Narrow Results

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

For a singular nonlinear traveling wave system of the first class, if there exist two node points of the associated regular system in the singular straight line, then the dynamics of the solutions of the singular system will be very complex. In this paper, two representative nonlinear traveling wave system models (namely, the traveling wave system of Green–Naghdi equations and the traveling wave system of the Raman soliton model for optical metamaterials) are investigated. It is shown that, if there exist two node points of the associated regular system in the singular straight line, then the singular system has no peakon, periodic peakon and compacton solutions, but rather, it has smooth periodic wave, solitary wave and kink wave solutions.

In 2014, Khalil et al. [2014] proposed the conformable fractional derivative, which obeys chain rule and the Leibniz rule. In this paper, motivated by the monograph of Jibin Li [Li, 2013], we study the exact traveling wave solutions for a class of third-order MKdV equations with the conformable fractional derivative. Our approach is based on the bifurcation theory of planar dynamical systems, which is much different from the simplest equation method proposed in [Chen & Jiang, 2018]. By employing the traveling wave transformation $u(x,t)=\varphi (\xi ),$$\xi =kx+v{t}^{\alpha }$, we reduce the PDE to an ODE which depends on the fractional order $\alpha$, then the analysis depends on the order $\alpha$. Moreover, as $\alpha =1$, the exact solutions are consistent with the integer PDE. However, in all the existing papers, the reduced ODE is independent of the fractional order $\alpha$. It is believed that this method can be applicable to solve the other nonlinear differential equations with the conformable fractional derivative.

The nonlinear Schrödinger equation with nonlinear dispersion is investigated. By using the bifurcation-theoretic method of planar dynamical systems, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons as well as compacton solutions for this planar dynamical system are obtained. Under different parameter conditions, solutions can be exactly obtained. Fifteen exact explicit solutions of the traveling wave system are derived.

The SKT cross-diffusion equation is proposed by N. Shigesada, K. Kawasaki and E. Teramoto in 1979 to investigate segregation phenomena of two competing species with each other in the same habitat area. The effect of cross-diffusion affects the population pressure between two different species. Lou and Ni derived limiting systems to see whether this effect may give rise to a spatial segregation or not, and to clarify its mechanism. In this paper, we introduce some new representation of solutions to a stationary limiting problem modified from representation by Lou, Ni and Yotsutani. We apply it to the numerical investigation of existence, non-existence, multiplicity and stability.

We study Bianchi I type brane cosmologies with scalar matter self-interacting through combinations of exponential potentials. Such models correspond in some cases to inflationary universes. We discuss in detail the conditions for accelerated expansion to occur: in particular, we show that the condition which is necessary and sufficient for inflation in the relativistic version of the models is not sufficient in the brane case. Another peculiar feature of the models is that the relationship between the value of the scale factor at the beginning of inflation and the equation of state is very different from what one finds in the relativistic framework. We also analyze the influence of the value of the anisotropy and the brane tension, and show that the associated effects become negligible in the late time limit, those related to the anisotropy disappearing earlier. This study focuses mainly on single field models, but we also consider a generalization yielding models with multiple non-interacting fields and examine their features briefly. We conclude that, in the brane scenario, an increase in the number of fields assists inflation, as happened in general relativity.

We present an inflationary solution of the early universe considering tachyon field. The technique of Zhuravlev and Chervon to obtain inflationary cosmological models without restrictions on a scalar field potential is employed here. We note that like the scalar field, the inflationary solution obtained here with tachyon field does not depend on the potential. However, unlike the scalar field, inflation with the tachyon field is obtained for restricted values of the field to begin with. We present the potential for which one gets inflation. Unlike the scalar field potential, the tachyonic potential is not regular at all values of the field. The solution obtained here with tachyon field is new.

One of the simplest ways to extend 4D cosmological models is to add another spatial dimension to make them 5D. In particular, it has been shown that the simplest of such 5D models, i.e. one in which the right-hand side of the Einstein equation is empty, induces a 4D nonempty universe. Accordingly, the origin of matter in a real 4D universe might be mathematically attributed to the existence of one (fictitious) extra spatial dimension. Here we consider the case of an empty 5D universe possessing a cosmological constant Λ and obtain exact solutions for both positive and negative values of the Λ. It is seen that such a model can naturally reduce to a power law ΛCDM model for the real universe. Further, it can be seen that the arbitrary constants and functions appearing in this model are endowed with definite physical meanings.

In general relativity, the gravitational field of an infinite rotating cylinder is globally (but not locally) different from that of a static cylinder. It is shown here that, for an infinite rigidly translating (non-rotating) cylinder of perfect fluid with a regular axis, there exists a (translating) frame of reference relative to which the gravitational field is static.

A class of solutions describing the interior of a static spherically symmetric compact anisotropic star is reported. The analytic solution has been obtained by utilizing the Finch and Skea [Class. Quantum Grav.6 (1989) 467] ansatz for the metric potential g_rr which has a clear geometric interpretation for the associated background spacetime. Based on physical grounds, appropriate bounds on the model parameters have been obtained and it has been shown that the model admits an equation of state (EOS) which is quadratic in nature.

The exact superposition of a central static black hole with surrounding thin disk in presence of a magnetic field is investigated. We consider two models of disk, one of infinite extension based on a Kuzmin–Chazy–Curzon metric and other finite based on the first Morgan–Morgan disk. We also analyze a simple model of active galactic nuclei (AGN) consisting of black hole, a Kuzmin–Chazy–Curzon disk and two rods representing jets, in presence of magnetic field. To explain the stability of the disks, we consider the matter of the disk made of two pressureless streams of counter-rotating charged particles (counter-rotating model) moving along electrogeodesic. Using the Rayleigh criterion, we derivate for circular orbits the stability conditions of the particles of the streams. The influence of the magnetic field on the matter properties of the disk and on its stability are also analyzed.

The Hertz potential is a powerful tool for the source-free electrodynamics. Especially for the algebraically special spacetime background, the Hertz potential formalism simplifies the Maxwell equations quite much. In astrophysics, strong electric–magnetic field is very common. Force-free electrodynamics is a good approximation for strong enough electric–magnetic field compared to the inertial energy of the involved plasma. For example, the force-free model has been extensively used to describe the magnetosphere of stars in the universe. In this paper, we extend the Hertz potential formalism to the force-free electrodynamics. The Hertz potential formalism simplifies the force-free dynamical equations as much as that in the source-free case. As an application, we use the Hertz potential formalism to the Schwarzschild background. And the Brennan–Gralla–Jacobson solutions are recovered straightforwardly.

We present new anisotropic models for Buchdahl [H. A. Buchdahl, Phys. Rev.116 (1959) 1027.] type perfect fluid solution. For this purpose, we started with metric potential $e^{\lambda }$ same as Buchdahl [H. A. Buchdahl, Phys. Rev.116 (1959) 1027.] and $e^{\nu }$ is monotonically increasing function as suggested by Lake [K. Lake, Phys. Rev. D67 (2003) 104015]. After that we determine the new pressure anisotropy factor $\mathrm{\Delta }$ with the help of both the metric potentials $e^{\lambda }$ and $e^{\nu }$ and propose new well behaved general solution for anisotropic fluid distribution. The physical quantities like energy density, radial and tangential pressures, velocity of sound and redshift etc. are positive and finite inside the compact star. In this connection, we have studied the stability of the models, which is most vital one and also we determined the equation of state $p=f(\rho )$ for the realistic compact star models. It is noted that the mass and radius of our models can represent the structure of realistic astrophysical objects such as Her X-1 and RXJ 1856-37.

We resolve a metric singularity at large $r$ that is due to the introduction of the cosmological constant $\mathrm{\Lambda }$ in simple static spherically symmetric systems in classical general relativity for a mass bounded within a radius $r_0$. For the metric to be nonsingular, we find that ordinary matter must exist beyond $r_0$, and that mass densities and $\mathrm{\Lambda }$ must have spatial ranges. These features can be developed covariantly and can ameliorate discrepancies between theoretical values of $\mathrm{\Lambda }$ and those derived from astronomical observations. Requiring a nonsingular metric in classical general relativistic modeling of this and other physical systems has the potential to offer suggestive insights into cosmological parameters.

In this paper, the Einstein–Maxwell spacetime is considered for compact stellar system. To find out solutions of the field equations, we adopt a finite and positive well-behaved metric potential. Under this particular choice, we therefore develop the expressions of the physical features, such as mass, charge, density and pressure, for stellar system in embedding class 1 spacetime. It is observed that all these features are physically viable. In the model, some known compact stars, viz. $4U$ 1820–30, $4U$ 1608–52 and $EXO$ 1745–248 $(I)\&(II)$ are studied successfully through physical analysis. It is also interesting to note that the obtained set of regular solutions to the Einstein–Maxwell equations represents an electromagnetic mass model for isotropic fluid without invoking any negative pressure.

Within the context of nonlinear electromagnetism, we consider the Yukawa extension of a Reissner–Nordström black hole. Exact solution is given which modifies certain characteristics of the latter. Some thermodynamical aspects are given for comparison. The model which is based on a nonpolynomial Lagrangian implicitly may be considered as a useful agent to describe a short-ranged, charged and massive interaction.

This paper formulates the exact static anisotropic spherically symmetric solution of the field equations through gravitational decoupling. To accomplish this work, we add a new gravitational source in the energy–momentum tensor of a perfect fluid. The corresponding field equations, hydrostatic equilibrium equation as well as matching conditions are evaluated. We obtain the anisotropic model by extending the known Durgapal and Gehlot isotropic solution and examined the physical viability as well as the stability of the developed model. It is found that the system exhibits viable behavior for all fluid variables as well as energy conditions and the stability criterion is fulfilled.

In this work, we study cosmological evolution in the mini creation event with anisotropic fluid under the Bianchi-I spacetime. After providing the basic mathematical formalism of the model, we find out the exact solutions in its particular form to the field equations. In order to get physical validity, we have presented elaborate discussions on the graphical results, especially nonsingular behaviour of the model. It is shown that the model under mini bang can successfully exhibit several interesting cosmological features which have specific signatures to tally with the observational evidences.

In this paper, we have obtained a new family of interior solutions of the Einstein–Maxwell field equations in general relativity for static, spherically symmetric distribution of charged fluid. The electric density is assumed to be $\frac{E^2}{C}=\frac{K}{2}x{(1+7x)}^{(\frac{2}{3})}$ where $K$ is a positive constant. The solution is well behaved for a wide range of $K$ hence, it is suitable for modeling a superdense star. The surface density is assumed to be ${\rho }_s=2\times 10^{14}{\mathrm{\text{g/cm}}}^3$, for the validation of the solution set present, a comparison has also been made with the stars EXO 1785-248, SMC X-1, Her X-1, 4U 1538-52. For these particular stars, the maximum value of surface redshift is obtained to be 0.3251 for the star EXO 1785-248 and the minimum value is obtained as 0.2027 for the star Her X-1. We subject our solution to rigorous physical viability tests. In the absence of charge, we obtain the regular and well-behaved sixth model of Durgapal.

Recent research works have shown the existence of super-massive neutron stars (NSs) with mass about $2.2M_{\odot }$ or even more. The query about those super-massive NSs inspires the researchers to analyze their features and structures immensely. Here, we have inspected the behavior and properties of some of those super-massive NSs in $f(T)$ modified gravity with $T=T+\alpha T^2$, where $T$ is the torsional scalar and $\alpha$ is a regulatory parameter. In this framework of teleparallel formalism of modified gravity, we obtain the equations of motion by considering quintessence field, modified Chaplygin gas (MCG) and electromagnetic field. For our model, we use matching conditions under spherical symmetry, in order to find out the numerical values of different unknown constants of our model. This helps us to acquire various physical quantities thoroughly and to understand about the nature of those super-massive NSs deeply and quite clearly. Moreover, from our work, we can also explain the role of quintessence field and MCG in case of massive compact stars. The mass–radius relationship curve of this model can effectively describe the mass of the heaviest NS (about $2.6M_{\odot }$) ever detected via gravitational wave detection. Again, we overall investigate the anisotropic behavior, density profile, pressure profile, core repulsive force, stability, equilibrium and energy conditions of those massive compact objects. We further analyze different important parameters like anisotropic stress, surface redshift, adiabatic behavior, compactness factor, sound speed, etc. in case of super-massive NSs for better realization and future study.