Volume 71, Issue 2

In this paper, a (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which is a generalisation of the NLS equation, is under investigation. The classical and generalised N -fold Darboux transformations are constructed in terms of determinant representations. With the non-vanishing background and iterated formula, a family of the analytical solutions of the (2+1)-dimensional NLS equation are systematically generated, including the bright-line solitons, breathers, and rogue waves. The interaction mechanisms between two bright-line solitons and among three bright-line solitons are both elastic. Several patterns for first-, second, and higher-order rogue wave solutions fixed at space are displayed, namely, the fundamental pattern, triangular pattern, and circular pattern. The two-dimensional space structures of first-, second-, and third-order rogue waves fixed at time are also demonstrated.

The modified simple equation method, the exp-function method, and the method of soliton ansatz for solving nonlinear partial differential equations are presented. Based on these three different methods, we obtain the exact solutions and the bright–dark soliton solutions with parameters of the long-short wave resonance equations which describe the resonance interaction between the long wave and the short wave. When these parameters take special values, the solitary wave solutions are derived from the exact solutions. We compare the results obtained using the three methods. Also, a comparison between our results and the well-known results is given.

The method of the reverberation-ray matrix has been developed and successfully applied to analyse the wave propagation in a multibranched framed structure or in a layered medium. However, the method is confined to the case of mechanical loads applied at the medium until now. This paper aims to extend the formulation of the reverberation-ray matrix to cases of thermal propagation and diffusion. The thermal response in functionally graded materials (FGM) with the non-Fourier heat conduction model is analysed. In the present work, it is assumed that the material properties of an FG plate vary only in the thickness direction by following the power law function. The effect of non-Fourier and material inhomogeneity in the plate subjected to a periodic thermal disturbance is investigated. The present approach is validated by comparing it with the solutions obtained by other methods.

By imposing reduction conditions on rational solutions for a system involving the Hirota–Miwa equation, rational solutions for lattice potential KdV equation are constructed. Besides, the rational solutions for two semi-discrete lattice potential KdV equations are also considered. All these rational solutions are in the form of Schur function type.

In this work, detailed first-principles calculations within the generalised gradient approximation (GGA) of electronic, structural, magnetic, and optical properties of Ni,Ti, and Al-based Heusler alloys are presented. The lattice parameter of C1 b with space group F 4̅3 m (216) NiTiAl alloys is predicted and that of Ni 2 TiAl is in close agreement with available results. The band dispersion along the high symmetry points W → L →Γ→ X → W → K in Ni 2 TiAl and NiTiAl Heusler alloys are also reported. NiTiAl alloy has a direct band gap of 1.60 eV at Γ point as a result of strong hybridization between the d state of the lower and higher valence of both the Ti and Ni atoms. The calculated real part of the dielectric function confirmed the band gap of 1.60 eV in NiTiAl alloys. The present calculations revealed the paramagnetic state of NiTiAl. From the band structure calculations, Ni 2 TiAl with higher Fermi level exhibits metallic properties as in the case of both NiAl and Ni 3 Al binary systems.

The structural, mechanical, electronic, dynamic, and optical properties of the ZrPdSn compound crystallising into the MgAgAs structure are investigated by the ab initio calculations based on the density functional theory. The lattice constant, bulk modulus, and first derivative of bulk modulus were obtained by fitting the calculated total energy-atomic volume results to the Murnaghan equation of state. These results were compared to the previous data. The band structure and corresponding density of states (DOS) were also calculated and discussed. The elastic properties were calculated by using the stress-strain method, which shows that the MgAgAs phase of this compound is mechanically stable. The presented phonon dispersion curves and one-phonon DOS confirms that this compound is dynamically stable. In addition, the heat capacity, entropy, and free energy of ZrPdSn were calculated by using the phonon frequencies. Finally, the optical properties, such as dielectric function, reflectivity function, extinction coefficient, refractive index, and energy loss spectrum, were obtained under different pressures.

This work is related to the dynamics of a discrete-time density-dependent generalised Beddington model. Moreover, we investigate the existence and uniqueness of positive equilibrium point, boundedness character, local and global behaviours of unique positive equilibrium point, and the rate of convergence of positive solutions that converge to the unique positive equilibrium point of this model. Numerical examples are provided to illustrate theoretical discussion.

In this work the nuclear level density parameters of 238 U have been extracted in the back-shifted Fermi gas model (BSFGM), as well as the constant temperature model (CTM), through fitting with the recent experimental data on nuclear level densities measured by the Oslo group. The excitation functions for 238 U(p,2n α ) 233 Pa, and 238 U(p,4n) 235 Np reactions and the fragment yields for the fragments of the 238 U(p,f) reaction have been calculated using obtained level density parameters. The results are compared to their corresponding experimental values. It was found that the extracted excitation functions and the fragment yields in the CTM coincide well with the experimental values in the low-energy region. This finding is according to the claim made by the Oslo group that the extracted level densities of 238 U show a constant temperature behaviour.

Lower and upper bounds for the total π -electron energy are obtained, which are applicable to conjugated π -electron systems with non-bonding molecular orbitals (NBMOs). These improve the earlier estimates, in which the number of NBMOs has not been taken into account.

Carbon nanotube (CNT) reinforced nanocrystalline aluminum matrix composites are fabricated by a simple and effective physical mixing method with sonication. In this study, the microstructural characterisations and property evaluations of the nanocomposites were performed. The structural characterisations revealed that CNTs were dispersed, embedded, and anchored within the metal matrix. A strong interfacial adhesion appeared between CNTs and nanocrystalline aluminum as a result of the fabrication process. Raman and Fourier transform infrared spectroscopic studies also confirmed the surface adherence of CNTs with nanocrystalline aluminum matrix during the fabrication process. Thermal expansion behaviour of CNT-reinforced aluminum matrix composites was investigated up to 240°C using a dilatometer. The coefficient of thermal expansion of the nanocomposites decreased continuously with the increasing content of CNTs. The maximum reduction of 82% was found for 4 wt% CNTs in the nanocomposite. The coefficient of thermal expansion variation with CNTs was also compared with the predictions from the thermoelastic models. The expansion behaviour of the nanocomposites was correlated to the microstructure, internal stresses, and phase segregations. The electrical and thermal conductivity was also studied and was observed to decrease for all reinforced CNT weight fractions.

Evolution of bright optical spatial solitons in biased photovoltaic photorefractive (PVPR) medium is investigated in the present work. The space-charge field developed in the medium is comprised of local and nonlocal parts. Lowest order charge drift results in the buildup of the local space-charge field, whereas higher order drift and charge diffusion are responsible for nonlocal field development. The dynamical equation for solitons in the closed circuit PVPR medium is obtained under Akhmanov’s paraxial ray approximation. Conditions for stationary propagation are obtained, and the path of soliton in the medium is examined. The asymmetry in the nonlinear refractive index introduced by nonlocal contribution to the space-charge field causes a soliton to deflect from its straight line path in the medium. The roles of charge diffusion and higher order drift on soliton trajectory are examined.

The transient natural convection of a viscous fluid in a heated vertical tube is studied using the two-dimensional differential transform method (DTM). A time-dependent Dirichlet boundary condition is imposed for tube wall temperature. The partial differential equations for the velocity and temperature fields within the tube are solved by the DTM while considering temperature-dependent viscosity and thermal conductivity of the fluid. As a result, tractable solutions in double-series form are derived for the temperature and flow velocity. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical results illustrate the effects of temperature-dependent properties on transient temperature and flow behaviour, including the Nusselt number and volumetric flow rate. The DTM gives accurate series solutions without any special functions for nonlinear transient heat transfer problems which are advantageous in finding the derivative or integral.