Entanglement is one of the most surprising features of composite quantum systems. Yet, challenges remain in our understanding and quantification of the entanglement. There is no unique degree of entanglement from various measures, as presented by numerous studies on quantifying entanglement. As indicated in this paper, any degree of entanglement for two qubits resulting from a particular measure can be detected in excess of onecorresponding pure state. Evidently, those identical entanglement pure states can be counted as a quantitative condition to be satisfied by other proper measures. The most popular measures of pure states for two qubits are based on the same structure, as indicated in this paper. Then, the algorithm to detect the identical entanglement pure states for two qubits is proposed based on randomness distillation. Eventually, two sets of identical entanglement states are listed for two qubits.

The bound state solutions to the radial Schrödinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvaluesof the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. $\rm{N}_{2}$,CO,NO and CH. In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.

This study investigates the temperature and concentration gradients on the transfer of heat and mass in the presence of Joule heating, viscous dissipation and time-dependent first-order chemical reaction in the flow of micropolar fluid. Governing boundary value problems are solved analytically and the effects of parameters involved are studied. The behaviour of the Nusselt number (at both disks) is noted and recorded in a tabular form. The present results have an excellent agreement with the already published results for a special case. The rate of transport of heat by concentration gradient and the diffusion of solute molecules by temperature gradient are increased. The concentration field is increased by constructive chemical reaction and it decreases when the rate of destructive chemical reaction is increased.

In this study, we explore the modified form of (1 + n)-dimensional Zakharov–Kuznetsov equation, which is used to investigate the waves in dusty and magnetised plasma. It is proved that the equation follows the property of nonlinear self-adjointness. Lie point symmetries are calculated and conservation laws in the framework of the new general conservation theorem of Ibragimov are obtained. The $(1/G'), (G'/G)$-expansion and modified Kudryshov methods are applied to extract exact analytical solutions. The so-called bright, dark and singular solutions are also found using the solitary wave ansatz method. The results obtained in this study are new and may be of significant importance where this model is used to study the waves in different plasmas.

In this paper, we analyse the chaotic satellite system using dissipativity, equilibrium points, bifurcation diagrams, Poincare section maps, Lyapunov exponents and Kaplan–Yorke dimension. We obtain the equilibrium points of chaotic satellite system and at each equilibrium point, we obtain the eigenvalue of Jacobian matrix of the satellite system to verify the unstable region.We calculate the Kaplan–Yorke dimension, which ensures the strange behaviour of the system. We observe closely the three-dimensional (3D) phase portraits of the satellite system at different parameter values. We plot the Lyapunov exponent graphs corresponding to every 3D phase portrait of satellite systems, to verify the chaoticity of satellite systems. We establish time-delay synchronisation for twoidentical satellite systems. The simulated and qualitative results are in an excellent agreement.

In this paper, we provide an operational criterion for controlled dense coding (CDC) with a general class of three-qubit partially entangled states. A general three-qubit pure entangled state can be classified into two inequivalent classes according to their genuine tripartite entanglement. We claim that if a three-qubit state shows entanglement characteristic similar to Greenberger–Horne–Zeilinger (GHZ)-class, then such non-trivial tripartitestates are useful in CDC whereas states belonging to the W-class are not useful for that. We start with a particularclass of non-trivial partially entangled states belonging to the GHZ-class and show that they are effective in CDC. Then we cite several other examples of different types of tripartite entangled states to support our conjecture.

In this paper, we present an algorithm by using the homotopy analysis method (HAM), Adomiandecomposition method (ADM) and variational iteration method (VIM) to find the approximate solutions of the space–time fractional cubic isothermal autocatalytic chemical system (STFCIACS). The HAM, ADM and VIMapproximate solutions are evaluated and compared by using the computation program Mathematica and excellent results are obtained.

In this paper, an improvement of the finite-difference time-domain (FDTD) method using a non-standard finite-difference scheme for solving the Schrödinger equation is presented. The standard numerical scheme for a second derivative in the spatial domain is replaced by a non-standard numerical scheme. In order to apply the non-standard FDTD (NSFDTD), first, the estimates of eigenenergies of a system are needed and computed bythe standard FDTD method. These first eigenenergies are then used by the NSFDTD method to obtain improvedeigenenergies. The NSFDTD method can be employed iteratively using the resulting eigenenergies to obtain moreaccurate results. In this paper, the NSFDTD method is validated using infinite square well, harmonic oscillator andMorse potentials. Significant improvements are found when using the NSFDTD method.

Potassium dihydrogen phosphate (KDP) family of hydrogen-bonded crystal constitutes an important family of crystals not only because of their immense importance in the field of nonlinear optics (NLOs) but also due to the fact that hydrogen bonds of these crystals provide us with a rare opportunity to understand the fundamental nature of hydrogen bonds, such as the effect of local chemical environment on the strength of hydrogen bonds and nuclear quantum effect on strong, moderate and weak hydrogen bonds. Keeping this aim in mind, we have undertaken detailed single-crystal neutron diffraction (SCND) investigations on ammonium dihydrogen phosphate(ADP) and KDP along with their deuterated analogue crystals under ambient conditions. Fine differences in thehydrogen bonds of the above-mentioned crystals are analysed in the light of a simple diabatic two-state theoreticalmodel for hydrogen bonds. It is proposed that the presence of a partially covalent $\rm{N–H_{N}–O}$ bond in ADP has a very significant effect on its O–H bond making it highly anharmonic. It is this higher bond anharmonicity in ADPthat is most likely responsible for its larger NLO coefficient compared to KDP.

In this study, the coupling between the stratospheric quasi-biennial oscillation (QBO) and the critical frequency of the ionospheric F2-layer (foF2) was analysed statistically. The multiple regression model was used as a statistical tool. The model was developed by adding the sunspot number (SSN), which affects the foF2 (measured for Madras, Kodaikanal, Bogota, Manila and Tahiti) in the ionosphere at a significant level. Four different ‘Dummy’sets of data were used in the model in order to observe the effect of the direction (east–west) and the magnitude (for both directions, between 0 and 15 m/s and between 0 and 16 m/s and the largest value) of QBO. It was observed that the variations of foF2 in the range of 60–78% in the model could be explained by SSN and $\rm{SSN^{2}}$. The change of 2–13% that occurred in foF2 could be explained by the whole set of QBO. It was also observed that the effect of the direction and magnitude of QBO on foF2 differed between the stations.

This study investigates the spinodal decomposition dynamics in binary mixtures containing mobile particles by combining the Cahn–Hilliard equation with Langevin dynamics for particles with Brownian motionchanges proportional to their mobility. We solve the Cahn–Hilliard equation numerically using a semi-implicit Fourier spectral method, and show that the domain growth rate first increases with the increase in particle mobility, and then decreases. The effect of filler particle concentration on the domain growth depends on its mobility: whenthe particle mobility is low, the domain growth rate decreases with the increase in particle concentration; whereas when the particle mobility is high, the domain growth rate decreases and then increases and finally decreases again with the increase in particle concentration. The proposed model suggests the possibility of controlling macroscopicbehaviour of binary alloys by altering filler particle properties.

In this work, we deal with a nonlinear wave equation, namely the Vakhnenko equation, which models the propagation of nonlinear wave in the barotropic relaxing media. Based on the homoclinic breather limit method, we seek rogue wave solution to the above equation. The results show that rogue wave or giant wave can exist insuch a medium.

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency $\Omega$, amplitude $F_{0}$ and phase $\phi$, i.e. the system with the Hamiltonian of $\hat{H} = (\hat{p}^{2}/2m) − (m\omega^{2}x^{2}/2) − F_{0}x sin(\Omega t + \phi)$. The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables $\xi = x \sqrt{m\omega/hbar}, f_{0} = F_{0}/\omega\sqrt{hbarm\omega}$ and $\tau = \omega t$. The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator.The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading $\sigma (\tau) = \sqrt{\overline{(\Delta\xi^{2}(\tau))}}$ which decreases first from quite macroscopic values of $\sigma_{0} = 2^{12,...,25}$ to minimal one of $\sim(1/\sqrt{2})$ at times $\tau$ < $\tau_{0} = 0.125 1n(16\sigma^{4}_{0} + 1)$ and then grows back unlimitedly. For certain phases $\phi$ depending on the $\Omega/\omega$ ratio and $n = log_{2}\sigma_{0}$, the mass centre of the packet $\xi_{av}(\tau) = \overline{\hat{x}(\tau)} · \sqrt{m\omega/hbar}$ delays approximately two natural ‘periods’ $\sim(4\pi/\omega)$ in the area of the stationary point and then escapes to ‘+’ or ‘−’ infinity in a bifurcating way. For ‘resonant’ $\Omega = \omega$, the bifurcation phases $\phi$ fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic $\phi(\Omega, n\rightarrow\infty)$ obeying the classical formula $\phi_{cl}(\Omega) = −arctg \Omega$ for initial energy $E = 0$ in the wide range of $\Omega = 2^{−4}, ..., 2^{7}$.

This study reveals the dark, bright, combined dark–bright, singular, combined singular optical solitons and singular periodic solutions to the conformable space–time fractional complex Ginzburg–Landau equation. We reach such solutions via the powerful extended sinh-Gordon equation expansion method (ShGEEM). Constraint conditions that guarantee the existence of valid solitary wave solutions are given. Under suitable choice of the parameter values, interesting three-dimensional graphs of some of the obtained solutions are plotted.

The extended definition of the polynomial B-splines may give a chance to improve the results obtained by the classical cubic polynomial B-splines. The optimum value of the extension parameter can be determined by scanning some intervals containing zero. This study aims to solve some initial boundary value problems constructed for the Gardner equation by the extended cubic B-spline collocation method. The test problems are derived from some analytical studies to validate the efficiency and accuracy of the suggested method. The conservation laws are also determined to observe whether the test problems remain constant as expected from the theoretical aspect. The stability of the proposed method is investigated by the von Neumann analysis.