W. J. Zakrzewski

General classical solutions in the CPN−1 model

We study the classical solutions with finite action of the CPn−1 non-linear σ model in two dimensions. The general solution can be expressed explicitly in terms of n rational analytic functions. All solutions which are neither instantons nor anti-instantons turn out to be saddle points of the action.

Acceleration-extended Galilean symmetries with central charges and their dynamical realizations

Abstract 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 , θ and 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.

Static solutions of a D-dimensional modified nonlinear Schrödinger equation

We study static solutions of a D-dimensional modified nonlinear Schrodinger equation (MNLSE) which was shown to describe, in two dimensions, the self-trapped (spontaneously localized) electron states in a discrete isotropic electron–phonon lattice [1, 2]. We show that this MNLSE, unlike the conventional nonlinear Schrodinger equation, possesses static localized solutions at any dimensionality when the effective nonlinearity parameter is larger than a certain critical value which depends on the dimensionality of the system under study. We investigate various properties of the equation analytically, using scaling transformations, within the variational scheme and numerically, and show that the results of these studies agree qualitatively and quantitatively. In particular, we prove that, for various values of D, when the coupling constant is larger than a certain critical value (which depends on D), this equation has two solutions, a stable (metastable) and an unstable one. We show that the solutions can be well approximated by a Gaussian ansatz and we also show that, in two dimensions, the equation possesses solutions with a nonzero angular momentum.

A modified Mottola-Wipf model with sphaleron and instanton fields

Abstract We modify the Mottola-Wipf O(3) sigma model by extending it with a Skyrme term. The resulting model supports a localised instanton solution, as well as sphaleron solution in the static limit.

Skyrmions and domain walls in dimensions

We study classical solutions of the vector O(3) sigma model in (2 + 1) dimensions, spontaneously broken to . 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.

Solitons inα-helical proteins

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 Davydovs 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 possesses the lowest energy and is a robust localized entity. We also demonstrate that this soliton is the result of a hybridization of the quasiparticle states from the two lowest degenerate bands and has an inner structure which can be described as a modulated multihump 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 previously been observed numerically [A. C. Scott, Phys. Rev. A 26, 578 (1982)]. Here we argue that the solitons studied by 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.

Shrinking of solitons in the (2 + 1)-dimensional sigma model

We study the shrinking of solitons in the original (2 + 1)-dimensional sigma model. We show that the rate of this shrinking for a single soliton can be estimated analytically. We perform numerical simulations and show that the time dependence of the shrinking of a single soliton is described by a power law.

Soliton scattering in the Skyrme model in (2+1) dimensions. I : Soliton-soliton case

The authors consider instanton and anti-instanton solutions of the O(3) sigma -model in two Euclidean dimensions modified by the addition of an appropriate potential and Skyrme-like terms as static solitons (and antisolitons) of the same model in (2+1) dimensions. They study various scattering properties of these structures, which they call skyrmions. Most of the work is numerical and uses two formulations of the model; the well-known formulation in terms of a real vector field phi (with three components restricted to phi . phi =1) and the formulation involving one complex field. They compare their results with similar results obtained in the pure O(3) sigma -model. They find that the potential and Skyrme terms which stabilize the skyrmions have relatively little effect on their scattering properties. In the scattering process initiated at low relative velocities the skyrmions bounce back while at velocities larger than some critical value they scatter at right angles. The value of this critical velocity depends on the values of the parameters of the potential and skyrme terms. The scattering is quasi-elastic and the skyrmions preserve their shape after the collision. They also analyse the scattering properties of rotating skyrmions and present their trajectories for some values of their rotation for several values of the parameters of the additional terms.

Euclidean supersymmetrization of instantons and self-dual monopoles

Abstract We show that N=1, D=4 euclidean supersymmetry leads to the complexification of gauge potentials, and that to have a supersymmetric euclidean D=4 theory which allows for the selfduality of real gauge potentials, the supersymmetry has to be extended to even N, i.e. at least N=2. We also discuss the supersymmetrizatin of Mantons procedure of using the dimensional reduction D=4→D=3 to derive self-dual monopoles from instantons.

Localized solutions in a two-dimensional Landau-Lifshitz model

Abstract We demonstrate the existence of stable time dependent solutions of the Landau-Lifshitz model for anisotropic ferromagnets. We find such solutions in all topological sectors, including N = 0. The non-topological structures can be madel to move with a constant velocity and in head-on collisions they scatter at 90°. The topological structures rotate around their centre of mass. All structures are stable with respect to small perturbations.