and G_0 is a discrete subgroup of order 4. Low energy two-lump dynamics is approximated by geodesic motion on M_2 with respect to a metric g defined by the restriction to M_2 of the kinetic energy functional of the model. This lump dynamics decouples into a trivial ``centre of mass'' motion and nontrivial relative motion on a reduced moduli space. It is proved that (M_2,g) is geodesically incomplete and has only finite diameter. A low dimensional geodesic submanifold is identified and a full description of its geodesics obtained.Comment: 22 pages, Latex, 7 postscript figure

A discrete phi^4 system without Peierls-Nabarro barrier

A discrete phi^4 system is proposed which preserves the topological lower bound on the kink energy. Existence of static kink solutions saturating this lower bound and occupying any position relative to the lattice is proved. Consequently, kinks of the model experience no Peierls-Nabarro barrier, and can move freely through the lattice without being pinned. Numerical simulations reveal that kink dynamics in this system is significantly less dissipative than that of the conventional discrete phi^4 system, so that even on extremely coarse lattices the kink behaves much like its continuum counterpart. It is argued, therefore, that this is a natural discretization for the purpose of numerically studying soliton dynamics in the continuum phi^4 model.Comment: 8 pages, LaTeX, 8 postscript figure