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W. J. Zakrzewski

Publications and source records attributed to W. J. Zakrzewski.

At least 37 records · Page 2Linked to original sources
arXiv2010

Canonical surfaces associated with projectors in grassmannian sigma models

We discuss the construction of higher-dimensional surfaces based on the harmonic maps of $S^2$ into $CP^{N-1}$ and other grassmannians. We show that there are two ways of implementing this procedure - both based on the use of the relevant projectors. We study various properties of such projectors and show that the Gaussian curvature of these surfaces, in general, is not constant. We look in detail at the surfaces corresponding to the Veronese sequence of such maps and show that for all of them this curvature is constant but its value depends on which mapping is used in the construction of the surface.

math-ph↗
arXiv2009

Can cosmic acceleration be caused by exotic massless particles?

To describe dark energy we introduce a fluid model with no free parameter on the microscopic level. The constituents of this fluid are massless particles which are a dynamical realisation of the unextended $D=(3+1)$ Galilei algebra. These particles are exotic as they live in an enlarged phase space. Their only interaction is with gravity. A minimal coupling to the gravitational field, satisfying Einstein's equivalence principle, leads to a dynamically active gravitational mass density of either sign. A two-component model containing matter (baryonic and dark) and dark energy leads, through the cosmological principle, to Friedmann-like equations. Their solutions show a deceleration phase for the early universe and an acceleration phase for the late universe. We predict the Hubble parameter $H(z)/H_0$ and the deceleration parameter $q(z)$ and compare them with available experimental data. We also discuss a reduced model (one component dark sector) and the inclusion of radiation. Our model shows no stationary modification of Newton's gravitational potential.

astro-ph.CO↗
arXiv2009

The scattering of a skyrmion configuration on asymmetric holes or barriers in a model Landau-Lifshitz equation

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.

nlin.PS↗
arXiv2008

Scattering of a two skyrmion configuration on potential holes of barriers in a model Landau-Lifshitz equation

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 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 terms' coefficient. In the barrier system, the normal circular path was deformed as the skyrmions traversed the barrier. During the same period, it was seen that the skyrmions sped up as they went over the barrier. For critical values of the barrier height and width, the skyrmions were no longer bound and were free to separate. In the case of a potential hole, the baby skyrmions no longer formed a bound state and moved asymptotically along the axis of the hole. It is shown how to modify the definition of the angular momentum to include the effects of the obstructions, so that it is conserved.

nlin.PS↗
arXiv2008

Nonrelativistic Dark-Energy Fluid in a Baby Universe

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.

gr-qc↗
arXiv2008

Some properties of deformed Sine Gordon models

We study some properties of the deformed Sine Gordon models. These models, presented by Bazeia et al, are natural generalisations of the Sine Gordon models in (1+1) dimensions. There are two classes of them, each dependent on a parameter n. For special values of this parameter the models reduce to the Sine Gordon one; for other values of n they can be considered as generalisations of this model. The models are topological and possess one kink solutions. Here we investigate the existence of other solutions of these models - such as breathers. The work is numerical and we find that the breathers, as such, probably do not exist. However, we show that some of these models, namely, the n = 1 of the first class possess breather-like solutions which are quasi-stable; ie these quasi-breathers exist for long periods of time (thousands of periods of oscillations). These results are found to be independent of the discretisation used in the numerical part of our work.

nlin.SI↗
arXiv2007

Acceleration-Enlarged Symmetries in Nonrelativistic Space-Time with a Cosmological Constant

By considering the nonrelativistic limit of de-Sitter geometry one obtains the nonrelativistic space-time with a cosmological constant and Newton-Hooke (NH) symmetries. We show that the NH symmetry algebra can be enlarged by the addition of the constant acceleration generators and endowed with central extensions (one in any dimension (D) and three in D=(2+1)). We present a classical Lagrangian and Hamiltonian framework for constructing models quasi-invariant under enlarged NH symmetries which depend on three parameters described by three nonvanishing central charges. The Hamiltonian dynamics then splits into external and internal sectors with new non-commutative structures of external and internal phase spaces. We show that in the limit of vanishing cosmological constant the system reduces to the one presented in [1] which possesses accelaration-enlarged Galilean symmetries.

hep-th↗
arXiv2007

Scattering of sine-Gordon Breathers on Potential Wells

We analyse the scattering of sine-Gordon breathers on a square potential well. We show that the scattering process depends not only on the vibration frequency of the breather and its incoming speed but also on its phase as well as the depth and width of the well. We show that the breather can pass through the well and exit with a speed different, sometime larger, from the initial one. It can also be trapped and very slowly decay inside the well or bounce out of the well and go back to where it came from. We also show that the breather can split into a kink and an anti-kink pair when it hits the well.

cond-mat.other↗
arXiv2007

Acceleration-Extended Galilean Symmetries with Central Charges and their Dynamical Realizations

We add to Galilean symmetries the transformations describing constant accelerations. The corresponding extended Galilean algebra allows, in any dimension $D=d+1$, the introduction of one central charge $c$ while in $D=2+1$ we can have three such charges: c, θand θ'. We present nonrelativistic classical mechanics models, with higher order time derivatives and show that they give dynamical realizations of our algebras. The presence of central charge $c$ requires the acceleration square Lagrangian term. We show that the general Lagrangian with three central charges can be reinterpreted as describing an exotic planar particle coupled to a dynamical electric and a constant magnetic field.

hep-th↗
arXiv2006

Scattering of Sine-Gordon kinks on potential wells

We study the scattering properties of Sine Gordon kinks on obstructions in the form of finite size potential `wells'. We model this by making the coefficient of the $\cos(ϕ)-1$ term in the Lagrangian position dependent. We show that when the kinks find themselves in the well they radiate and then interact with this radiation. As a result of this energy loss the kinks become trapped for small velocities while at higher velocities they are transmitted with a loss of energy. However, the interaction with the radiation can produce `unexpected' reflections by the well. We present two simple models which capture the gross features of this behaviour. Both involve standing waves either at the edges of the well or in the well itself.

hep-th↗
arXiv2006

Surfaces in ${\mathbb R}^{N^2-1}$ based on harmonic maps $S^2\to CP^{N-1}$

We show that many surfaces in $\R^{N^2-1}$ can be generated by harmonic maps of $S^2\to CP^{N-1}$. These surfaces are based on the projectors in $CP^{N-1}$ which describe maps of $S^2\to CP^{N-1}$. In the case when these maps form the Veronese sequence all the surfaces have constant curvature.

math.DG↗
arXiv2006

Dynamical properties of a Soliton in a Potential Well

We analyse the scattering of a two-dimensional soliton on a potential well. We show that this soliton can pass through the well, bounce back or become trapped and we study the dependence of the critical velocity on the width and the depth of the well. We also present a model based on a pseudo-geodesic approximation to the full system which shows that the vibrational modes of the soliton play a crucial role in the dynamical properties of its interactions with potential wells.

hep-th↗
arXiv2006

Galilean Exotic Planar Supersymmetries and Nonrelativistic Supersymmetric Wave Equations

We describe the general class of $N$-extended $D=(2+1)$ Galilean supersymmetries obtained, respectively, from the $N$-extended D=3 Poincaré superalgebras with maximal sets of central charges. We confirm the consistency of supersymmetry with the presence of the `exotic' second central charge $θ$. We show further how to introduce a N=2 Galilean superfield equation describing nonrelativistic spin 0 and spin 1/2 free particles.

hep-th↗
arXiv2006

Susy CP^(N-1) model and surfaces in R^(N^2-1)

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.

math-ph↗
arXiv2005

Exotic Galilean Conformal Symmetry and its Dynamical Realisations

The six-dimensional exotic Galilean algebra in (2+1) dimensions with two central charges $m$ and $θ$, is extended when $m=0$, to a ten-dimensional Galilean conformal algebra with dilatation, expansion, two acceleration generators and the central charge $θ$. A realisation of such a symmetry is provided by a model with higher derivatives recently discussed in \cite{peterwojtek}. We consider also a realisation of the Galilean conformal symmetry for the motion with a Coulomb potential and a magnetic vortex interaction. Finally, we study the restriction, as well as the modification, of the Galilean conformal algebra obtained after the introduction of the minimally coupled constant electric and magnetic fields.

hep-th↗
arXiv2005

Scattering of topological solitons on holes and barriers

We study the scattering properties of topological solitons on obstructions in the form of holes and barriers. We use the 'new baby Skyrme' model in (2+1) dimensions and we model the obstructions by making the coefficient of the baby skyrme model potential - position dependent. We find that that the barrier leads to the repulsion of the solitons (for low velocities) or their complete transmission (at higher velocities) with the process being essentially elastic. The hole case is different; for small velocities the solitons are trapped while at higher velocities they are transmitted with a loss of energy. We present some comments explaining the observed behaviour.

hep-th↗
arXiv2005

Surfaces immersed in su(N+1) Lie algebras obtained from the CP^N sigma models

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.

math.DG↗