Narrow Results

This research studies primarily centers around a specific vacuum solution of the Einstein’s field equations with a negative cosmological constant, known as a type-D vacuum solution in Anti-de Sitter (AdS) background. The key finding of this study is that the determinant of the Ricci tensor ($R_{\mu \nu }$) possesses a nonzero value. This observation ensures that this vacuum spacetime can be investigated within the framework of Ricci-inverse gravity as well. In fact, it has demonstrated that this type-D vacuum solution serves as an exact solution in Ricci inverse gravity in the background of AdS space. Notably, the negative cosmological constant undergoes modification by the coupling constant within this context. Moreover, we show that this gravity theory permits the formation of closed time-like curves at a specific moment in time analogue to the general relativity case. Consequently, this theory introduces an intriguing time-machine model within the domain of Ricci-inverse gravity theory.

We find new classes of the dynamical black hole solutions in five- or higher-dimensional Einstein–Maxwell-dilaton theory, in which the dilaton field interacts nontrivially with the Maxwell field, and the cosmological constant, with two arbitrary coupling constants. The solutions which are nonstationary, and almost conformally regular everywhere, are constructed by embedding the four-dimensional self-dual geometries. We consider three different situations, where the two coupling constants are not equal, are nonzero equal and finally both are zero. We also consider the cases where one of the metric functions, depends on one coordinate only, and another interesting case where it depends on two spatial coordinates. We also find that the cosmological constant can take all possible negative, zero and positive values, hence our solutions are asymptotically anti-de Sitter, flat and de Sitter. We discuss the properties of the solutions which are the analytic description of coalescing black holes in any dimensions, greater or equal to five.

The fractional mKDV–Zakharov–Kuznetsov (FmKDV-ZK) equation with M-truncated derivatives (MTDs) is considered. To obtain a novel rational, trigonometric, hyperbolic, and elliptic, solutions for FmKDV-ZK, we use two different methods including the modified mapping technique and generalized Riccati equation method. These methods are effective and simple. Also, they give us different kinds of solutions such as elliptic, trigonometric, hyperbolic, and rational solutions. The FmKDV-ZK equation is widely used in engineering, geophysics, meteorology, ocean dynamics, and plasma physics, therefore the obtained solutions may be applied to a wide range of important physical phenomena. Moreover, we display a wide range of solutions in two-dimensional and three-dimensional spaces to examine the effects of the MTDs operator. We conclude that the surface expands as the derivative order decreases.

In this work, the dynamical structure for the extended equation is analyzed through unified Riccati equation expansion (UREE) and the Lie isomorphism method for the (3+1)-dimensional Wazwaz–Benjamin–Bona–Mahony (WBBM) equation. This equation represents the unidimensional propagation of short amplitude long waves on the water’s surface in a medium. These employed techniques are the most powerful and effective ways to obtain different sets of new and more generalized exact soliton solutions of the WBBM equation. Furthermore, what distinguishes this study from other studies is that it not only acquires a variety of analytical wave solutions for the studied models but also, demonstrates the interaction phenomena for these results as they propagate over time. Also, shows various meaningful graphs of the processes that provide valuable wisdom for understanding their behavior. The UREE method directly provides various new exact soliton solutions with some novel dynamical properties. We perform a detailed Lie symmetry analysis to governing equation that leaves the system invariant. The Lie group method explores six Lie isomorphism groups to study the WBBM equation. First, we find infinitesimal transformations employing the one-parameter Lie symmetry method. Second, we solve the infinitesimal generator and reduce the order of the equation. Moreover, we illustrate some two-dimensional (2D), three-dimensional (3D), and contour diagrams of the obtained results and compute the exact analytical solution utilizing the used methods. To find novel solutions, the Adomian method is also used, where the Adomian polynomials are utilized to deal with nonlinear terms. Variety of new analytical solutions with different types of dynamical behavior are analyzed by utilizing the computational software like Mathematica. These new analytical exact wave solutions are demonstrated in various dynamical structures of periodic wave soliton, interaction periodic wave and kink wave soliton, lump wave soliton, doubly soliton, multi-wave soliton, kink periodic, parabolic wave, multisoliton, traveling wave, and standing wave-shaped profiles.

In this paper, the time-fractional two-dimensional unsteady convection diffusion system governed by nonlinear partial differential equations captured the time evaluation of disturbance and addressing modeling real-world phenomena such as turbulence, traffic flow, heat and fluid transport, and gas dynamics. Fractional derivatives are introduced to enable a more generalized representation of diffusion behaviors. To establish solution the unique existence, we apply a fixed-point strategy via Lipschitz continuity. By employing the generalized exponential rational function method, we derive exact solitary wave solutions in the forms of dark, singular, complex, combo, trigonometric, and rational waves. Furthermore, 3D plots illustrate solution dynamics under various parameter choices providing deep exploration into the system’s behavior and demonstrating the applicability of fractional order models in advanced technological and scientific contexts.

We look for "static" spherically symmetric solutions of Einstein's Equations for perfect fluid source with equation of state p = wρ, for constant w. We consider all four cases compatible with the standard ansatz for the line element, discussed in previous work. For each case, we derive the equation obeyed by the mass function or its analogs. For these equations, we find all finite-polynomial solutions, including possible negative powers.

For the standard case, we find no significantly new solutions, but show that one solution is a static phantom solution, another a black hole-like solution. For the dynamic and/or tachyonic cases we find, among others, dynamic and static tachyonic solutions, a Kantowski–Sachs (KS) class phantom solution, another KS-class solution for dark energy, and a second black hole-like solution.

The black hole-like solutions feature segregated normal and tachyonic matter, consistent with the assertion of previous work. In the first black hole-like solution, tachyonic matter is inside the horizon, in the second, outside.

The static phantom solution, a limit of an old one, is surprising at first, since phantom energy is usually associated with super-exponential expansion. The KS-phantom solution stands out since its "mass function" is a ninth order polynomial.

After formulating the notion of integrability for axially symmetric harmonic maps from ${ℝ}^3$ into symmetric spaces, we give a complete and rigorous proof that, subject to some mild restrictions on the target, all such maps are integrable. Furthermore, we prove that a variant of the inverse scattering method, called vesture (dressing) can always be used to generate new solutions for the harmonic map equations starting from any given solution. In particular, we show that the problem of finding $N$-solitonic harmonic maps into a non-compact Grassmann manifold $SU(p,q)/S(U(p)\times U(q))$ is completely reducible via the vesture (dressing) method to a problem in linear algebra which we prove is solvable in general. We illustrate this method, and establish its agreement with previously known special cases, by explicitly computing a 1-solitonic harmonic map for the two cases $(p=1,q=1)$ and $(p=2,q=1)$ and showing that the family of solutions obtained in each case contains respectively the Kerr family of solutions to the Einstein vacuum equations, and the Kerr–Newman family of solutions to the Einstein–Maxwell equations in the hyperextreme sector of the corresponding parameters.

The article presents a series of numerical simulations of exact solutions of the Einstein equations performed using the Cactus code, a complete three-dimensional machinery for numerical relativity. We describe an application ("thorn") for the Cactus code that can be used for evolving a variety of exact solutions, with and without matter, including solutions used in modern cosmology for modeling the early stages of the universe. Our main purpose has been to test the Cactus code on these well-known examples, focusing mainly on the stability and convergence of the code.

In this paper, a generalized extended tanh-function method for constructing exact solutions of nonlinear evolution equations (NEEs) is presented and implemented in a computer algebraic system. Applying the generalized method, with the aid of Maple, we consider the system of the approximate equations for long water waves and find the new more general explicit exact solutions. The properties of the new solitary wave solutions are shown by some figures.

The Painlevé property of coupled, nonautonomous Korteweg–de Vries (KdV) type of systems is studied. The conditions under which the systems pass the Painlevé test for integrability are obtained. For some of the integrable cases, exact solutions are given.

Numerical schemes that are implemented by interpolation of exact solutions to a differential equation naturally preserve geometric properties of the differential equation. The solution interpolation method can be used for development of a new class of geometric integrators, which generally show better performances than standard method both quantitatively and qualitatively. Several examples including a linear convection equation and a nonlinear heat equation are included.

In this paper, we show how computerized symbolic computations can be used to find an auto-Bäcklund transformation and a family of exact analytical solutions to the variant Boussinesq model for water waves. Sample explicit solutions are presented, which are respectively solitonic and rational.

We report on a new two-parameter class of cosmological solutions of the Einstein–Maxwell equations. The solutions have everywhere regular curvature invariants. We prove that the solutions are geodesically complete and globally hyperbolic.

Anisotropic cosmological models such as the Gödel universe and its extensions – Gödel type solutions, are embedded on a visible brane in the model 1 of Randall Sundrum. The size of the extra dimension is stabilized by tuning the rotation parameter to a very small value so that hierarchy problem can be solved. A limiting case also yields the Randall Sundrum 2 model. The angular velocity ω ~ 1 / a would be of O (e^k|y|) which would be unacceptably large if e^k|y| were adjusted to solve the hierarchy problem as normally formulated. Should one choose to reverse the choice of the fundamental and generated scales, e^k|y| could turn out to be O (10^-16) which would fall in the right range.

We prove a theorem that characterizes a two-parameter family of solutions to Einstein equations with a negative cosmological constant, representing, in general, non-spherical radiating black holes in an anti-de Sitter background. It is shown that the best known non-spherical radiating black hole solutions are particular cases and static non-spherical black hole solutions, for Type I fluid, are also retrieved. A brief discussion on the energy conditions, singularities and horizons is provided.

We obtain the most general static cylindrically symmetric vacuum solutions of the Einstein field equations in (4 + N) dimensions. Under the assumption of separation of variables, we construct a family of Levi–Civita–Kasner vacuum solutions in (4 + N) dimensions. We discuss the dimensional reduction of the static solutions. Depending on the reduction procedure, they can be interpreted either as a scalar-vacuum generalization of Levi–Civita spacetimes, or as the effective 4D vacuum spacetime outside of an idealized string in braneworld theory.

Starting with a Zipoy–Voorhees line element, we construct and study the three-parameter family of solutions describing a deformed black string with arbitrary tension.

We present the topological solutions of Einstein gravity in the presence of a non-Abelian Yang–Mills field. In (n+1) dimensions, we consider the SO(n(n-1)/2-1, 1) semisimple group as the Yang–Mills gauge group, and introduce the black hole solutions with hyperbolic horizon. We argue that the four-dimensional solution is exactly the same as the four-dimensional solution of Einstein–Maxwell gravity, while the higher-dimensional solutions are new. We investigate the properties of the higher-dimensional solutions and find that these solutions in five dimensions have the same properties as the topological five-dimensional solution of Einstein–Maxwell (EM) theory although the metric function in five dimensions is different. But in six and higher dimensions, the topological solutions of EYM and EM gravities with non-negative mass have different properties. First, the singularity of EYM solution does not present a naked singularity and is spacelike, while the singularity of topological Reissner–Nordström solution is timelike. Second, there are no extreme six or higher-dimensional black holes in EYM gravity with non-negative mass, while these kinds of solutions exist in EM gravity. Furthermore, EYM theory has no static asymptotically de Sitter solution with non-negative mass, while EM gravity has.

In this work we solve the Dirac equation by constructing the exact bound state solutions for a mixing of scalar and vector spherically asymmetrical singular oscillators. This is done provided that the vector potential is equal to the scalar potential. The spinor wave functions and bound state energy levels are presented. The case V(r) = -S(r) is also considered.

Motivated by studies on gravitational lenses, we present an exact solution of the field equations of general relativity, which is static and spherically symmetric, has no mass but has a nonvanishing spacelike components of the stress–energy–momentum tensor. In spite of its strange nature, this solution has nontrivial descriptions of gravitational effects. We show that the main aspects found in the dark matter phenomena can be satisfactorily described by this geometry. We comment on the relevance it could have to consider nonvanishing spacelike components of the stress–energy–momentum tensor ascribed to dark matter.