International Journal of Geometric Methods in Modern Physics

In this paper, the coupled system nonlinear (2$+$1)-dim Broer–Kaup–Kupershmit (BKK) equation under consideration is based on extension of modified rational expansion method. The various kinds of multiple soliton solutions named anti-kink soliton, traveling wave solutions, periodic solution, dark soliton, kink soliton, bright soliton, anti-kink bright solitons, kink bright solitons for nonlinear BKK system are successfully constructed under symbolic computation Mathematica. The calculated results are very interesting, different and novel which have not been investigated in past literature. The graphical demonstration of constructed solutions show by contour, 2D and 3D shape by using computer software Mathematica. The constructed multiple soliton solutions prove that EMRE approach is very straightforward, efficient, reliable and powerful for the investigation of other nonlinear partial differential equations. The investigated results will be described and play a keen role in the study of nonlinear physical phenomena in the area of physics and engineering.

In this paper, we construct two kinds of $q$-differential integrable hierarchies. The first one is the $q$-universal character of B-type (BUC) and lattice $q$-BUC hierarchies expressed by $q$-shift operators. It is also shown that the lattice $q$-BUC hierarchy reduces to the $q$-BUC hierarchy with certain conditions. The other one is the $q$-deformed generalized symplectic Schur functions and the $q$-deformed generalized Q-functions defined by $q$-derivative operators. By means of the $q$-deformed vertex operators, the $q$-deformed symplectic universal character (SUC) and BUC hierarchies have been presented. Moreover, it is proved that $q$-deformed generalized symplectic Schur functions and generalized Q-functions are solutions of the $q$-deformed SUC and BUC hierarchies, respectively. Meanwhile, by using multi-component tau functions, we establish the multi-component $q$-deformed SUC and BUC hierarchies. In addition, based upon the discrete tau functions, the multi-component modified $q$-deformed SUC and BUC hierarchies have been discussed.

In this study, we study two generalized stochastic Korteweg-de Vries (KdV) equations. The Painlevé property of these nonlinear models is tested using Kruksal’s method, which establishes the model’s integrability. As a result, using Hirota’s bilinear approach and symbolic computation, the N-soliton solutions are constructed. In addition, the extended hyperbolic function method (EHFM), the modified Kudryashov method (MKM), and the sub-equation method (SEM) are used to acquire the bright soliton, dark soliton, singular soliton, periodic, rational, and exponential solutions. To help understand the dynamic features of the derived soliton solutions, we present a number of 2D, 3D, and contour graphs using appropriate parametric values.

In this paper, we perform a complete non-relativistic study of the improved class of inversely quadratic Yukawa plus Hulthén potential (ICIQYHP) model in the context of three-dimensional non-relativistic non-commutative quantum phase-space (3D-NRNCPS) symmetries impacted by perturbed spin–orbit interaction and the external magnetic fields for the homogeneous (N₂ and O₂) and heterogeneous (CO and NO) diatomic molecules and the heavy meson systems, such as charmonium (cc) and bottomonium (bb) using generalized Bopp’s shifts method and standard perturbation theory with the Greene–Aldrich approximation to the centrifugal barrier. The new non-relativistic energy equation and eigenfunction for the ICIQYHP in the presence of deformation phase–space are obtained to be sensitive to the atomic quantum numbers ($j,l,s,$ and $m$), the mixed potential depths ($V_{01},V_{21},V_{11}$ and $V_{31}$), the screening parameter ${\delta }_0$, and non-commutativity parameters ($\mathrm{\Phi }/\overline{\eta },\chi /\overline{\chi }$ and $\zeta /\overline{\zeta }$). The critical particular cases in 3D-NRNCPS symmetries have been obtained by adjusting the parameters of the ICIQYHP, such as the improved Hulthén potential, the improved Coulomb potential, and the improved inversely quadratic Yukawa potential. We have also studied the spin-averaged mass spectra of the heavy mesons and the thermo-magnetic properties under the class of inversely quadratic Yukawa plus Hulthén potential model in 3D-NRQM and 3D-NRNCPS symmetries. This research can potentially be applied to atomic, condensed matter, nuclear, molecular physics, and chemical physics.

A spatially homogeneous and anisotropic Bianchi type-VI cosmological model in the framework of $f(Q,T)$ gravity has been studied by considering three different functional forms of $f(Q,T)$ i.e. $f(Q,T)=aQ+bT$, $f(Q,T)=a{Q}^k+bT$ and $f(Q,T)=-aQ-b{T}^2$. The field equation of $f(Q,T)$ gravity has been solved with the help of the Bianchi type-VI line element. The cosmological parameters of the models are evaluated, and the dynamical aspects of the model have been discussed. The result agrees with the observations of accelerated expansion of the universe.

In this paper, we present the coisotropic embedding theorem as a tool to provide a solution for the inverse problem of the calculus of variations for a particular class of implicit differential equations, namely the equations of motion of free Electrodynamics.

The aim of this paper is to investigate Lie symmetries including Killing, homothetic and conformal symmetries of Lemaitre–Tolman–Bondi (LTB) spacetime metric. To find all LTB metrics admitting these three types of symmetries, we have analyzed the set of symmetry equations by a Maple algorithm that provides some restrictions on the functions involved in LTB metric under which this metric admits the three mentioned symmetries. The solution of symmetry equations under these restrictions leads to the explicit form of symmetries. The stress–energy tensor is calculated for all the obtained metrics in order to discuss their physical significance. It is noticed that most of these metrics satisfy certain energy conditions and correspond to anisotropic fluids.

This study is inspired by the generalized holographic cut-off proposed by Nojiri and Odintsov (S. I. Nojiri and S. D. Odintsov, Unifying phantom inflation with late-time acceleration: Scalar phantom–non-phantom transition model and generalized holographic dark energy, Gen. Relativ. Gravit. 38 (2006) 1285–1304) as it aims to have an understanding of the Kaniadakis holographic dark fluid, a particular representative of Nojiri–Odintsov holographic dark energy. To realize inflation and late time acceleration of the universe, we have considered this generalized form of holographic fluid as the fluid for realization of inflationary scenario. Here, inflation has a holographic origin, namely that its source is the Kaniadakis holographic energy density. The inflationary cosmology of Kaniadakis holographic dark fluid has been have explored by assuming that the scale factor is an intermediate scale factor. Several constraints have been derived throughout the study in relation to late time acceleration and early inflation. To study the warm inflation with this holographic origin, we have computed tensor-to-scalar ratio for different values of the number of e-fold and warm inflation has been realized in a high dissipative regime.

This paper presents an analytical study of the behavior of radial free-geodesics in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime within the Lambda Cold Dark Matter ($\mathrm{\Lambda }$CDM) model. Using the radial free motion solutions, we provide two methods for characterizing the geodesics and defines a general formula that encapsulates all possible solutions, determined by two initial conditions. We show that the past light cone, event horizon, and particle horizon, can be considered as special cases of this overarching formula. Furthermore, the paper explores the free geodesics within the currently accepted cosmological model based on the recent Planck results, thoroughly examining the various possible geodesic scenarios.

A brief introduction of the Extended Relativity Theory in Clifford Spaces ($C$-space) paves the way to the explicit construction of the generalized relativistic transformations of the Clifford multivector-valued coordinates in $C$-spaces. The most general transformations furnish a full mixing of the grades of the multivector-valued coordinates. The transformations of the multivector-valued momenta follow leading to an invariant generalized mass ${ℳ}$ in $C$-spaces which differs from $m$. No longer the proper mass appearing in the relativistic dispersion relation $E^2-p\cdot p=m^2$ remains invariant under the generalized relativistic transformations. It is argued how this finding might shed some light into the cosmological constant problem, dark energy, and dark matter. We finalize with some concluding remarks about extending these transformations to phase spaces and about Born reciprocal relativity. An appendix is included with the most general (anti) commutators of the Clifford algebra multivector generators.

In this paper, we study the cosmic dynamics of varying vacuum models where the dark matter interacts with the vacuum energy. We consider the homogeneous and isotropic spacetime with spatial curvature and apply the dynamical system technique to the varying vacuum models by specifying the form of energy exchange rate $(Q)$ between the dark energy and dark matter. Further, we utilize the cosmographic parameters and statefinder parameters in the terms of dynamical variables of the cosmological dynamical system to explore the cosmic dynamics of the universe. The Milne universe solution exists as a consequence of spatially curved geometry in the model. The models also yield radiation- matter- and dark energy-dominated phases in order and thus explain the late-time accelerated expansion of the universe. The strength of interaction terms will affect the existence of cosmological phases in the model but there will always be an attractor corresponding to the accelerating universe. We numerically solve the system to illustrate the evolution of cosmological quantities and dynamical variables in the models. The role of curvature is visible during the transition from the decelerated phase into the accelerating phase. The numerical solutions affirm that the cosmological parameters in the models are consistent with their corresponding observational values.

A set of position-dependent noncommutative algebra in two dimension (2D) that describes the space near the Planck scale had been introduced [J. Phys. A: Math. Theor. 53 (2020) 115303]. This algebra predicted the existence of maximal length of graviton measurable at low energy. From this algebra, we deduce in the present paper, a new noncommutative algebra that is compatible with the deformed algebra proposed by Costa Filho et al. [Phys. Rev. A 84 (2011) 050102] to describe the Position-Dependent Mass (PDM) system in 1D. To this aim, we derive from the momentum operators, the Schrödinger-like equation which describes PDM system in null quantum well potential. The spectrum of this system is asymetrically deformed and exhibits different behaviours from the one obtained by Costa Filho et al. Thus, we observe that by increasing the PDM, the energy decreases and this decrease is more pronounced as the quantum number increases. Finally, we evaluate the thermodynamic quantities within the canonical ensemble and we show that these results are consistent with the ones recently obtained by Bensalem and Bouaziz [Phys. A Stat. Mech. Appl. 523 (2019) 583–592].

In this paper, we propose a specialized parameterization for the Hubble parameter, inspired by $\mathrm{\Lambda }$CDM cosmology, to investigate the cosmic expansion history of the Universe. This parameterization is employed to analyze the universe’s late-time behavior within the context of $Q^n$ gravity, where $Q$ represents non-metricity. By using data from 57 Hubble data points, 1048 supernova (SNe) data points, and six Baryon Acoustic Oscillation (BAO) data points, we determine the optimal values for the model parameters. Additionally, we explore three distinct cosmological models based on the parameter $n$, specifically when it takes on the values of $0.55$, $1.5$ and $2.0$. The results of our analysis indicate that our proposed parameterization, along with the associated models for different values of $n$, predicts an accelerated cosmic expansion phase.

Differentiation of the scalar Feynman propagator with respect to the spacetime coordinates yields the metric on the background spacetime that the scalar particle propagates in. Now Feynman propagators can be modified in order to include quantum-gravity corrections as induced by a zero-point length $L>0$. These corrections cause the length element $\sqrt{s^2}$ to be replaced with $\sqrt{s^2+4L^2}$ within the Feynman propagator. In this paper, we compute the metrics derived from both the quantum-gravity free propagators and from their quantum-gravity corrected counterparts. We verify that the latter propagators yield the same spacetime metrics as the former, provided one measures distances greater than the quantum of length $L$. We perform this analysis in the case of the background spacetime ${ℝ}^D$ in the Euclidean sector.

In this paper, we investigate in this paper the Type IV singular bouncing in the framework of $f(R,{𝒢})$ theory of gravity where $R$ and ${𝒢}$ mean the curvature scalar and the Gauss–Bonnet invariant, respectively. Cosmological $f(R,{𝒢})$ models constrained by the slow-roll evolution is reconstructed and their explicit forms are provided near the bounce and far away from it. One obtains finally two models whose stability is numerically analyzed in this work. Our results show that the stability of the reconstructed models is very affected by their parameters. The model far from the singularity recovers stability quickly than the model near the singularity.

A generalized Robertson–Walker spacetime is not, in general, a perfect fluid spacetime and the converse is not, in general, true. In this paper, we show that if a perfect fluid spacetime admits an $\eta$–$\rho$-Einstein soliton, then the integral curves generated by the velocity vector field $u$ are geodesics and the acceleration vector vanishes. Also, we show that if a perfect fluid spacetime with Killing velocity vector field admits a gradient $\eta$–$\rho$-Einstein soliton, then the spacetime represents either dark matter era or, the acceleration vector vanishes. Next, we show that if a generalized Robertson–Walker spacetime admits an $\eta$–$\rho$-Einstein soliton, then it becomes a perfect fluid spacetime. Finally, we prove that in a dark matter fluid with vanishing vorticity satisfying gradient $\eta$–$\rho$-Einstein soliton in $f({ℛ})$-gravity with constant Ricci scalar, either the energy density and isotropic pressure are constant or, the potential function $\psi$ remains invariant under the velocity vector $u$. Also, for two models $f({ℛ})=e^{\alpha {ℛ}}+\alpha log{ℛ}$ and $f({ℛ})={{ℛ}}^n$ ($\alpha$ = constant and ${ℛ}$ is the Ricci scalar of the spacetime), various energy conditions in terms of the Ricci scalar are examined and state that the Universe is in an accelerating phase and satisfies the weak, null and dominant energy conditions, but violate the strong energy condition.

In this paper, we investigate null Cartan geodesic isophote curves in the Minkowski $3$-space, and give examples where such curves actually exist. By categorizing the types of light vectors, we characterize different types of null Cartan geodesic isophote curves. Moreover, we present the relationship between null Cartan geodesic isophote curves and other special curves.

The general relativity unification and quantum theory is a significant open problem in theoretical physics. This problem arises from the fact that these two fundamental theories, which describe gravity and the behavior of particles at the microscopic level, respectively, are currently incompatible. The unification of these theories is crucial for a complete comprehension of the fundamental forces and the nature of the universe. In this regard, the quantum properties of a Black Hole result in fundamental importance. By analyzing such properties in quantum field theory, in the first step, the gravity enters as a classical background. In semi-classical approximation, Black Holes will emit Hawking radiation with an almost thermal spectrum, while Black Hole’s entropy is proportional to the Black Hole’s horizon. Besides, Hawking’s temperature and Black Hole entropy should follow the first law of Black Hole thermodynamics. Also, Jacobson [Thermodynamics of spacetime: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260, doi:10.1103/PhysRevLett.75.1260] showed shown that there is a connection between Black Hole thermodynamics and Einstein’s equation that opens the root of a potential thermodynamic nature of gravity. This issue opened a new impressive research framework in which the Einstein field equation can be expressed as a form of the first law of thermodynamics and vice versa. In this study, it is assumed that the universe has a fractal structure. Accordingly, modified Friedmann’s equations and the Black Holes thermodynamics in a fractal universe have been examined. The fractal framework shows what features and changes occur in the description of the universe, particularly in studying the thermodynamics of a Black Hole. However, the paper strategy is organized as follows: in the beginning, we consider the first thermodynamic law in a fractal universe. Then, we investigate the Friedmann equation of the fractal universe in the form of the entropy balance, this means $dQ=T_{h}d{S}_h$, where $dQ$ and $T_h$ are the thermal energy and horizon temperature. We consider the entropy $S_h$ have two terms; (1) obeys the usual area law and (2) the entropy production term due to the non-equilibrium thermodynamics of a fractal universe. Therefore, in a fractal universe, a term with non-equilibrium thermodynamics of spacetime may be needed. Also, we study the generalized second law of thermodynamics in a fractal universe. When the temperature of the apparent horizon and the temperature of the matter fields inside the horizon are equal, i.e. $T=T_h$, the second law of generalized thermodynamics can be obtained according to the state parameter range equation, which is consistent with the recent observations. Finally, in Sec. 6, based on the mathematical calculations, we study the various cosmological parameters such as the Hubble parameter, scale factor, deceleration parameter and equation of state parameter.