Analytic solutions of the space–time conformable fractional Klein–Gordon equation in general form

ABSTRACT

The Klein–Gordon equation plays an important role in mathematical physics. In this paper, a direct method which is very effective, simple, and convenient, is presented for solving the conformable fractional Klein–Gordon equation. Using this analytic method, the exact solutions of this equation are found in terms of the Jacobi elliptic functions. This method is applied to both time and space fractional equations. Some solutions are also illustrated by the graphics.     

Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes

For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to \({\frac{\left|am\right|}{2Mr_+}}\) . In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation.     

Generalized Klein–Gordon field equations with octonion space-time (OST) algebra

The concept of space-time representation is redefined using the octonion space-time (OST) algebra. In this study, describing the properties of octonions and their possible connection with Euclidean space-times, the internal and external space-time events are represented within the OST algebra. Keeping in mind the octonionic dual-Euclidean space-times, we express the homogeneous field equations which leads to the symmetrical nature of internal and external space-times. We derive the generalized Proca–Maxwell equations for massive-dyons in the case of the OST algebra. Accordingly, we have obtained a new set of octonionic Klein–Gordon potential (KGP) and Klein–Gordon field (KGF) equations for massive dyons from the generalized Proca–Maxwell equations. This formalism demonstrates that the octonionic KGP and KGF equations can be expressed in a single equation and it is equivalent to energy-momentum relation for dyons. As such, we have made an attempt to write the conservation of Noetherian current from the octonionic Klein–Gordon equations.     

Fractional Klein–Gordon Equations and Related Stochastic Processes

This paper presents finite-velocity random motions driven by fractional Klein–Gordon equations of order $\alpha \in (0,1]$ . A key tool in the analysis is played by the McBride’s theory which converts fractional hyper-Bessel operators into Erdélyi–Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein–Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein–Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha )$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha )$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of direction. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$ -dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler–Poisson–Darboux equation to which our theory applies.     

The modified simple equation method for solving some fractional-order nonlinear equations

Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinear fractional Klein–Gordon equation by using the modified simple equation method.     

Solitons and periodic solutions to a couple of fractional nonlinear evolution equations

This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.     

Loop Quantum Gravity à la Aharonov–Bohm

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bi?ák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.     

A Bohmian approach to the perturbations of non-linear Klein–Gordon equation

In the framework of Bohmian quantum mechanics, the Klein–Gordon equation can be seen as representing a particle with mass m which is guided by a guiding wave ?(x) in a causal manner. Here a relevant question is whether Bohmian quantum mechanics is applicable to a non-linear Klein–Gordon equation? We examine this approach for ?⁴(x) and sine-Gordon potentials. It turns out that this method leads to equations for quantum states which are identical to those derived by field theoretical methods used for quantum solitons. Moreover, the quantum force exerted on the particle can be determined. This method can be used for other non-linear potentials as well.     

Velocities of Longitudinal and Transverse Elastic Vibrations in Superionic Silver Sulfide

JETP Letters - A system consisting of material particles and a field is studied. The latter is described by the nonlinear Klein—Gordon equation. A modified Klein—Gordon equation, which...     

On precise center stable manifold theorems for certain reaction–diffusion and Klein–Gordon equations

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.     

A singularity theorem for Einstein–Klein–Gordon theory

Hawking’s singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking’s hypotheses, an important example being the massive Klein–Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein–Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein–Klein–Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. These results remain true in the presence of additional matter obeying both the strong and weak energy conditions.     

The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations

In this paper, we implemented the functional variable method and the modified Riemann–Liouville derivative for the exact solitary wave solutions and periodic wave solutions of the time-fractional Klein–Gordon equation, and the time-fractional Hirota–Satsuma coupled KdV system. This method is extremely simple but effective for handling nonlinear time-fractional differential equations.     

Massless Spin–Zero Particle and the Classical Action via Hamilton–Jacobi Equation in G?del Universe

In this letter we investigate the separability of the Klein–Gordon and Hamilton–Jacobi equation in G?del universe. We show that the Klein–Gordon eigen modes are quantized and the complete spectrum of the particle’s energy is a mixture of an azimuthal quantum number, m and a principal quantum number, n and a continuous wave number k. We also show that the Hamilton–Jacobi equation gives a closed function for classical action. These results may be used to calculate the Casimir vacuum energy in G?del universe.     

The symmetries and conservation laws of some Gordon-type equations in Milne space-time

In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for the Klein–Gordon equation in one case. Finally, the existence of higherorder variational symmetries of a projection of the Klein–Gordon equation is investigated using the multiplier approach.     

Scalar bosons under the influence of noninertial effects in the cosmic string spacetime

In this paper we present two different classes of solutions for the Klein–Gordon equation in the presence of a scalar potential under the influence of noninertial effects in the cosmic string spacetime. We show that noninertial effects restrict the physical region of the spacetime where the particle can be placed, and furthermore that the energy levels are shifted by these effects. In addition, we show that the presence of a Coulomb-like scalar potential allows the formation of bound states when the Klein–Gordon equation is considered in this kind of spacetime.     

Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons

We consider a damped, parametrically driven discrete nonlinear Klein–Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein–Gordon equation.     

The discrete variational derivative method based on discrete differential forms

As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete variational derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn–Hilliard equation and the Klein–Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H¹-stable under some assumptions. In particular, one for the nonlinear Klein–Gordon equation is obtained by combination of the energy conservation property and the discrete Poincaré inequality, which are the temporal and spacial structures that are preserved by the above methods.     

Analytical solutions in a cosmic string Born–Infeld-dilaton black hole geometry: quasinormal modes and quantization

Chargeless massive scalar fields are studied in the spacetime of Born–Infeld dilaton black holes (BIDBHs). We first separate the massive covariant Klein–Gordon equation into radial and angular parts and obtain the exact solution of the radial equation in terms of the confluent Heun functions. Using the obtained radial solution, we show how one gets the exact quasinormal modes for some particular cases. We also solve the Klein–Gordon equation solution in the spacetime of a BIDBHs with a cosmic string in which we point out the effect of the conical deficit on the BIDBHs. The analytical solutions of the radial and angular parts are obtained in terms of the confluent Heun functions. Finally, we study the quantization of the BIDBH. While doing this, we also discuss the Hawking radiation in terms of the effective temperature.