International Journal of Geometric Methods in Modern Physics

In this paper, the effects of peristaltic transport with double-diffusive convection in nanofluids through an asymmetric channel with different waveforms is presented. Mathematical modeling for two-dimensional and two-directional flows of a Jeffery fluid model along with double-diffusive convection in nanofluids are given. Exact solutions are obtained for nanoparticle fraction field, concentration field, temperature field, stream functions, pressure gradient and pressure rise in terms of axial and transverse coordinates under the restrictions of long wavelength and low Reynolds number. With the help of computational and graphical results, the effects of Brownian motion, thermospheres, Dufour, Soret and Grashof numbers (thermal, concentration, nanoparticles) on peristaltic flow patterns with double-diffusive convection are discussed.

This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants.

In this paper, we study a special type of magnetic trajectories associated with a magnetic field ${ℬ}$ defined on a 3D Riemannian manifold. First, we assume that we have a moving charged particle which is supposed to be under the action of a gravitational force $G$ in the magnetic field ${ℬ}$ on the 3D Riemannian manifold. Then, we determine trajectories of the charged particle associated with the magnetic field ${ℬ}$ and we define gravitational magnetic curves ($G$-magnetic curves) of the magnetic vector field ${ℬ}$ on the 3D Riemannian manifold. Finally, we investigate some geometrical and physical features of the moving charged particle corresponding to the $G$-magnetic curve. Namely, we compute the energy, magnetic force, and uniformity of the $G$-magnetic curve.

The Heisenberg supermagnet model is an important supersymmetric integrable system which is the super extension of the Heisenberg ferromagnet model. By virtue of introducing the general auxiliary matrix variables, we construct a new $(2+1)$-dimensional generalized integrable Heisenberg supermagnet models under two constraints. Meanwhile, we establish their corresponding gauge equivalent counterparts. Moreover, we derive new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

From the recent observations, it is well known that the expansion rate of our universe varies with time (early decelerating and accelerating in the present epoch) which is an unsolved problem. This motivated to us to consider this paper and so we have developed a new cosmological model in Einstein’s modified gravity theory using two types of modifications: (i) Geometrical modification, in which we have used Lyra’s geometry in the curvature part of the Einstein field equations (EFE) and (ii) Modification in gravity (energy momentum tensor) on right hand side of EFE, as per the Brans–Dicke model. With these two modifications, we have obtained the exact solutions of Einstein Brans–Dicke field equations in Lyra’s geometry for a spatially homogeneous Bianchi type-I space-time with time variable deceleration parameter (DP). We have calculated various physical parameters for the model and found them consistent with recent observations. We have also examined the energy conditions for the model and found them satisfactory. We have found that the scalar field of Brans–Dicke theory behaves like a best fit dark energy candidate in the reference of Lyra’s geometry.

In connection with several attempts to extend Onsager’s (1944) crystal statistics to three dimensions keeping the spirit of the original Ising’s (1925) model, we suggest to involve binary and ternary extension of triangles in connection with fractals and chaos, following the ideas of Kikuchi (1951). We apply ternary Clifford algebras in addition to the usual (binary) Clifford algebras and identify two ternaries with three ternaries with the help of noncommutative Galois extension. In some sense, the paper continues the ideas of Ławrynowicz, Marchiafava and Nowak-Kȩpczyk (2006), and Kerner and Suzuki (2012) published in this journal.

The cosmological dynamics of a non-locally corrected gravity theory, involving a power of the inverse d’Alembertian, is investigated. Casting the dynamical equations into local form, the fixed points of the models are derived, as well as corresponding de Sitter and power-law solutions. Necessary and sufficient conditions on the model parameters for the existence of de Sitter solutions are obtained. The possible existence of power-law solutions is investigated, and it is proven that models with de Sitter solutions have no power-law solutions. A model is found, which allows to describe the matter-dominated phase of the Universe evolution.

In this paper, we have analyzed the equation of state parameter for dark energy in the spatially homogeneous and anisotropic Bianchi type-III spacetime filled with a barotropic fluid and dark energy by considering a variable deceleration parameter. To solve the Einstein field equations, we assume that the expansion scalar is proportional to the shear scalar. We find that during the evolution of the Universe, the equation of state and the cosmological parameters are positive decreasing functions of cosmic time $t$. Also, we discuss the physical parameters, as well as the jerk and cosmic span parameters, which predict that the Universe in this model approaches the $\mathrm{\Lambda }$CDM model at late times.

We propose an extension of $n$-ary Nambu–Poisson bracket to superspace ${ℝ}^{n|m}$ and construct by means of superdeterminant a family of Nambu–Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace ${ℝ}^{n|m}$. We prove in the case of the superspaces ${ℝ}^{n|1}$ and ${ℝ}^{n|2}$ that our $n$-ary bracket, defined with the help of superdeterminant, satisfies the conditions for $n$-ary Nambu–Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov–Jacobi identity (fundamental identity). We study the structure of $n$-ary bracket defined with the help of superdeterminant in the case of superspace ${ℝ}^{n|2}$ and show that it is the sum of usual $n$-ary Nambu–Poisson bracket and a new $n$-ary bracket, which we call $\chi$-bracket, where $\chi$ is the product of two odd degree smooth functions.

The general form of Noether symmetries admitted by Lagrangians corresponding to a diagonal metric are determined. We apply this general result in order to classify different metric functions for the determination of Noether generators for the equations of motion. For the two broad cases considered, we identify symmetry algebras up to dimension thirteen.

The problem of classifying boundary points of space-time, for example singularities, regular points and points at infinity, is an unexpectedly subtle one. Due to the fact that whether or not two boundary points are identified or even “nearby” is dependent on the way the space-time is embedded, difficulties occur when singularities are thought of as an inherently local aspect of a space-time, as an analogy with electromagnetism would imply. The completion of a manifold with respect to a pseudo-Riemannian metric can be defined intrinsically. This was done by Scott–Szekeres via an equivalence relation, formalizing which boundary sets cover other sets. This paper works through the possibilities, providing examples to show that all covering relations not immediately ruled out by the definitions are possible.

A classification of static spherically symmetric space-times in $f(R)$ theory of gravity according to their conformal vector fields (CVFs) is presented. For this analysis, a direct integration technique is used. This study reveals that for static spherically symmetric space-times in $f(R)$ theory of gravity, CVFs are just Killing vector fields (KVFs) or homothetic vector fields (HVFs). For this classification, six cases have been discussed out of which there exists only one case for which CVFs become HVFs while in the rest of the cases CVFs become KVFs.

The paper deals with the investigation of a homogeneous and anisotropic space-time described by Bianchi type-III metric with perfect fluid in Lyra geometry setting. Exact solutions of Einstein’s field equations have been obtained under the assumption of quadratic equation of state (EoS) of the form $p=a{\rho }^2-\rho$, where $a$ is a constant and strictly $a>0$. The physical and geometrical aspects are also examined in detail.

Let $r(s)$ be a parameter and an asymptotic curve on a surface $P(s,t).$ We obtain conditions for offsets $\overline{P}(s,t)$ of $P(s,t)$ such that the image $\beta (s)$ of the curve $r(s)$ is a common asymptotic on each offset. We illustrate the method with an example.

The aim of this work is to study affine translation surfaces in the Euclidean 3-space with density. We completely classify affine translation surfaces with zero weighted mean curvature.

The deformation retract is, by definition, a homotopy between a retraction and the identity map. We show that applying this topological concept to Ricci-flat wormholes/black holes implies that such objects can get deformed and reduced to lower dimensions. The homotopy theory can provide a rigorous proof to the existence of black holes/wormholes deformations and explain the topological origin. The current work discusses such possible deformations and dimensional reductions from a global topological point of view, it also represents a new application of the homotopy theory and deformation retract in astrophysics and quantum gravity.