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We look at BPS systems involving two interacting Sine-Gordon like fields both when one of them has a kink solution and the second one either a kink or an antikink solution. The interaction between the two fields is controlled by a parameter $\lambda$ which has to satisfy $| \lambda|< 2$. We then take these solitonic static solutions (with solitons well localised) and construct from them systems involving two solitons in each field (kinks and antikinks) and then use them as initial conditions for their evolution in Lorentz covariant versions of such models. This way we study their interactions and compare them with similar interactions involving only one Sine-Gordon field. In particular, we look at the behaviour of two static kinks in each field (which for one field repel each other) and of a system involving kinks and anti-kinks (which for one field attract each other) and look how their behaviour depends on the strength of the interaction $\lambda$ between the two fields. Our simulations have led us to look again at the static BPS solutions of systems involving more fields. We have found that such ostensibly 'static' BPS solutions can exhibit small motions due to the excitation of their zero modes. These excitations arise from small unavoidable numerical errors (the overall translation is cancelled by the conservation of momentum) but as systems of two or more fields have more that one zero mode such motions can be generated and are extremely small. The energy of our systems has been conserved to within $10^{-5}\%$., Comment: 32 pages, 40 figures

We study the gauge invariance of the supersymmetric grassmannian sigma model $G(M,N)$. It is richer then its purely bosonic submodel and we show how to use it in order to reduce some constant curvature holomorphic solutions of the model into simpler expressions., 8 pages, no figures

We look at solutions (both BPS and non-BPS) of the $\mathbb{CP}^{N-1}$ model on $\mathbb{R} \times S^1$ (with twisted boundary conditions), in particular by using a conformal mapping technique, and we show how to interpret these solutions by decomposing them into expressions describing constituent solitons. We point out the problems that may arise (for non-BPS solutions) when one naively looks at the clustering properties of these solutions. This could lead to misunderstandings when studying extrapolations between small and large compactification radii., Comment: 21 pages, 4 figures; v2: published version

The BPS baby Skyrme models are submodels of baby Skyrme models, where the nonlinear sigma model term is suppressed. They have skyrmion solutions saturating a BPS bound, and the corresponding static energy functional is invariant under area-preserving diffeomorphisms (APDs). Here we show that the solitons in the BPS baby Skyrme model, which carry a nontrivial topological charge $Q_{b} \in \pi_2(S^2)$ (a winding number), are dual to vortices in a BPS vortex model with a topological charge $Q_{v}\in \pi_1(S^1)$ (a vortex number), in the sense that there is a map between the BPS solutions of the two models. The corresponding energy densities of the BPS solutions of the two models are identical. A further consequence of the duality is that the dual BPS vortex models inherit the BPS property and the infinitely many symmetries (APDs) of the BPS baby Skyrme models. Finally, we demonstrate that the same topological duality continues to hold for the U(1) gauged versions of the models., Comment: Latex, 18 pages, references added, typos corrected

We construct new classes of vortex-like solutions of the CP^N model in (3+1) dimensions and discuss some of their properties. These solutions are obtained by generalizing to (3+1) dimensions the techniques well established for the two dimensional CP^N models. We show that as the total energy of these solutions is infinite, they describe evolving vortices and anti-vortices with the energy density of some configurations varying in time. We also make some further observations about the dynamics of these vortices., 23 pages, 30 figures

We present a class of solutions of the CPN model in (3+1) dimensions. We suggest that they represent vortex-like configurations. We also discuss some of their properties. We show that some configurations of vortices have a divergent energy per unit length while for the others such an energy has a minimum for a very special orientation of vortices. We also discuss the Noether charge densities of these vortices., Comment: 14 pages, 12 figures (pdf)

This current work is an extension to work previously done by the authors. The dynamics of a baby skyrmion configuration, in a model Landau-Lifshitz equation, was studied in the presence of various potential obstructions. The baby skyrmion configuration was constructed from two Q=1 hedgehog solutions to the new baby Skyrme model in $(2+1)$ dimensions. The potential obstructions were created by introducing a new term into the Lagrangian which resulted in a localised inhomogeneity in the potential term's coefficient. In the barrier system the normal circular path was deformed as the skyrmions traversed the barrier, after which the skyrmions orbited the boundary of the system. For critical values of the barrier height and width the skyrmions were no longer bound although the unbound behaviour is not clearly distinct from the bound. In the case of a potential hole the dynamics of baby skyrmions is dependent upon the binding energy of the system. Depending upon its value, the skyrmions' behaviour varies. The angular momentum must be modified to ensure overall conservation. We show that there exists a link between the oscillation in the skyrmion's energy density and the periods of non-conservation of the angular momentum in Landau-Lifshitz models., 40 pages, 15 figures

We show that the dynamical realization of the acceleration-enlarged Galilean symmetry leads to nonrelativistic massless particles whose energy may be negative. We present a fluid mechanical generalisation of this observation and use it to contruct a nonrelativistic two-dimensional fluid model which possesses solutions with a negative energy density. Considering this model as describing dark energy in a baby universe (two space dimensions) we show that its negative energy density leads to a repulsive gravitational interaction of the fluid with any test body.

We describe surfaces in R^{N^2-1} generated by the holomorphic solutions of the supersymmetric CP^{N-1} model. We show that these surfaces are described by the fundamental projector constructed out of the solutions of this model and that in the CP^{N-1} case the corresponding surface is a sphere. Although the coordinates of the sphere are superfields the sphere's curvature is constant. We show that for N>2 the corresponding surfaces can also be constructed from the similar projector.

We study here polaron (soliton) states of electrons or holes in a model describing carbon-type nanotubes. In the Hamiltonian of the system we take into account the electron-phonon interaction that arises from the deformation dependencies of both the on-site and the hopping interaction energies. Using an adiabatic approximation, we derive the equations for self-trapped electron states in zigzag nanotubes. We find the ground states of an electron in such a system and show that the polaron states can have different symmetries depending on the strength of the electron-phonon coupling. Namely, at relatively weak coupling the polarons possess quasi-one-dimensional (quasi-1D) properties and have an azimuthal symmetry. When the coupling constant exceeds some critical value, the azimuthal symmetry breaks down and the polaron spreads out in more than one dimension. We also study polarons that are formed by the electrons in the conducting band (or by holes in the valence band) in semiconducting carbon nanotubes. We show that their properties are more complex than those of quasi-1D ground state polarons. In particular, polarons in semiconducting carbon nanotubes possess an inner structure: being self-trapped along the nanotube axis they exhibit some modulations around the nanotube., 27 pages 1 figure

A simple phenomenological model for describing the conformational dynamics of biological macromolecules via the nonlinearity-induced instabilities is proposed. It is shown that the interaction between charges and bending degrees of freedom of closed molecular aggregates may act as drivers giving impetus to conformational dynamics of biopolymers. It is demonstrated that initially circular aggregates may undergo transformation to polygonal shapes and possible application to aggregates of bacteriochlorophyl a molecules is considered., REVTeX, 19 pages, 9 figures

We study some geometrical aspects of two dimensional orientable surfaces arrising from the study of CP^N sigma models. To this aim we employ an identification of R^(N(N+2)) with the Lie algebra su(N+1) by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss-Weingarten and the Gauss-Codazzi-Ricci equations are expressed in terms of the solution of the CP^N model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in su(2) and su(3) Lie algebras., Comment: 32 pages, 1 figure; changes: major revision of presentation, clarifications added

A model describing the dynamics of crystal exciton states on a hexagonal lattice has been studied. The hamiltonian includes terms describing the movement of the exciton throughout the crystal. Here we generalize the hamiltonian by including first order couplings of such terms to the lattice phonon fields. A discussion of the ground state configuration of the exciton field as a function of the system couplings is given. Finally we analyse the stability of certain excited states under perturbations., 28 pages, 11 figures

We investigate some aspects of the soliton dynamics in an alpha-helical protein macromolecule within the steric Davydov-Scott model. Our main objective is to elucidate the important role of the helical symmetry in the formation, stability and dynamical properties of Davydov's solitons in an alpha-helix. We show, analytically and numerically, that the corresponding system of nonlinear equations admits several types of stationary soliton solutions and that solitons which preserve helical symmetry are dynamically unstable: once formed, they decay rapidly when they propagate. On the other hand, the soliton which spontaneously breaks the local translational and helical symmetries possess the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of an hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multi-hump amplitude distribution of excitations on individual spines. The complex and composite structure of the soliton manifests itself distinctly when the soliton is moving and some interspine oscillations take place. Such a soliton structure and the interspine oscillations have previosly been observed numerically in [A.C. Scott, Phys.Rev. A 26 578 (1982)]. Here we argue that the solitons studied by A. Scott, are hybrid solitons and that the oscillations arise due to the helical symmetry of the system and result from the motion of the soliton along the $\alpha$-helix. The frequency of the interspine oscillations is shown to be proportional to the soliton velocity., Comment: 26 pages, 8 figures

With the help of numerical simulations we study N-soliton scattering (N=3,4) in the (2+1)-dimensional CP^1 model with periodic boundary conditions. When the solitons are scattered from symmetrical configurations the scattering angles observed agree with the earlier \pi/N predictions based on the model on R_2 with standard boundary conditions. When the boundary conditions are not symmetric the angles are different from \pi/N. We present an explanation of our observed patterns based on a properly formulated geodesic approximation., Comment: Latex format, 23 pages including 10 figures

We describe the supersymmetrization of two formulations of free noncommutative planar particles -- in coordinate space with higher order Lagrangian [1] and in the framework of Faddeev and Jackiw [2,3], with first order action. In nonsupersymmetric case the first formulation after imposing subsidiary condition eliminating internal degrees of freedom provides the second formulation. In supersymmetric case one can also introduce the split into ``external'' and ``internal'' degrees of freedom both describing supersymmetric models., LaTeX, 6 pages, to appear in "Group24: Physical and Mathematical Aspects of Symmetry", the Proceedings of Group24 XXIV-th International Colloqium on Group-Theoretic Methods in Physics, Paris, 15 21.07.2002, Ed. J.P. Gazeau, R. Kerner et. al. IOP Publishing House, Bristol, 2002

We study classical solutions of a particular version of the modified Skyrme model in (3+1) dimensions. The model possesses Skyrmion solutions as well as stable domain walls that connect different vacua of the theory. We show that there is an attractive interaction between Skyrmions and domain walls. Thus Skyrmions can be captured by the domain walls. We show also that, when the mass term is of a special type, the model possesses bound states of Skyrmions and of the domain wall. They look like deformed 2-dimensional Skyrmions captured by the wall. The field configurations of these solutions can interpreted as having come from the evolution of the 3-dimensional Skyrmions captured by the domain wall. For more conventional choices of the mass term of the model in the model the attraction between the Skyrmions and the wall leads to the capture of the Skyrmions which are then turned into topological waves which spread out on the wall. We have observed, numerically, such captures and the emission of the waves. We speculate that this observation may be useful in the explanation of the problem of baryogenesis and baryon-antibaryon asymmetry of the Universe.

Due to a computational mistake this paper has been withdrawn., ascii : withdrawing the paper

Due to a computational mistake this paper has been withdrawn., ascii : withdrawing the paper

We study the diffusion and deformation of classical solitons coupled to thermal noise. The diffusion coefficient for kinks in the $\phi^4$ theory is predicted up to the second order in $kT$. The prediction is verified by numerical simulations. Multiskyrmions in the vector O(3) sigma model are studied within the same formalism. Thermal noise results in a diffusion on the multisoliton collective coordinate space (moduli space). There are entropic forces which tend, for example, to bind pairs of solitons into bi-solitonic molecules., Comment: 5 Revtex pages, 2 .ps files

We show that any solution of the SU(2) Skyrme model can be used to give a topologically trivial solution of the SU(4) one. In addition, we extend the method introduced by Houghton et al. and use harmonic maps from S2 to CP(N-1) to construct low energy configurations of the SU(N) Skyrme models. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only marginally higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon (and energy) densities of the SU(3) configurations with baryon number B=2-4 are more symmetrical than their SU(2) analogues. We also present the baryon densities for the B=5 and B=6 configurations and discuss their symmetries., Comment: latex : 25 pages, 9 Postscript figures, uses epsf

The $CP^2$ model, with and without a generalized Hopf term, is studied using the collective coordinate approximation. In the spirit of this approximation, an ansatz is given which in previous numerical studies was seen to give a good parameterization of the numerical solution. The equations of motion for the collective coordinates are then solved analytically, for solitons close together and for solitons far apart. The solutions show how the generalized Hopf term changes the scattering angle which in its absence is $90^{\circ}$., 15 pages, Latex

We study the 3+1 dimensional Skyrme model with a mass term different from the usual one. We show that this new model possesses domain walls solutions. We describe how, in the equivalent 2+1 dimensional model, the Skyrmion is absorbed by the wall., latex, 5 pages. Proceedings of the Soliton conference (Kingston 1997)

A simple method how to study response of solitons in dissipative systems to external impulsive perturbations is developed. Thanks to nontrivial choice of small parameter, the perturbative scheme captures genuine nonlinear phenomena. The method is developed and tested by numerical simulations for kinks in 1+1 dimensions and for skyrmions in 2+1 dimensions. Extension to models including second order time derivatives is discussed., 6 pages, 6 figures, RevTex+epsf

We report our observations that the baby-Skyrme model in (2+1) dimensions possesses non-topological stationary solutions which we call pseudo-breathers. We discuss their properties and present our results on their interaction with the topological skyrmions., Comment: latex, 4 pages. Proceedings of the Soliton conference (Kingston 1997)

We study classical solutions of the vector O(3) sigma model in (2+1) dimensions, spontaneously broken to O(2)xZ2. The model possesses Skyrmion-type solutions as well as stable domain walls which connect different vacua. We show that different types of waves can propagate on the wall, including waves carrying a topological charge. The domain wall can also absorb Skyrmions and, under appropriate initial conditions, it is possible to emit a Skyrmion from the wall., Comment: plain tex : 15 pages, 21 Postscript figures, uses epsf.tex

We demonstrate the existence of stable time dependent solutions of the Landau-Lifshitz model with a constant external magnetic field. We find such solutions in all topological sectors, including N=0. We discuss some of their properties., Comment: plain tex : 14 pages, 9 Postscript figures, uses epsf.tex

In the spirit of previous papers, but using more general field configurations, the non-linear O(3) model in (2+1)-D, modified by the addition of both a potential-like term and a Skyrme-like term, is considered. The instanton solutions are numerically evolved in time and some of their stability properties studied. They are found to be stable, and a repulsive force is seen to exist among them. These results, which are restricted to the case of zero speed systems, confirm those obtained in previous investigations, in which a similar problem was studied for a different choice of the potential-like term., 10 pages, 4 figures, latex

Field-theoretic models for fields taking values in quantum groups are investigated. First we consider $SU_q(2)$ $��$ model ($q$ real) expressed in terms of basic notions of noncommutative differential geometry. We discuss the case in which the $��$ models fields are represented as products of conventional $��$ fields and of the coordinate-independent algebra. An explicit example is provided by the $U_q(2)$ $��$ model with $q\sp{N}=1$, in which case quantum matrices $U_q(2)$ are realised as $2N\times 2N$ unitary matrices. Open problems are pointed out., 20 pages