Volume 71, Issue 6

In this article, we studied the Klein–Gordon equation in a generalised uncertainty principle (GUP) framework which predicts a minimal uncertainty in position. We considered a spinless particle in this framework in the presence of a magnetic field, applied in the z-direction, which varies as 1x2.${1 \over {{x^2}}}.$ We found the energy eigenvalues of this system and also obtained the correspounding eigenfunctions, using the numerical method. When GUP parameter tends to zero, our solutions were in agreement with those obtained in the absence of GUP.

In this article, we give the definition of the multi-component constrained CKP (McCKP) and two-component constrained CKP (cCKP) hierarchies (under the condition N =2). Then we give recursion operators for the two-component cCKP hierarchy. At last, we give the constrained condition from the two-component cCKP hierarchies to cCKP hierarchy.

Nacreous composite materials have excellent mechanical properties, such as high strength, high toughness, and wide phononic band gap. In order to research band structure characteristics of nacreous composite materials with various defects, supercell models with the Brick-and-Mortar microstructure are considered. An efficient multi-level substructure algorithm is employed to discuss the band structure. Furthermore, two common systems with point and line defects and varied material parameters are discussed. In addition, band structures concerning straight and deflected crack defects are calculated by changing the shear modulus of the mortar. Finally, the sensitivity of band structures to the random material distribution is presented by considering different volume ratios of the brick. The results reveal that the first band gap of a nacreous composite material is insensitive to defects under certain conditions. It will be of great value to the design and synthesis of new nacreous composite materials for better dynamic properties.

In this article, we apply the singularity structure analysis to test an extended 2+1-dimensional fifth-order KdV equation for integrability. It is proven that the generalized equation passes the Painlevé test for integrability only in three distinct cases. Two of those cases are in agreement with the known results, and a new integrable equation is first given. Then, for the new integrable equation, we employ the Bell polynomial method to construct its bilinear forms, bilinear Bäcklund transformation, Lax pair, and infinite conversation laws systematically. The N-soliton solutions of this new integrable equation are derived, and the propagations and collisions of multiple solitons are shown by graphs.

The flow of nanofluids past a stretching sheet has attracted much attention owing to its wide applications in industry and engineering. Numerical solution has been discussed in this article for studying the effect of suction (or injection) on flow of nanofluids past a stretching sheet. The numerical results carried out using Chebyshev collocation method (ChCM). Useful results for temperature profile, concentration profile, reduced Nusselt number, and reduced Sherwood number are discussed in tabular and graphical forms. It was also demonstrated that both temperature and concentration profiles decrease by an increase from injection to suction. Moreover, the numerical results show that the temperature profiles decrease at high values of Prandtl number Pr. Finally, the present results showed that the reduced Nusselt number is a decreasing function, whereas the reduced Sherwood number is an increasing function at fixed values of Prandtl number Pr, Lewis number Le and suction (or injection) parameter s for variation of Brownian motion parameter Nb, and thermophoresis parameter Nt.

By utilizing the first-principles method, the pressure-induced effects on phase transition, mechanical stability, and elastic properties of WS 2 /CrS 2 are investigated in the pressure range from 0 to 80 GPa. Transitions from 2H c to 2H a for WS 2 and CrS 2 are found to occur at 17.5 and 25 GPa, respectively. It is found that both 2H a and 2H c phases of WS 2 and CrS 2 meet the mechanical stability criteria up to 80 GPa, suggesting that those structures are mechanically stable. The bulk and shear modulus anisotropy of the two phases of WS 2 and CrS 2 decrease rapidly under pressure and, finally, trend to isotropy. With increasing pressure, the elastic moduli ( Y , B , and G ), sound velocities ( v s , v p , v m ), and Debye temperatures (Θ) of 2H a and 2H c of WS 2 and CrS 2 increase monotonously. Moreover, the Debye temperature (Θ) of 2H c phase is higher than that of 2H a phase for both WS 2 and CrS 2 . The bulk, shear, and Young’s modulus, Poisson coefficient, and brittle/ductile behaviour are estimated. The percentages of anisotropy in compressibility and shear and the ratio of bulk to shear modulus ( B / G ) are also studied.

In this article, the propagation and collision of vector solitons are investigated from the 3-coupled variable-coefficient nonlinear Schrödinger equations, which describe the amplification or attenuation of the picosecond pulses in the inhomogeneous multicomponent optical fibre with different frequencies or polarizations. On the basis of the Lax pair, infinitely-many conservation laws are obtained. Under an integrability constraint among the variable coefficients for the group velocity dispersion (GVD), nonlinearity and fibre gain/loss, and two mixed-type (2-bright-1-dark and 1-bright-2-dark) vector one- and two-soliton solutions are derived via the Hirota method and symbolic computation. Influence of the variable coefficients for the GVD and nonlinearity on the vector soliton amplitudes and velocities is analysed. Through the asymptotic and graphic analysis, bound states and elastic and inelastic collisions between the vector two solitons are investigated: Not only the elastic but also inelastic collision between the 2-bright-1-dark vector two solitons can occur, whereas the collision between the 1-bright-2-dark vector two solitons is always elastic; for the bound states, the GVD and nonlinearity affect their types; with the GVD and nonlinearity being the constants, collision period decreases as the GVD increases but is independent of the nonlinearity.

The aim of this paper is to calculate the analytical form of the equation of state for dilute relativistic plasma. We obtained the excess free energy and pressure in the form of a convergent series expansion in terms of the thermal parameter μ where μ = mc2KT,$\mu \, = \,{{m{c^2}} \over {KT}},$ m is the mass of charge, c is the speed of light, K is the Boltzmann’s constant, and T is the absolute temperature. The results are discussed and compared with previous work of other authors.

The Mhd slip flow and heat transfer of stagnation point Jeffrey fluid over deformable surfaces are the state of the art of this article. Following an analytical approach, the existence, uniqueness, and possible multiplicity of the physical solutions affected by several physical parameters are investigated. Particularly, magnetic interaction and slip factor are shown to much influence the structure of the solutions regarding both momentum and thermal boundary layers. The presented exact solutions not only provide a clear understanding of fruitful physical mechanisms present in this nonlinear flow problem but they have also merits in calculations by means of numerous numerical schemes aiming to explore further complex phenomena.

Based on the modified direct method, the variable-coefficient perturbed mKdV equation is changed to the constant-coefficient perturbed mKdV equation. The truncated Painlevé method is applied to obtain the nonlocal symmetry of the constant-coefficient perturbed mKdV equation. By introducing one new dependent variable, the nonlocal symmetry can be localized to the Lie point symmetry. Thanks to the localization procedure, the finite symmetry transformation is presented by solving the initial value problem of the prolonged systems. Furthermore, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, and Painlevé II solutions are obtained using the symmetry reduction method to the enlarged systems. Two special concrete soliton-cnoidal interaction solutions are studied in both analytical and graphical ways.

Based on numerous experimental data on inelastic neutron and X-ray scattering in liquids, we assert that the phonon–roton spectrum of collective excitations, predicted by Landau for superfluid helium, is a universal property of the liquid state. We show that the existence of the roton minimum in the spectrum of collective excitations is caused by the short-range order in liquids. Using the virial theorem, we assume that one more branch of excitations should exist in He II, whose energy spectrum differs from the phonon–roton spectrum. Such excitations are associated with the pole of single-particle Green function, which can have a gap at small values of momenta.