Orthogonal polynomials with weight exp[$−NV (x)$] are studied where $V (x) = \sum_{k=1}^{d} a_{2k} x^{2k}$ is a polynomial of order $2d$. The generalized Freud’s equations for $d = 3, 4$ and 5 are derived and using this $R_{\mu} = h_{\mu} / h{\mu−1}$ is obtained, where $h_{\mu}$ is the normalization constant for the corresponding orthogonal polynomials. Moments of the density functions, expressed in terms of $R_{\mu}$ , are obtained using Freud’s equation and using this, explicit results of level densities as $N \rightarrow \infty$ are derived using the method of resolvents. The results are compared with those using Dyson–Mehta method.

In this paper, we obtain the 1-soliton solutions of the $(3 + 1)$-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and the generalized Benjamin equation. By using two solitary wave ansatz in terms of sech$^{p}$ and tanh$^{p}$ functions, we obtain exact analytical bright and dark soliton solutions for the considered model. These solutions may be useful and desirable for explaining some nonlinear physical phenomena in genuinely nonlinear dynamical systems.

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = af(y(t - \tau)) - {\text{by}} (t)$, where $D^{\alpha}$ is a Caputo fractional derivative of order $0 < \alpha \leq 1$. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay.

The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the $(2 + 1)$-dimensional Camassa–Holm Kadomtsev–Petviashvili equation and the higher-order nonlinear Schrödinger equation. By using this useful method, we found some exact solutions of the above-mentioned equations. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. It is shown that the proposed method is effective and general.

Quantum Hamilton–Jacobi formalism is used to give a proof for Gozzi’s criterion, which states that for eigenstates of the supersymmetric partners, corresponding to the same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We also show that this proof is also applicable to the case, where isospectral deformation is involved.

The Coxeter–Weyl groups $W(A_{4})$, $W(B_{4})$ and $W(D_{4})$ have proven very useful for two-qubit systems in quantum information theory. A simple technique is employed to construct the unitary matrix representations of the groups, based on quaternionic transformation of the usual reflection matrices. The von Neumann entropy of each reduced density matrix is calculated. It is shown that these unitary matrix representations are naturally related to various universal quantum gates and they lead to entangled states. Canonical decomposition of generators in terms of fundamental gate representations is given to construct the quantum circuits.

Matter-wave bright solitons in bichromatic lattice potentials are considered and their dynamics for different lattice environments are studied. Bichromatic potentials are created from superpositions of (i) two linear optical lattices and (ii) a linear and a nonlinear optical lattice. Effective potentials are found for the solitons in both bichromatic lattices and a comparative study is done on the dynamics of solitons with respect to the effective potentials. The effects of dispersion on solitons in bichromatic lattices are studied and it is found that the dispersive spreading can be minimized by appropriate combinations of lattice and interaction parameters. Stability of nondispersive matter-wave solitons is checked from phase portrait analysis.

Static spherically symmetric space-time is studied to describe dense compact star with quark matter within the framework of MIT Bag Model. The system of Einstein’s field equations for anisotropic matter is expressed as a new system of differential equations using transformations and it is solved for a particular general form of gravitational potential with parameters. For a particular parameter, as an example, it is shown that the model satisfies all major physical features expected in a realistic star. The generated model also smoothly matches with the Schwarzschild exterior metric at the boundary of the star. It is shown that the generated solutions are useful to model strange quark stars.

Multiplicity distributions of shower particles and target fragments in $^{7}$Li–Em (emulsion) collisions at 3 A GeV/c are experimentally studied. In the framework of the multisource thermal model, the multicomponent Erlang distribution is used to describe the experimental multiplicity distributions of shower particles, grey fragments, black fragments, and heavily ionized fragments. The correlations between these multiplicities are experimentally reported. With the increase of impacting centrality (or the target fragment multiplicity), a saturation phenomenon for shower particle multiplicity is observed in the experiment.

Theoretical and experimental studies of temporal dynamics of grazing incidence grating (GIG) cavity, single-mode dye laser pumped by high repetition rate copper vapour laser (CVL) are presented. Spectral chirp of the dye laser as they evolve in the cavity due to transient phase dynamics of the amplifier gain medium is studied. Effect of grating efficiency, focal spot size, pump power and other cavity parameters on the temporal behaviour of narrow band dye laser such as build-up time, pulse shape and pulse width is studied using the four level dye laser rate equation and photon evolution equation. These results are compared with experimental observations of GIG single-mode dye laser cavity. The effect of pulse stretching of CVL pump pulse on the temporal dynamics of the dye laser is studied.

The quantum capacitance, an important parameter in the design of nanoscale devices, is derived for armchair-edge single-layer graphene nanoribbon with semiconducting property. The quantum capacitance oscillations are found and these capacitance oscillations originate from the lateral quantum confinement in graphene nanoribbon. Detailed studies of the capacitance oscillations demonstrate that the local channel electrostatic potential at the capacitance peak, the height and the number of the capacitance peak strongly depend on the width, especially a few nanometres, of the armchair-edge graphene nanoribbon. It implies that the capacitance oscillations observed in the experiments can be utilized to measure the width of graphene nanoribbon. The results also show that the capacitance oscillations are not seen when the width is larger than 30 nm.

Well-dispersed undoped and Mg-doped ZnO nanoparticles with different doping concentrations at various annealing temperatures are synthesized using basic chemical solution method without any capping agent. To understand the effect of Mg doping and heat treatment on the structure and optical response of the prepared nanoparticles, the samples are characterized using X-ray diffraction (XRD), energy-dispersive X-ray (EDX), UV–Vis optical absorption, photoluminescence (PL), scanning electron microscopy (SEM) and transmission electron microscopy (TEM) measurements. The UV–Vis absorbance and PL emission show a blue shift with increasing Mg doping concentration with respect to bulk value. UV–Vis spectroscopy is also used to calculate the band-gap energy of nanoparticles. X-ray diffraction results clearly show that the Mg-doped nanoparticles have hexagonal phase similar to ZnO nanoparticles. TEM image as well as XRD study confirm the estimated average size of the samples to be between 6 and 12 nm. Furthermore, it is seen that there was an increase in the grain size of the particles when the annealing temperature is increased.

Nanostructured ZnO:Mn thin films have been prepared by sol–gel dip coating method. The content of Mn in the sol was varied from 0 to 12 wt%. The effect of Mn concentration on the optical, structural, and morphological properties of ZnO thin films were studied by using Fourier transform infrared (FTIR), UV–visible and photoluminescence (PL) spectroscopy, X-ray diffraction (XRD) and scanning electron microscopy (SEM). XRD results showed that the films have hexagonal wurtzite structure at lower content of Mn. The diffraction peaks corresponding to ZnO disappeared and two diffraction peaks of MnO₂ and Mn₃O₄ appeared at the highest value of doping concentration (viz., 12 wt$\%$). SEM results revealed that the surface smoothness of the films improved at higher content of Mn. The optical band gap of the films decreased from 3.89 to 3.15 eV when the Mn concentration increased from 0 to 12 wt$\%$. The PL spectra of the films showed the characteristic peaks linked to band-to-band, green and yellow emissions. Besides, the PL intensity of the samples decreased with increase in Mn concentration.

The GRAPES-3 experiment at Ooty contains a large-area (560 m$^{2}$) tracking muon detector. This detector consists of 16 modules, each 35 m$^{2}$ in area, that are grouped into four supermodules of 140 m$^{2}$ each. The threshold energy of muons is $\sec(\theta)$ GeV along a direction with zenith angle $\theta$ and the angular resolution of the muon detector is $6^{\circ}$. Typically, it records $\sim 4 \times 10^{9}$ muons every day. The muon detector has been operating uninterruptedly since 2001, thus providing a high statistics record of the cosmic ray flux as a function of time over one decade. However, prior to using these data, the muon rate has to be corrected for two important atmospheric effects, namely, variations in atmospheric pressure and temperature. Because of the near equatorial location of Ooty ($11.4^{\circ}$N), the seasonal variations in the atmospheric temperature are relatively small and shall be ignored here. Due to proximity to the equator, the pressure changes at Ooty display a dominant 12 h periodic behaviour in addition to other seasonal changes. Here, we discuss various aspects of a novel method for accurate pressure measurement and subsequent corrections applied to the GRAPES-3 muon data to correct these pressure-induced variations. The pressure-corrected muon data are used to measure the profile of the solar diurnal anisotropy during 2006. The data, when divided into four segments, display significant variation both in the amplitude ($\sim 45\%$) and phase ($\sim42$ m) of the solar diurnal anisotropy during 2006, which was a period of relatively low solar activity.

In this article, the eigenvalues for the three-body interactions on the line and the Landau levels in the presence of topological defects have been regenerated by the Nikiforov–Uvarov (NU) method. Two exhaustive lists of such exactly solvable potentials are given.