Regular Series


Vol. 42 (2011), No. 8, pp. 1741 – 1890


Bright and Dark Solitons of an Integrable Equation Governing Short Waves in a Long-wave Model with Perturbation Terms

abstract

We consider an integrable equation governing short waves in a long-wave model, derived recently by Faquir et al. [M.J. Faquir, M.A. Manna, A. Neveu, Proc. R. Soc. A463, 1939 (2007)]. The study is conducted in presence of perturbation terms. The perturbation terms that are considered are non-linear dispersion terms and fourth order dispersion. The solitary wave Ansatz is used to carry out the integration of the considered perturbed evolution equation. Both bright and dark solitons solutions are obtained. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The conditions of the existence of the derived solitons are derived.


Bender–Dunne Orthogonal Polynomials, Quasi-Exact Solvability and Asymptotic Iteration Method for Rabi Hamiltonian

abstract

We present a method for obtaining the quasi-exact solutions of the Rabi Hamiltonian in the framework of the asymptotic iteration method (AIM). The energy eigenvalues, the eigenfunctions and the associated Bender–Dunne orthogonal polynomials are deduced. We show (i) that orthogonal polynomials are generated from the upper limit (i.e., truncation limit) of polynomial solutions deduced from AIM, and (ii) prove to have nonpositive norm.


On the Gravito-Electromagnetic Analogy

abstract

Earlier research by Zel’manov and by Hönl and Dehnen has shown how the geodesic equation for a charged test particle can be written as a Lorentz force law in which the four-velocity \(u^i\) of an observer in the physical three-space \(\gamma _{\alpha \beta }=-g_{\alpha \beta } + g_{0\alpha }g_{0\beta }\)/\(g_{00}\) is regarded as a gravitational vector potential. Analysing this analogy further, we write the four \(\binom {i}{0}\) components of the Einstein equations in a form resembling a non-linear Maxwell system, which, for a stationary field, is most clearly understood from the Kaluza–Klein perspective, the projection being from four dimensions to three, rather than from five dimensions to four. For the vacuum theory defined by vanishing energy-momentum tensor, \(T_{ij}=0\), these equations exhibit the structure of a non-linear sigma model, found by Ernst, and investigated by Gibbons and Hawking and by Sanchez, the scalar potentials of which we here relate to the gravito-electromagnetic fields. The non-stationary gravitational field is also considered in the normal coordinate system introduced by de Donder and Lanczos, in which case a gravitational displacement current occurs in the three \(\binom {\alpha }{0}\) field equations, converting the (\((_{00})\)/\(g_{00}\)) and \(\binom {\alpha }{0}\) components into a dynamical system. Finally, we discuss the vacuum degeneracy of the superstring theory, arguing from the quantum-gravitational path-integral method that Minkowski space is favoured probabilistically over the stringy vacuum state, in agreement with observation.

See Erratum Acta Phys. Pol. B 52, 1283 (2021)


Trace Operators for the State Labelling Problem in the Exceptional Lie Algebra \(F_{4}\)

abstract

The Casimir operators of the exceptional Lie algebra \(F_{4}\) are constructed using the method of trace operators. Taking into account a decomposition of matrices related to the embedding \(F_{4}\supset \mathfrak {so}(9)\), subgroup scalars for the corresponding state labelling problem are determined as traces of powers of the components, enabling us to propose an orthonormal basis of states for each generic irreducible representation (IRREP) of \(F_{4}\). The basis of eigenstates for the IRREP \([1000]\) of \(F_{4}\) is explicitly given.


Twisted Acceleration-Enlarged Newton–Hooke Space-Times and Conservative Force Terms

abstract

There are analyzed two classical systems defined on twist-deformed acceleration-enlarged Newton–Hooke space-times — non-relativistic particle moving in constant field force \(\vec {F}\) and harmonic oscillator model. It is demonstrated that only in the case of canonical twist deformation the force terms generated by space-time non-commutativity remain conservative for both models.


On the Permutational Symmetry of the Hubbard Model

abstract

The application of the Jucys–Murphy operators, generating a maximal Abelian subalgebra in the group algebra of the symmetric group, in the process of immediate diagonalisation of the one-dimensional Hubbard Hamiltonian is demonstrated. The way of construction of appropriate projection operators of the Young orthogonal basis is pointed out, and the fact that these operators play a role of eigenvectors for Jucys–Murphy operators in the group algebra of the symmetric group is underlined. It is indicated that this operator technique is competitive to the Kostka matrix at the level of bases, which yields matrices of appropriate Clebsh–Gordan coefficients. The permutational symmetry of the lattice chain with \(N\) sites occupied by two electrons is discussed in detail.


Implementing the Frequency Filtering into a Dictionary Image

abstract

In our previous research we have proposed new method for image compression which was based on LZ77 dictionary algorithm. We introduced two modifications such as inaccurate color matching and noise acceptance. Experimental results presented in that paper proved that the new method of image compression gives promising results as compared with original LZ77 dictionary algorithm. In this paper, we propose to supplement the previous algorithm with frequency content reduction. To this end we propose an interpolation based scaling and a compression along the vertical axes. The obtained results are compared to the previously proposed method.


The Heavy Baryon Masses in Variational Approach and Spin–Isospin Dependence

abstract

By using the variational approach, we have studied the strange, charmed and beauty baryons masses. The considered potential is Coulomb as well as linear confining terms and the spin–isospin dependent potential is regarded as a perturbation, too. Some numerical results are given for spectra of heavy baryons and compared with experiments or other works.


Exclusive \(\pi ^+\pi ^-\) Production at the LHC with Forward Proton Tagging

abstract

A process of central exclusive \(\pi ^+\pi ^-\) production in proton–proton collisions and its theoretical description is presented. A possibility of its measurement, during the special low luminosity LHC runs, with the help of the ATLAS central detector for measuring pions and the ALFA stations for tagging the scattered protons is studied. A visible cross-section is estimated to be 21 \(\mu \)b for \(\sqrt {s}=7\) TeV, which gives over 2000 events for 100 \(\mu \)b\(^{-1}\) of integrated luminosity. Differential distributions in pion pseudorapidities, pion and proton transverse momenta as well as \(\pi ^+\pi ^-\) invariant mass are shown and discussed.


Predictions for a Superheavy Element 120

abstract

Predictions are made for the decay chains of the nuclei \(^{298}\)120 and \(^{299}\)120, i.e. for two isotopes of the not-yet-observed superheavy element 120. These nuclei are planned to be synthesized in the nuclear reaction \(^{54}\)Cr + \(^{248}\)Cm, in an experiment to be performed in Darmstadt (Germany). We predict that at least four \(\alpha \) decays in both the \(^{298}\)120 and \(^{299}\)120 chains should be observed. This means that at least six new superheavy nuclides and one new superheavy element (120) should be seen, if the cross section for the reaction is sufficiently large. The predicted half-lives: 11 \(\mu \)s and 15 \(\mu \)s for the nuclei \(^{298}\)120 and \(^{299}\)120, respectively, indicate that we are not far from the lower limit of the half-life (about 1 \(\mu \)s) for a nucleus to be observable. Due to this, the planned experiment will be an important step towards answering the essential question: where is the limit of the periodic table of the elements?


Spatial Solitons in Nonlinear Schrödinger Equation with Variable Nonlinearity and Quadratic External Potential

abstract

An improved self-similar transformation is used to construct exact solutions of the nonlinear Schrödinger equation with variable nonlinearity and quadratic external potential, which both depend on the distance of propagation and the transverse spatial coordinate. By means of analytical and numerical methods we reveal the main features of the spatial solitons found. We focus on the most important optical examples, where the applied optical field is a function of both linearly or periodically varying distance and spatial coordinate. In the case of periodically varying nonlinearity, the variations of confining external potential are found to be sign-reversible (periodically attractive and repulsive) and thus supporting the soliton management.


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