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      Volume 87, Issue 2

      August 2016

    • The ($G' /G, 1/G$)-expansion method for solving nonlinear space–time fractional differential equations


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      In this work, we present ($G' /G, 1/G$)-expansion method for solving fractional differential equations based on a fractional complex transform. We apply this method for solving space–time fractional Cahn--Allen equation and space--time fractional Klein–Gordon equation. The fractional derivatives are described in the sense of modified Riemann--Lioville. As a result of some exact solution in the form of hyperbolic, trigonometric and rational solutions are deduced. The obtained solutions may be used for explaining of some physical problems.The($G' /G, 1/G$)-expansion method has a wider applicability for nonlinear equations. We have verified all the obtained solutions with the aid of Maple.

    • First integrals and analytical solutions of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient


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      Fin materials can be observed in a variety of engineering applications. They are used to ease the dissipation of heat from a heated wall to the surrounding environment. In this work, we consider a nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient. The equation(s) under study are highly nonlinear. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. Firstly, we consider the Lie group analysis for different cases of thermal conductivity and the heat transfer coefficients. These classifications are obtained from the Lie group analysis. Then, the first integrals of the nonlinear straight fin problem are constructed by three methods, namely, Noether’s classical method, partial Noether approach and Ibragimov’s nonlocal conservation method. Some exact analytical solutions are also constructed. The obtained result is also compared with the result obtained by other methods.

    • Angular momentum and the electromagnetic top


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      The electric charge–magnetic dipole interaction is considered. If $\Gamma_{\rm em}$ is the electromagnetic and $\Gamma_{\rm mech}$ the mechanical angular momentum, the conservation law for the total angular momentum $\Gamma_{\rm tot}$ holds: $\Gamma_{\rm tot}$ =$\Gamma_{\rm em}$ + $\Gamma_{\rm mech}$ = ${\rm const.}$, but when the dipole moment varies with time, $\Gamma_{\rm mech}$ is not conserved. We show that the non-conserved $\Gamma_{\rm mech}$ of such a macroscopic isolated system might be experimentally observable. With advanced technology, the strength of the interaction hints to the possibility of novel applications for gyroscopes, such as the electromagnetic top.

    • Nonlinear Rayleigh--Taylor instability of the cylindrical fluid flow with mass and heat transfer


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      The nonlinear Rayleigh--Taylor stability of the cylindrical interface between the vapour and liquid phases of a fluid is studied. The phases enclosed between two cylindrical surfaces coaxial with mass and heat transfer is derived from nonlinear Ginzburg--Landau equation. The F-expansion method is used to get exactsolutions for a nonlinear Ginzburg--Landau equation. The region of solutions is displayed graphically.

    • Brief report: Volume dependence of Grüneisen parameter for solids under extreme compression


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      The Nie expression is amended in such a way that the expression follows the infinite pressure behaviour, i.e., P → ∞or V → 0. A new empirical relationship is developed to predict the values of volume dependence of Grüneisen parameter. NaCl and ε-Fe have been employed to test the suitability of the expression.The results obtained reveal that the relationship is reliable as there is a good agreement between the calculated and the experimental data

    • Investigations on the local structure and the spin-Hamiltonian parameters for the tetragonal $Cu^{2+}$ centre in $ZnGeF_{6}·6H_{2}O$ crystal


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      The spin-Hamiltonian parameters ($g$ factors $g_{||}, g{|perp}$ and hyperfine structure constants $A_{||}$, $A{|perp}$) and the local structure for the tetragonal $Cu^{2+}$ centre in trigonal $ZnGeF_{6}·6H_{2}O$ crystal are theoretically studied using the perturbation formulae of these parameters for a 3d9 ion in tetragonally elongated octahedra. In the calculations, the contributions to the spin-Hamiltonian parameters from ligand orbital and spin-orbit coupling are included on the basis of the cluster approach in view of moderate covalency of the studied systems, and the required crystal field parameters are obtained using the superposition model and the local structures of the studied $[Cu(H_{2}O)_{6}]^{2+}$ cluster. According to the calculations, the ligand octahedra around $Cu^{2+}$ suffer relative elongation$\tau{\sim 0.085 \AA) along the [0 0 1] (or $C_4$) axis for the tetragonal $Cu^{2+}$ centres in $ZnGeF_{6}·6H_{2}O$ crystal, due to the Jahn--Teller effect. The calculated results show good agreement with the experimental data.

    • A Bohmian approach to the perturbations of non-linear Klein--Gordon equation


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      In the framework of Bohmian quantum mechanics, the Klein--Gordon equation can be seen as representing a particle with mass m which is guided by a guiding wave $\phi(x)$ in a causal manner. Here a relevant question is whether Bohmian quantum mechanics is applicable to a non-linear Klein--Gordon equation? We examine this approach for $\phi_{4}(x)$ and sine-Gordon potentials. It turns out that this method leads to equations for quantum states which are identical to those derived by field theoretical methods used for quantum solitons. Moreover, the quantum force exerted on the particle can be determined. This method can be used for other non-linear potentials as well.

    • Chaotic behaviour from smooth and non-smooth optical solitons under external perturbation


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      Smooth and non-smooth optical solitons in the nonlinearly dispersive Schrödinger equation are given by phase portraits. The Melnikov technique is used to detect conditions for chaotic motion of this deterministic system and to analyse conditions for the suppression of chaos. Our results show that the system is in a state of Melnikov chaos by external disturbances. After the implementation of the controlled system, the optical solitons can transmit in a stable station for a long time. Numerical simulation also shows that maximum interference frequency of the system enables the dynamic behaviour to be more complex. The effect of controller parameter on phase portraits as well as on the numerical simulations of bifurcation diagram and maximum Lyapunov exponents are also investigated.

    • End modes in arrays of modulated Su--Schrieffer--Heeger chains


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      In this article, an extended and modulated version of the classic Su--Schrieffer--Heeger model is analysed. The nature of the end modes and the effect of cyclic modulation of the hopping parameters are studied in detail. The analysis is extended to the case of an array of linear chains described by the Su--Schrieffer--Heeger model, where the robustness of the end states for a large range of coupling strengths between the chains is found.

    • Effects of particle size and laser wavelength on heating of silver nanoparticles under laser irradiation in liquid


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      Laser energy absorption results in significant heating of metallic nanoparticles and controlling the heating of nanoparticles is one of the essential stages of selective cell targeting. It is necessary to note that the laser action should be done by laser pulses with a wavelength that is strongly absorbed by the particles and it is important to select wavelengths that are not absorbed by the medium. Laser pulse duration must be chosen sufficiently short to minimize heat flow emitted from absorbing particles. Numerical calculations based on Mie theory were used to obtain the effect of laser wavelength and particle size on absorption factor for colloidal silver nanoparticles with radii between 5 and 50 nm. Calculations for acquiring temperatures under irradiations of pulsed KrF laser and pulsed Nd:YAG laser were performed. We showed that for low wavelengths of the laser, smaller nanoparticles have larger absorption efficiency compared to larger nanoparticles and in high wavelengths, temperature of all particles increased in the same way.

    • On phase-space representations of quantum mechanics using Glauber coherent states


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      A phase-space formulation of quantum mechanics is proposed by constructing two representations (identified as $pq$ and $qp$) in terms of the Glauber coherent states, in which phase-space wave functions (probability amplitudes) play the central role, and position $q$ and momentum $p$ are treated on equal footing. After finding some basic properties of the $pq$ and $qp$ wave functions, the quantum operators in phase-space are represented by differential operators, and the Schrödinger equation is formulated in both pictures. Afterwards, the method is generalized to work with the density operator by converting the quantum Liouville equation into $pq$ and $qp$ equations of motion for two-point functions in phase-space. A coordinate transformation between those points allows one to construct a cell in phase-space, whose central point can be treated as a parameter. In this way, one gets equations of motion describing the evolution of one-point functions in phase-space. Finally, it is shown that some quantities obtained in this paper are related in a natural way with cross-Wigner functions, which are constructed with either the position or the momentum wave functions.

    • Momentum distribution of charged particles in jets in dijet events and comparison to perturbative QCD predictions


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      Inclusive momentum distributions of charged particles are measured in dijet events. Events were produced at the AMY detector with a centre of mass energy of 60 ${\rm GeV}$. Our results were compared, on the one hand to those obtained from other $e^+ e^-$, $ep$ as well as CDF data, and on the other hand to the perturbative QCD calculations carried out in the framework of the modified leading log approximation (MLLA) and assuming local parton--hadron duality (LPHD). A fit of the shape of the distributions yields $\scr Q_{eff} = 263 \pm 13 {\rm MeV}$ for the AMY data. In addition, a fit to the evolution of the peak position with dijet mass using all data from different experiments gives $\scr Q_{eff} = 226 \pm 18 {\rm MeV}$. Next, αs was extracted using the shape of the distribution at the Z0 scale, with a value of 0.118 \pm 0.013. This is consistent, within the statistical errors, with many accurate measurements. We conclude that it is the success of LPHD + MLLA that the extracted value of $\alpha_{s}$ is correct. Possible explanations for all these features will be presented in this paper.

    • Quantum ring states in magnetic field and delayed half-cycle pulses


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      The present work is dedicated to the time evolution of excitation of a quantum ring in external electric and magnetic fields. Such a ring of mesoscopic dimensions in an external magnetic field is known to exhibit a wide variety of interesting physical phenomena. We have studied the dynamics of the single electron quantum ring in the presence of a static magnetic field and a combination of delayed half-cycle pulse pair. Detailed calculations have been worked out and the impact on dynamics by variation in the ring radius, intensity of external electric field, delay between the two pulses, and variation in magnetic field have been reported. A total of 19 states have been taken and the population transfer in the single electron quantum ring is studied by solving the time-dependent Schrödinger equation (TDSE), using the efficient fourth-order Runge--Kutta method. Many interesting features have been observed in the transition probabilities with the variation of magnetic field, delay between pulses and ring dimensions. A very important aspect of the present work is the persistent current generation in a quantum ring in the presence of external magnetic flux and its periodic variation with the magnetic flux, ring dimensions and pulse delay.

    • Structural, photoconductive, thermoelectric and activation energy measurements of $V$-doped transparent conductive $SnO_{2}$ films fabricated by spray pyrolysis technique


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      This report investigated the structural, optical and electrical properties of V-doped $SnO_{2}$ thin films deposited using spray pyrolysis technique. The $SnO_{2}$:$V$ films, with different $V$-content, were deposited on glasssubstrates at a substrate temperature of $550\deg C$ using an aqueous ethanol solution consisting of tin and vanadium chloride. X-ray diffraction studies showed that the $SnO_{2}$:$V$ films were polycrystalline only with tin oxide phasesand the preferred orientations are along (1 1 0), (1 0 1), (2 1 1) and (3 0 1) planes. Using Scherrer formula, the grain sizes were estimated to be within the range of 25--36 nm. The variation in sheet resistance and optical direct band gap are functions of vanadium doping concentration. Field emission scanning electron microscopy (FESEM) revealed the surface morphology to be very smooth, yet grainy in nature. Optical transmittance spectra of the films showed high transparency of about $\approx 69--90%$ in the visible region, decreasing with increase in $V$-doping. The direct band gap for undoped $SnO_{2}$ films was found to be 3.53 eV, while for higher V-doped films it shifted toward lower energies in the range of 3.27--3.53 eV and then increased again to 3.5 eV. The Hall effect and Seebeck studies revealed that the films exhibit n-type conductivity. The thermal activation energy, Seebeck coefficient and maximum of photosensitivity in the films were found to be in the range of 0.02--0.82 eV (in thelow-temperature range), $0.15--0.18 {\rm mV K^{−1}}$ (at $T = 350 K$) and 0.96--2.84, respectively.

    • Kink degeneracy and rogue potential solution for the (3+1)-dimensional B-type Kadomtsev--Petviashvili equation


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      In this paper, we obtained the exact breather-type kink soliton and breather-type periodic soliton solutions for the (3+1)-dimensional B-type Kadomtsev--Petviashvili (BKP) equation using the extended homoclinic test technique. Some new nonlinear phenomena, such as kink and periodic degeneracies, are investigated. Using the homoclinic breather limit method, some new rational breather solutions are found as well. Meanwhile, we also obtained the rational potential solution which is found to be just a rogue wave. These results enrich thevariety of the dynamics of higher-dimensional nonlinear wave field.

    • Dynamics of ‘quantumness’ measures in the decohering harmonic oscillator


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      We studied the behaviour under decoherence of four different measures of the distance between quantum states and classical states for the harmonic oscillator coupled to a linear Markovian bath. Three of these are relative measures, using different definitions of the distance between the given quantum states and the set of all classical states. The fourth measure is an absolute one, the negative volume of the Wigner function of the state. All four measures are found to agree, in general, with each other. When applied to the eigenstates $|n\ rangle$, all four measures behave non-trivially as a function of time during dynamical decoherence. First, we find that the first set of classical states to which the set of eigenstate evolves is (by all measures used) closest to the initial set. That is, all the states decohere to classicality along the ‘shortest path’. Finding this closest classical set of states helps improve the behaviour of all the relative distance measures. Second, at each point in time before becoming classical, all measures have a state $n*$ with maximal quantum-classical distance; the value $n*$ decreases as a function of time. Finally, we explore the dynamics of these non-classicality measures for more general states.

    • L subshell fluorescent $X$-ray measurements to study $CK$ transitions in the $66\leqZ\leq83$ region


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      L subshell fluorescent X-rays in Dy, Ho, Er, Lu, Ta, W, Pt, Au, Hg, Pb and Bi have been measured using synchrotron with selective creation of electron vacancies in individual subshells. Coster--Kronig (CK) yields were derived from the measured intensities. Present measurements have been made at photon energies above the edges where differences between measured and theoretical attenuation coefficients are almost negligible. Parametric trends for the results with $Z$ were developed to cover all $Zs$ in the range of 66--83. The trends predict the switching-off of L1--L2, N1 transition at Z = 67. The extent of fall/rise of $f_Lij$ values corresponding to off/on of certain transitions is found inversely proportional to the difference in binding energies of two consecutive subshells involved in the transition. For $Zs$ above/below these rises/falls, $fL_13$ and $fL_12$ values are almost constants. $f_L23$ values involving no break at $Zs$ follow the general photoionization behaviour that ionization probability is highest at the edge energy and decreases with photon energy. Yield measurements with synchrotron radiation for Dy, Ho, Lu, Hg and Bi and experimental values for $f_L23$, $f_L12$ in Lu and for $f_L13$ in Ta are being quoted for the first time.

    • Hermite-distributed approximating functional-based formulation of multiconfiguration time-dependent Hartree method: A case study of quantum tunnelling in a coupled double-well system


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      We propose a variant of the multiconfiguration time-dependent Hartree (MCTDH) method within the framework of Hermite-distributed approximating functional (HDAF) method. The discretized Hamiltonian is a highly banded Toeplitz matrix which significantly reduces computational cost in terms of both storage and number of operations. The method proposed is employed to carry out the study of tunnelling dynamics of two coupled double well oscillators. We have calculated the orthogonality time \tau , which is a measure of the time interval for an initial state to evolve into its orthogonal state. It is observed that the coupling has a significant effect on \tau .

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