International Journal of Geometric Methods in Modern Physics

This paper investigates the $(3+1)$-dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the $(3+1)$-dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie’s continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver’s standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov’s generalized theorem and quasi-self-adjointness condition.

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space–space have recently become the object of renewed interest. In the context of three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries with arbitrary spin-orbit coupling quantum number $k$, we approximate to solve the deformed Dirac equation for a new suggested new generalized Morse potential and a class of Yukawa potential including improved Coulomb-like tensor interaction (N(GMP-CYP) plus ICLTI). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp’s shift method and standard perturbation theory using Greene–Aldrich approximation to nonlinear and exponential terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers ($j,k,l/\tilde{l},s/\tilde{s},m/\tilde{m}$), the mixed potential depths ($D_e$, $r_e$, $a$, $b$, $c$), the range of the potential $\alpha$ and noncommutativity parameters ($\mathrm{\Theta }$,$\tau$,$\chi$). We have studied the nonrelativistic limit of new spin symmetry under the N(GMP-ICYP) model, we will also treat some important special cases such as the new generalized Morse potential, the new class of Yukawa potential, the new Hellmann potential, the new inversely quadratic Yukawa potential, the new Hulthén potential and new Coulomb potential. Finally, we studied a case of composite systems.

In this paper, we construct second type frame affiliated with the nonlinear heat system. This work is an important study to present a spacelike Landau–Lifshitz second type ${𝒩}{ℒ}{\mathscr{H}}{𝒮}$ viscosity model. Then, we introduce antiferromagnetic spacelike Landau–Lifshitz second type ${𝒩}{ℒ}{\mathscr{H}}{𝒮}$ viscosity model electromotive nonlinear heat microscales. Also, we obtain optical magnetic spacelike second type ${𝒩}{ℒ}{\mathscr{H}}{𝒮}$ microscales in Minkowski space. Finally, we design antiferromagnetic viscosity model for optical Landau–Lifshitz second type ${𝒩}{ℒ}{\mathscr{H}}{𝒮}$ optimistic density.

The object of this paper is to study mixed quasi-Einstein manifolds of dimension $n$ (for short, ${M(\mathrm{\text{QE}})}_n$) and their geometric properties. We discuss some geometric properties of ${M(\mathrm{\text{QE}})}_n$ such as Ricci semisymmetric, concircular Ricci pseudosymmetric and $W_i$ Ricci pseudosymmetric. Furthermore, we discuss the mixed quasi-Einstein manifolds with Ricci soliton and Riemann soliton, respectively. Finally, we construct an example of mixed quasi-Einstein manifold.

‘Internal spacetime’ is a modification of general relativity that was recently introduced as an approximate spacetime geometric model of quantum nonlocality. In an internal spacetime, time is stationary along the worldlines of fundamental (dust) particles. Consequently, the dimensions of tangent spaces at different points of spacetime vary, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this paper, we develop spinors on an internal spacetime, and construct a new Dirac-like Lagrangian ${ℒ}=\bar{\psi }(i\partial /-\widehat{\omega })\psi$ whose equations of motion describe their couplings and interactions. Furthermore, we show that hidden within ${ℒ}$ is the entire standard model: ${ℒ}$ contains precisely three generations of quarks and leptons, the electroweak gauge bosons, the Higgs boson, and one new massive spin-$2$ boson; gluons are considered in a companion paper. Specifically, we are able to derive the correct spin, electric charge, and color charge of each standard model particle, as well as predict the existence of a new boson.

Aramids are man-made high performance fibers admitting useful industrial applications. Aramids can be classified into para-aramids and meta-aramids. Kevlar is a para-aramid and Nomex is a meta-aramid. This work is devoted to compute the empirical formula for the Hosoya polynomial of these aramids. The closed form of a number of distance-related topological indices (TIs) is the famous distance-based Hosoya polynomial. These are Weiner, hyper-Weiner and Tratch–Stankevitch–Zafirov indices. Results exhibit that para-aramid and meta-aramid possess different Hosoya polynomials and corresponding distance-based TIs. Further, distance-related TIs derived from Hosoya polynomial for the para-aramid admit larger values as compared to those of the meta-aramid.

In this paper, we investigate an inflation scenario and dark energy in a scaled gravity, $R+\beta R$ where $\beta$ is a dimensionless gravity parameter, with a kinetic term. To reduce the associated moduli space, we implement such a parameter in a specific scalar potential given by $V(\varphi )=M^4(1-{(\frac{\varphi }{\mu })}^{\beta })$. By computing the relevant cosmological observables including the spectral index $n_s$ and the tensor-to-scalar ratio $r$ via the slow-roll analysis, we provide certain inflationary constraints on $\beta$ from the Planck observational and the recent released BICEP/Keck data. By means of a possible Higgs-inflaton coupling, we discuss dark energy in the proposed scaled gravity within such an inflation model.

A differential geometric statement of the noncommutative topological index theorem is worked out for covariant star products on noncommutative vector bundles. To start, a noncommutative manifold is considered as a product space $X=Y\times Z$, wherein $Y$ is a closed manifold, and $Z$ is a flat Calabi–Yau $m$-fold. Also, a semi-conformally flat metric is considered for $X$ which leads to a dynamical noncommutative spacetime from the viewpoint of noncommutative gravity. Based on the Kahler form of $Z,$ the noncommutative star product is defined covariantly on vector bundles over $X$. This covariant star product leads to the celebrated Groenewold–Moyal product for trivial vector bundles and their flat connections, such as $C^{\infty }(X)$. Hereby, the noncommutative characteristic classes are defined properly and the noncommutative Chern–Weil theory is established by considering the covariant star product and the superconnection formalism. Finally, the index of the ⋆-noncommutative version of elliptic operators is studied and the noncommutative topological index theorem is stated accordingly.

In this paper, an attempt is made to construct a Friedmann–Lemaitre–Robertson–Walker model in $f(R,T)$ gravity with a perfect fluid that yields acceleration at late times. We take $f(R,T)$ as $R+8\pi \mu T$. As in the $\mathrm{\Lambda }$CDM model, we take the matter to consist of two components, viz., ${\mathrm{\Omega }}_m$ and ${\mathrm{\Omega }}_{\mu }$ such that ${\mathrm{\Omega }}_m+{\mathrm{\Omega }}_{\mu }=1$. The parameter ${\mathrm{\Omega }}_m$ is the matter density (baryons $+$ dark matter), and ${\mathrm{\Omega }}_{\mu }$ is the density associated with the Ricci scalar $R$ and the trace $T$ of the energy–momentum tensor, which we shall call dominant matter. We find that at present ${\mathrm{\Omega }}_{\mu }$ is dominant over ${\mathrm{\Omega }}_m$, and that the two are in the ratio 3:1–3:2 according to the three data sets: (i) 77 Hubble OHD data set, (ii) 580 SNIa supernova distance modulus data set and (iii) 66 pantheon SNIa data which include high red shift data in the range $0\le z\le 2.36$. We have also calculated the pressures and densities associated with the two matter densities, viz., $p_{\mu }$, ${\rho }_{\mu }$, $p_m$ and ${\rho }_m$, respectively. It is also found that at present, ${\rho }_{\mu }$ is greater than ${\rho }_m$. The negative dominant matter pressure $p_{\mu }$ creates acceleration in the universe. Our deceleration and snap parameters show a change from negative to positive, whereas the jerk parameter is always positive. This means that the universe is at present accelerating and in the past it was decelerating. State finder diagnostics indicate that our model is at present a dark energy quintessence model. The various other physical and geometric properties of the model are also discussed.

In this paper, new solutions of the time-fractional Hirota–Satsuma coupled KdV equation model the intercommunication between two long waves that have well-defined dispersion connection received successfully by the unified method, the improved $F$-expansion method and the homogeneous balance method. In contrast, these methods are simple and efficient, and can obtain different exact solutions to this equation. By symbolic calculation, polynomial solutions, hyperbolic function solutions, trigonometric function solutions, rational function solutions, etc. are acquired. Furthermore, we plot and analyze some solutions.

In this paper, the traces of conformal twisted signature operators and noncommutative residue for compact manifolds with boundary are considered, the classic gravity representation for manifolds with boundary associated with conformal twisted signature operators and Kastler–Kalau–Walze-type theorems is explored.

In this work, we study the (2+1)-dimensional nonlinear Schrödinger-type equation that is related to many physical phenomena in nonlinear optical fibers and water waves. Some properties of the (2+1)-dimensional nonlinear Schrödinger-type equation are considered. We determine the infinitesimal generators, an optimal system and a commutator table of the Lie algebra by using Lie symmetry analysis. Also the conservation laws of the equation are obtained using the new conservation theorem proposed by Ibragimov.

In this paper, we characterize three-dimensional trans-Sasakian manifolds of type $(\alpha ,\beta )$ admitting *-$\eta$-Ricci-solitons and gradient almost *-$\eta$-Ricci-solitons. First, we prove that a trans-Sasakian three-manifold of type $(\alpha ,\beta )$ admitting a *-$\eta$-Ricci soliton reduces to a $\beta$-Kenmostu manifold, provided potential vector field $V$ is Jacobi along a geodesic of $\xi$. Next, we show that the trans-Sasakian metric as a gradient almost *-$\eta$-Ricci soliton is either flat $or$ *-$\eta$-Einstein $or$ a $\beta$-Kenmotsu manifold.

The prime object of this paper is to study the perfect fluid spacetimes obeying $f({ℛ})$-gravity, when $\eta$-Ricci solitons, gradient $\eta$-Ricci solitons, gradient Einstein solitons and gradient $m$-quasi Einstein solitons are its metrics. At first, the existence of the $\eta$-Ricci solitons is proved by a non-trivial example. We establish conditions for which the $\eta$-Ricci solitons are expanding, steady or shrinking. Besides, in the perfect fluid spacetimes obeying $f({ℛ})$-gravity, when the potential vector field of $\eta$-Ricci soliton is of gradient type, we acquire a Poisson equation. Moreover, we investigate gradient $\eta$-Ricci solitons, gradient Einstein solitons and gradient $m$-quasi Einstein solitons in $f({ℛ})$-gravity, respectively. As a result, we establish some significant theorems about dark matter era.

In this paper, we study the semi-symmetry and pseudo-symmetry of almost cosymplectic $3$-manifolds. First, we prove that an $H$-almost cosymplectic $3$-manifold $M$ is semi-symmetric if and only if it is cosymplectic. Here by an $H$-almost cosymplectic $3$-manifold, we mean an almost cosymplectic $3$-manifold whose characteristic vector field $\xi$ is a harmonic unit vector field. If an almost cosymplectic $3$-manifold $M$ whose fundamental endomorphism field $h$ is parallel in the direction of the characteristic vector field $\xi$ (${\nabla }_{\xi }h=0$), then it is $H$-almost cosymplectic. In particular, an almost cosymplectic $3$-manifold $M$ satisfying ${\nabla }_{\xi }h=0$ is semi-symmetric if and only if it is cosymplectic. Next, we prove that pseudo-symmetric $H$-almost cosymplectic $3$-manifolds are certain generalized almost cosymplectic $(\kappa ,\mu ,\nu )$-spaces.

The primary goal of this paper is to examine spacetimes admitting semiconformal curvature tensor in $f({ℛ})$ modify gravity. The semiconformal flatness of general spacetime and spacetime in $f({ℛ})$ gravity with perfect fluid, has been analyzed. For this consideration, we generate the forms of isotropic pressure $p$ and energy density $\sigma$. After that, a few energy conditions are taken into account. Finally, we study the divergence-free semiconformal curvature tensor in $f({ℛ})$ gravity in presence of perfect fluid. We emphasize that for recurrent or bi-recurrent energy–momentum tensor, Ricci tensor of this spacetime is semi-symmetric and consequently, the resulting spacetimes either accomplish inflation or possess fixed isotropic pressure and energy density.

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator ${\square }^{-1}$ are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.

This paper is aimed at studying the thermodynamic properties of quantum gases confined to a torus. To do that, we consider noninteracting gases within the grand canonical ensemble formalism. In this context, fermions and bosons are taken into account and the calculations are properly provided in both analytical and numerical manners. In particular, the system turns out to be sensitive to the topological parameter under consideration: the winding number. Furthermore, we also derive a model in order to take into account interacting quantum gases. To corroborate our results, we implement such a method for two different scenarios: a ring and a torus.