Klein-Gordon Research Articles

Analysis of quantum systems subject to inverse trigonometry Scarf plus Coulomb potential in N-dimensions

We present approximate solutions of the N-dimensional Klein–Gordon equation in the presence of a superposition of inverse trigonometry Scarf potential and Coulomb potential through asymptotic iteration approach. With the purpose of dealing with the -state, we use Greene-Aldrich approximation which is valid for the small values of the screening parameter. The analytical expressions for the relativistic energy eigenvalues and the corresponding eigenfunctions are established in hyperspherical coordinates. The normalized hyper-radial wavefunction is expressed in terms of hypergeometric and Jacobi polynomials. We analyze the dependences of the relativistic energy levels on the dimensions. We also discuss the effect of the screening parameter on the relativistic energy spectrum. The expectation values of inverse position the square of inverse position kinetic energy and square of momentum in N-dimensional space are reported using the Hellmann-Feynmann theorem. Discussing the special cases of the relevant potential, we show the accuracy of our results.

A trial solution for imposing boundary conditions of partial differential equations in physics-informed neural networks

This article proposes an auxiliary function for imposing the boundary and initial conditions in physics-informed neural network models in a hard manner that accelerates the learning process. This auxiliary function consists of two pre-trained neural networks and a main deep neural network with trainable parameters. The novelty of this new auxiliary function is the input of main deep neural network, which takes the outputs of distance function and boundary function as inputs in addition to the spatiotemporal variables. We demonstrate the efficacy and general applicability of the proposed model by applying it to several benchmark-forward problems namely, advection, Helmholtz and Klein-Gordon equations. The accuracy of predictions is examined by comparison with exact solutions. Our findings imply the superiority of the proposed model because of the improvement of the loss convergence to lower values by one order of magnitude for the same number of epochs. In the case of the advection equation, the relative L2 error has been reduced from 0.025 to 0.0201, and from 0.016 to 0.0152. When applied to the Helmholtz equation, our novel model achieved an error of 8.67 × 10−3, surpassing the conventional model's performance, which yielded an error of 6.04 × 10−2. Furthermore, for the Klein-Gordon equation, our new model led to a remarkable reduction in the relative L2 error, from 0.18 to an impressive 4.2 × 10−2.

On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method

Abstract This investigation explores two numerical approaches: the optimal auxiliary function method (OAFM) and the new iterative method (NIM). These techniques address the physical fractional-order Klein-Gordon equations (FOKGEs), a class of partial differential equations (PDEs) that model various physical phenomena in engineering and diverse plasma models. The OAFM is a recently introduced method capable of efficiently solving several nonlinear differential equations (DEs), whereas the NIM is a well-established method specifically designed for solving fractional DEs. Both approaches are utilized to analyze different variations in FOKGE. By conducting numerous numerical experiments on the FOKGE, we compare the accuracy, efficiency, and convergence of these two proposed methods. This study is expected to yield significant findings that will help researchers study various nonlinear phenomena in fluids and plasma physics.

Exact analytical quasibound states of a scalar particle around a slowly rotating black hole

In this work, behaviour of massive and massless scalar particle around a slowly rotating black hole in 3+1 dimension is investigated. Exact solutions of the Klein-Gordon equation, i.e. the wave functions and energy levels are discovered. The quasibound state has complex-valued energy levels E=ER+iEI where the real part ER is directly related to the particle's energy while the imaginary part represents a decay. Using the obtained exact wave solutions near the exterior horizon of the black hole, the Hawking radiation is discussed.

The partial null conditions and global smooth solutions of the nonlinear wave equations on [formula omitted] with d = 2,3

In this paper, we investigate the fully nonlinear wave equations on the product space R3×T with quadratic nonlinearities and on R2×T with cubic nonlinearities, respectively. It is shown that for the small initial data satisfying some space-decay rates at infinity, these nonlinear equations admit global smooth solutions when the corresponding partial null conditions hold and while have almost global smooth solutions when the partial null conditions are violated. Our proof relies on the Fourier mode decomposition of the solutions with respect to the periodic direction, the efficient combinations of time-decay estimates for the solutions to the linear wave equations and the linear Klein-Gordon equations, and the global weighted energy estimates. In addition, an interesting auxiliary energy is introduced. As a byproduct, our results can be applied to the 4D irrotational compressible Euler equations of polytropic gases or Chaplygin gases on R3×T, the 3D relativistic membrane equation and the 3D nonlinear membrane equation on R2×T.

Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

Infinite-time blowup and global solutions for a semilinear Klein–Gordan equation with logarithmic nonlinearity

In this article, our focus lies in investigating the existence of global solutions and the occurrence of infinite-time blowup for a nonlinear Klein–Gordon equation characterized by logarithmic nonlinearity, specifically in the form . Notably, our inquiry involves handling logarithmic functions within the reaction terms and accommodating the functions and in the boundary terms. Consequently, a pivotal task is to establish blowup conditions that intricately hinge on the characteristics of the domains and the boundary conditions. It is of significant note that our exploration incorporates domain and boundary information into the formulation of blowup conditions. By employing a combination of the potential well technique and energy estimation methodology, we delve into scenarios of low initial energy and critical initial energy. In doing so, we derive a set of sufficient conditions that encompass both the global existence and the potential explosive behaviour of solutions pertaining to this variant of the Klein–Gordon equation. This work contributes to enhancing our understanding of the intricate interplay between logarithmic nonlinearity, domain characteristics, and boundary conditions in shaping the behaviour of solutions in the realm of nonlinear wave equations.

Inverse Scattering Method via Riemann–Hilbert Problem for Nonlinear Klein–Gordon Equation Coupled with a Scalar Field

Inverse Scattering Method via Riemann–Hilbert Problem for Nonlinear Klein–Gordon Equation Coupled with a Scalar Field

Greybody factor for a non accelerated charged modified black hole in anti-de sitter regime

This paper investigates the greybody factor for a non accelerated black hole with modified Maxwell electrodynamics in an anti-de Sitter regime. For this purpose, we compute the radial equation for a massless scalar field with the help of Klein–Gordon equation. We then formulate effective potential by transforming this equation into Schrodinger wave equation. We analyze the graphical behavior of effective potential for different values of mass, parameter characterizing the modified Maxwell theory, anti-de Sitter radius and electromagnetic charge parameters. The exact solutions are computed at two different horizons, i.e., event and cosmological horizons through the radial equation. Furthermore, we match the obtained solutions in an intermediate regime to enhance feasibility of the greybody factor over the entire domain and check its behavior graphically. It is found that the greybody factor has a direct relation with the radius, electromagnetic charge as well as angular momentum of the black hole and an inverse relation with the anti-de Sitter radius and modification parameter. We conclude that the modified Maxwell solution reduces the emission rate of the black hole.

Radio Pulsars Resonantly Accelerating Electrons

Based on the recently demonstrated resonant wave–wave process, it is shown that electrons can be accelerated to ultra-relativistic energies in the magnetospheres of radio pulsars. The energization occurs via the resonant interaction of the electron wave (described by the Klein–Gordon (KG) equation) moving in unison with an intense electromagnetic (EM) wave; the KG wave/particle continuously draws energy from EM. In a brief recapitulation of the general theory, the high-energy (resonantly enhanced) electron states are investigated by solving the KG equation, minimally coupled to the EM field. The restricted class of solutions that propagate in phase with EM radiation (functions only of ζ=ωt−kz) are explored to serve as a possible basis for the proposed electron energization in the radio pulsars. We show that the wave–wave resonant energization mechanism could be operative in a broad class of radio pulsars with periods ranging from milliseconds to normal values (∼1 s); this could drive the magnetospheric electrons to acquire energies from 100 s of TeVs (millisecond pulsars) to 10 ZeVs (normal pulsars).

Applications of the Klein–Gordon equation in the Feshbach–Villars representation in the non-inertial cosmic string space–time

We study the relativistic quantum motion of a spinless particle using the Feshbach–Villars (FV) formalism in the spinning cosmic string spacetime. The equations of motion are derived using the first-order FV formulation of the Klein–Gordon (KG) equation. We apply the equation of motion (a) to study the motion of the particle confined to a rigid-wall potential, (b) motion in the presence of a Coulomb-type potential, and (c) particle interacting with the Feshbach–Villars oscillator (FVO). The energy levels and wave functions are obtained for the three cases. Our study focused on the impact of rotation and curvature on the energy levels of the particle.

Decay and Strichartz estimates for Klein–Gordon equation on a cone I: Spinless case

Abstract We consider the solutions of the Klein–Gordon equation in the 2 + 1 2{+}1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ ℤ {\mathbb{R}_{t}\times\mathbb{R}_{+}\times\mathbb{R}/2\pi\alpha\mathbb{Z}} created by a massive point particle. Using the Schwartz kernel of resolvent and spectral measure for Schrödinger operator on the spinless cone, we prove the dispersive estimates and Strichartz estimates for the Klein–Gordon equation. In a future paper, we will consider the problem on the spinning cone.

Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: Error analysis and algorithms

We consider the approximation of a class of dynamic partial differential equations (PDEs) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the nonlinear Klein-Gordon equation and the linear elastodynamic equation. Our analyses show that, with feed-forward neural networks having two hidden layers and the tanh activation function, the PINN approximation errors for the solution field, its time derivative and its gradient field can be effectively bounded by the training loss and the number of training data points (quadrature points). Our analyses further suggest new forms for the training loss function, which contain certain residuals that are crucial to the error estimate but would be absent from the canonical PINN loss formulation. Adopting these new forms for the loss function leads to a variant PINN algorithm. We present ample numerical experiments with the new PINN algorithm for the wave equation, the Sine-Gordon equation and the linear elastodynamic equation, which show that the method can capture the solution well.

Asymptotics of the Solution of an Initial–Boundary Value Problem for the One-Dimensional Klein–Gordon Equation on the Half-Line

Asymptotics of the Solution of an Initial–Boundary Value Problem for the One-Dimensional Klein–Gordon Equation on the Half-Line

A quantum analog model for a scalar particle interacting with a Kerr–de Sitter black hole

We study the thin-layer quantization of a Schrödinger particle constrained to move along an oblate spheroidal surface embedded in Euclidean space as an analog model to a particle in the background of a black hole and of a wormhole. The Schrödinger equation for the superficial wavefunction leads to an ordinary differential equation with five singularities — four regular and one irregular. We manage to transform this equation into a generalized Heun’s equation, with three regular singular points and one irregular singularity at infinity. This equation was studied by Schäfke and Schmidt back in 1980. By means of an external potential, we are able to reproduce the equations of motion of a particle close both to a black hole and a wormhole, thus creating an effective analog model. Additionally, we present correct and exact solutions for the radial part of the Klein–Gordon equation in the Kerr–de Sitter metric.

Path integral treatment of a Klein Gordon particle with generalized inverse Quadratic Yukawa potential

We solved the Klein–Gordon equation for a generalized inverse quadratic Yukawa (GIQY) potential via path integrals approach. We applied approximations to deal with the terms 1/r 2 and 1/r. A path integral representation of Green’s function relating to a particle moving in a mixture of equal vector and scalar potentials was established. To integrate Green’s function, a space-time transformation was successfully used, and the present problem was reduced to a previously known modified Pöschl-Teller potential problem. The integrated Green’s function contains important information about the considered system, from which we obtained energy eigenvalues and the corresponding normalized eigenfunctions for various values of n and ℓquantum numbers. Numerical results, Schrödinger solutions, and special cases such as the modified screened Coulomb plus inversely quadratic Yukawa potential, Kratzer potential, Yukawa potential, and Coulomb potential were also considered. These results are largely consistent with previous studies.

Computational analysis of thin-shell with scalar field for class of new black hole solutions in metric-affine gravity

The aim of this study is to investigate the geometric structure of a thin-shell within the context of metric-affine gravity. To accomplish this, we employ a cut and paste approach to match the inner Minkowski spacetime and an outer newly calculated class of black hole solutions in metric-affine gravity. These solutions are Reissner–Nordström-like black holes with shear, spin, dilation, electric, magnetic charges, and a cosmological constant. Using the Klein–Gordon equation of motion, we analyze the dynamics of the thin-shell configuration under both massive and massless scalar fields. We find that the scalar field affects thin-shell dynamics and can lead to interesting effects such as dynamics and stability configurations. We demonstrate that the linearized radial perturbation approach employing a phantom-like equation of state, such as quintessence, dark energy, and phantom energy, reveal stable configurations of the thin-shell located beyond the expected position of the event horizon of the exterior manifold. We also find that the stability of the thin-shell depends strongly on the parameters of the black hole solutions. Our results shed light on the dynamics of thin-shells in the context of metric affine gravity with scalar fields and provide new insights into the behavior of black hole solutions in this theory. It is concluded that the thin-shell composed of quintessence and dark energy shows the more stable configuration for the choice of Reissner-Nordström-like black holes with shear, spin, and dilation charges.

Peculiarities of the Universe and COVID-19 Virus Vital Functioning Discovered by Solving Klein-Gordon Equation

Peculiarities of the Universe and COVID-19 Virus Vital Functioning Discovered by Solving Klein-Gordon Equation

Fock Quantization of a Klein–Gordon Field in the Interior Geometry of a Nonrotating Black Hole

We study the canonical quantization of a scalar field in Kantowski–Sachs spacetime. For simplicity, we consider compactified spatial sections, since this does not affect the ultraviolet behavior. A time-dependent canonical transformation is performed prior to quantization. As in previously studied cases, the purpose of this canonical transformation is to identify and extract the background contribution to the field evolution which is obstructing a unitary implementation of the field dynamics at the quantum level. This splitting of the time dependence into a background piece and the part to be seen as true quantum evolution is, to a large extent, determined by the unitarity requirement itself. The quantization is performed in the usual setup of Fock representations, demanding the preservation of the spatial symmetries. Under the joint requirements of quantum unitary dynamics and compatibility with those classical symmetries, the quantization is shown to be unique, in the sense that any two representations with these properties are unitarily equivalent. This confirms the validity of our conditions as criteria to discriminate among possibly inequivalent quantum descriptions. The interest of this analysis goes beyond cosmological applications since the interior of a nonrotating black hole has a geometry of the Kantowski–Sachs type.

Quasi-periodic waves and asymptotic behavior for a coupled nonlinear Klein-Gordon equation

With the Hirota’s bilinear method, N N -periodic wave solutions to the coupled nonlinear Klein-Gordon equation are constructed in terms of theta function. The asymptotic behaviors under small amplitude limits are also deduced strictly for the one and two-periodic waves. Moreover, a numerical scheme is presented for the case of N ≥ 3 N\geq 3 . Some three-periodic wave solutions are calculated numerically as examples.