International Journal of Geometric Methods in Modern Physics

The focus of this paper is the examination of self-similar vector fields (SSVFs) within Bianchi type I spacetime utilizing the Rif tree approach. The paper employs a computer algorithm to convert the equations into a reduced involutive form, dividing the integration problem into multiple cases represented in a tree structure. These cases are subject to specific constraints on the metric functions. By employing these constraints, the set of equations governing self-similar symmetries is solved, yielding explicit expressions for the metrics and their associated self-similar vector fields.

In this paper, we give Weierstrass-type representation formulas for the null curves and for the minimal Lorentz surfaces in the Minkowski 3-space ${ℝ}_1^3$ using real-valued functions. Applying the Weierstrass-type representations for the null curves, we find a correspondence between the null curves in ${ℝ}_2^4$ and the pairs of null curves in ${ℝ}_1^3$. Based on this correspondence, we obtain a relation between the minimal Lorentz surfaces in ${ℝ}_2^4$ and the pairs of minimal Lorentz surfaces in ${ℝ}_1^3$. As an application, we give examples showing how we can obtain a minimal Lorentz surface in ${ℝ}_2^4$ using a pair of minimal Lorentz surfaces in ${ℝ}_1^3$.

In this study, we investigate the behavior of the Dirac–Born–Infeld (DBI) scalar field as a candidate for the nature of dark energy in a non-minimal coupling scenario with the scalar torsion within the framework of teleparallel gravity. We present numerical solutions for the equations of motion and analyze the behavior of various cosmological parameters for different values of the parameter $\tilde{v}$ and $\tilde{f}$. Our findings reveal that the DBI scalar field can contribute significantly to the energy density of the universe and drive the late-time acceleration of the universe, with a transition from deceleration to acceleration occurring within a redshift range consistent with recent observational bounds. We show that the present-day value of the deceleration parameter is consistent with recent observational constraints, and the DBI scalar field behaves like a cosmological constant model. Our results suggest that the DBI scalar field can play a significant role in driving the late-time acceleration of the universe and that it has the potential to be a viable alternative to the cosmological constant model. Further investigations are necessary to better understand the nature of the dark energy component and its implications for cosmology within the framework of teleparallel gravity.

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation $D_t^{\alpha }u=a{u}^p{u}_{xx}+b{u}^q{u}_x^2$ is studied by the symmetry analysis method. Liu et al. obtained two group generators $X_1=\frac{\partial }{\partial x},$ $X_2=\frac{2t}{\alpha }\frac{\partial }{\partial t}+x\frac{\partial }{\partial x}$ and only one trivial solution $u(t,x)=f(t)=\frac{C_0}{\mathrm{\Gamma }(\alpha )}{t}^{\alpha -1}$ for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case $p=q+1$, we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

In this paper, we thoroughly study $\varrho$-Ricci–Bourguignon almost soliton and gradient $\varrho$-Ricci–Bourguignon almost soliton in the paracontact geometry, precisely, on $K$-paracontact and para-Sasakian manifolds. Here, we prove that if the metric $g$ of the $K$-paracontact manifold endows a $\varrho$-Ricci–Bourguignon almost soliton with the nonzero potential vector field $V$ parallel to $\xi$, then the manifold is an Einstein with Einstein constant $-2n$. Next, we show that if a para-Sasakian manifold represents a gradient $\varrho$-Ricci–Bourguignon almost soliton, then the manifold is an Einstein with constant scalar curvature $-2n(2n+1)$. We also discuss $\varrho$-Ricci–Bourguignon almost soliton on $(\kappa ,\mu )$-paracontact manifold.

The aim of this paper is to explore the Clairaut anti-invariant Riemannian maps from/to Kähler manifolds admitting Ricci solitons. We find the curvature relations and calculate the Ricci tensor under different conditions. We discuss the condition under which range space becomes $\alpha$-Ricci soliton. We obtain conditions for the range and kernel spaces of these maps to be Einstein. Next, we find the scalar curvature for range space. Further, we give the relation between Ricci curvature and Lie derivative under these maps. Moreover, we find the condition for a vertical potential vector field on target manifold to be conformal vector field on range space of these maps. Finally, we give non-trivial examples of such maps.

The wave function of the quantum cosmological model of gravity coupled to a scalar field has been found at very early times. Corrections resulting from quadratic curvature terms may be evaluated and the functional dependence is found to have a similar form consistent with the inflationary epoch. The fundamental string theories would determine the gravitational action, and effective actions for both $\mathrm{\text{N}}=1$ and $\mathrm{\text{N}}=2$ string theories are examined, establishing a connection with topological string models. The nonlinear differential equation for the free energy of the 10-dimensional theory describing the compactification over a Calabi–Yau space of three complex dimensions is formulated, and it is transformed into a linear differential equation for the product of a function of the scaled string coupling and the partition function. A solution given by a sum over the genus is found for the series expansion of the partition function that can be matched in a bounded range for the coupling, and the scale factor, with the quantum cosmological wave function of the quadratic gravity model viable during the inflationary epoch.

In this paper, we first obtain a new identity for time scales by using a weighted kernel. Then, by using this equality, we prove a weighted Čeby šev inequality. Moreover, we establish a weighted Ostrowski-type inequality by using a method which is different from among in the literature.

In this work, we have studied how incorporating viscous fluids leads to exact bounce cosmological solutions in general relativity (GR) framework. Specifically, we propose a novel parameterization of bulk viscosity coefficient of the form $\zeta ={\zeta }_0{(t-t_0)}^{-2n}H$, where ${\zeta }_0$, with $n$ being some positive constants and $t_0$ being the bounce epoch. We investigate how this form of bulk viscosity may assist in explaining the early universe’s behavior, with a particular focus on non-singular bounce scenario by studying the various energy conditions and other related cosmological observables and how the model parameters affect the evolution of the Universe. We demonstrate that the null energy conditions (NEC) and strong energy conditions (SEC) violation occurs at the bounce point while dominant energy conditions (DEC) is satisfied. Finally, we carried out a stability check based on linear order perturbation to the Hubble parameter. We found that the perturbation vanishes asymptotically at later times, which indicates a stable behavior of the bounce scenario.

A higher-order spectral problem is introduced, based on a special Lie subalgebra of the general linear algebra. An associated matrix Liouville integrable hierarchy, each of which consists of four submatrix equations, is generated by means of the zero curvature formulation. The corresponding Hamiltonian structure is established by the trace variational identity and two integrable reductions, of which one is real and the other is complex, are constructed under similarity transformations.

The modified theories of gravity, especially the $f(Q)$ theory, have attracted much attention in recent years. This study delves into non-static plane-symmetric solutions within the framework of this theory, employing a metric approach. The field equations are addressed with the incorporation of the Hubble parametrization $H=\frac{H_0}{\sqrt{2}}{[1+{(1+z)}^{2(1+\beta )}]}^{1/2}$. The determination of optimal values for free parameters is achieved through the analysis of 57 data points of the Hubble parameter $H(z)$. Additionally, the study explores the behavior of various cosmological parameters, including the deceleration parameter $(q)$, energy density $(\rho )$, pressure $(p)$, and equation of state (EoS) parameter $(\omega )$, comparing them with observational findings. To validate the results, energy conditions and state finder analysis are undertaken. Ultimately, it is established that our model exhibits behavior akin to the $\mathrm{\Lambda }$CDM model. Finally, we found that our model behaves similar to the $\mathrm{\Lambda }$CDM model.

A new mechanism based on space-time noncommutativity has shown to produce anisotropy and an axial-like interaction, yielding leptonic asymmetry for fermionic particles propagating in a noncommutative curved FRW universe. As a byproduct, an analytical expression of this asymmetry for ultra-relativistic particles, such as neutrinos, is derived explicitly. Constraints and bounds from the cosmological parameters are also discussed.

In this paper, we use the Lie symmetry analysis to obtain the closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili Benjamin–Bona–Mahony (gKP-BBM) equation. We obtain the infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for the gKP-BBM equation. Also, we evaluated the interactions of soliton solutions of the gKP-BBM equation. It is observed that multi-soliton complexes are self-localized in state wherein, several fundamental solitons including bright solitons and dark solitons are nonlinearly superimposed. A notable observation is the existence of topological defects in soliton interactions, where two solitons appear to be separated, with the adjoining structure being “out of phase” with each other. The kink, acting as a twist in the soliton’s value, overcomes topological defects, leading to a transition from one phase to another. This comprehensive analysis contributes to a deeper understanding of the dynamics and interactions of solitons in the (3+1)-dimensional gKP-BBM equation.

With a Kiselev spacetime around Schwarzschild black holes that have undergone quantum correction, the goal of this work is to examine the geometric structure of a thin-shell. In order to do so, we take into consideration black hole solutions and match the inner Minkowski spacetime using a cut and paste method. Our findings indicate that the use of a phantom-like equation of state, such as quintessence, dark energy, and phantom energy, in the linearized radial perturbation technique shows stable thin-shell configurations that lie beyond the outer manifold’s predicted position of the event horizon. Moreover, we discover that the characteristics of the black hole solutions have a significant impact on the thin-shell stability. We examine the behavior of a thin-shell configuration with massless and massive scalar fields by using Klein–Gordon’s equation of motion. According to our findings, the thin-shell system dynamics are significantly influenced by the scalar field, resulting in interesting phenomena including expansion and collapse. These results provide fresh light on the behavior of black hole solutions by providing important insights into the interaction between quantum correction factors related to scalar fields and thin-shell dynamics. In particular, our research suggests that the quantum correction parameter decreases the stable configuration of a Schwarzschild black hole encircled by a quintessence field.

The inverse scattering transform is a tool for solving the initial value problem of nonlinear equations, and the solution of the initial-value problem by inverse scattering transform proceeds in three steps: direct scattering, time evolution and inverse problem. In this paper, we discuss the higher-order integrable discrete nonlinear Schrödinger equation which is subjected to inverse scattering transform under nonzero boundary conditions, and the corresponding soliton solutions are obtained and illustrated.

Investigating the Tachibana and Vishnevskii operators used on ${}^{vv}X$ and ${}^{cc}\tilde{X}$ in accordance with the almost paracontact structure in the semi-tangent bundle $t(M)$ is the primary goal of this work. Additionally, in the almost paracontact structure, these relationships will be examined for some special vector fields. Studying the Lie derivatives of the almost paracontact structure and almost contact structure on the semi-tangent bundle $t(M)$ with regard to ${}^{vv}X$ and ${}^{HH}\tilde{X}$ is the secondary goal of this paper.

This paper established a functional design of the spectral Tau method (STM) upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to execute the Tau integration analytically. Moreover, explicit algebraic systems for solving the Lane–Emden and Riccati equations and the contamination of a system of three artificial lakes are introduced. The convergence and error analyses are explored in depth. Some enlightening boundary value problems (BVPs) for real-life and physical applications, as singularly perturbed equations, are provided to guarantee that this approach is authentic, reliable, and appropriate. Accurate results are obtained using only a few number of retained nodes. The different outcomes, patterns, and better results showed that the FDCHPs differ from Chebyshev polynomials of the second kind.

This paper presents a pure geometric field theory within the framework of Parameterized Absolute Parallelism geometry, aiming to exceed the limitations of both Riemannian and Absolute Parallelism geometries. Following the Unification Principle, the theory employs Parameterized Absolute Parallelism geometry, which involves a linear connection with nonvanishing torsion, nonvanishing curvature and nonvanishing W-tensor. The presented theory is constructed following the Dolan–McCrea variational method to derive the field equations. The theory is shown to incorporate gravity and electromagnetic features. By studying the connection between physical quantities and geometric objects, Poisson and Laplace equations are obtained in the linearized version of the theory. In free space, the Schwarzschild exterior solution of general relativity holds in the present theory.