Klein-Gordon Research Articles

Numerical solutions for the f(R)-Klein–Gordon system

We construct a numerical relativity code based on the Baumgarte–Shapiro–Shibata–Nakamura (BSSN) formulation for the gravitational quadratic f(R) Starobinsky model. By removing the assumption that the determinant of the conformal 3-metric is unity, we first generalize the BSSN formulation for general f(R) gravity theories in the metric formalism to accommodate arbitrary coordinates for the first time. We then describe the implementation of this formalism to the paradigmatic Starobinsky model. We apply the implementation to three scenarios: the Schwarzschild black hole solution, flat space with non-trivial gauge dynamics, and a massless Klein–Gordon (KG) scalar field. In each case, long-term stability and second-order convergence is demonstrated. The case of the massless KG scalar field is used to exercise the additional terms and variables resulting from the f(R) contributions. For this model, we show for the first time that additional damped oscillations arise in the subcritical regime as the system approaches a stable configuration.

M-WDRNNs: Mixed-Weighted Deep Residual Neural Networks for Forward and Inverse PDE Problems

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in recent years. But studies have shown that there is a gradient pathology in PINNs. That is, there is an imbalance gradient problem in each regularization term during back-propagation, which makes it difficult for neural network models to accurately approximate partial differential equations. Based on the depth-weighted residual neural network and neural attention mechanism, we propose a new mixed-weighted residual block in which the weighted coefficients are chosen autonomously by the optimization algorithm, and one of the transformer networks is replaced by a skip connection. Finally, we test our algorithms with some partial differential equations, such as the non-homogeneous Klein–Gordon equation, the (1+1) advection–diffusion equation, and the Helmholtz equation. Experimental results show that the proposed algorithm significantly improves the numerical accuracy.

Criteria for finite time blow up for a system of Klein–Gordon equations

We give three conditions on initial data for the blowing up of the corresponding solutions to some system of Klein–Gordon equations on the three dimensional Euclidean space. We first use Levine's concavity argument to show that the negativeness of energy leads to the blowing up of local solutions in finite time. For the data of positive energy, we give a sufficient condition so that the corresponding solution blows up in finite time. This condition embodies datum with arbitrarily large energy. At last we use Payne–Sattinger's potential well argument to classify the datum with energy not so large (to be exact, below the ground states) into two parts: one part consists of datum leading to blowing-up solutions in finite time, while the other part consists of datum that leads to the global solutions.

Single-particle quantum mechanics of the free Klein–Gordon equation with Lorentz violation

Single-particle quantum mechanics of the free Klein–Gordon equation with Lorentz violation

Asymptotic behavior of 2D wave–Klein-Gordon coupled system under null condition

We study the 2D coupled wave–Klein-Gordon systems with semilinear null nonlinearities Q0 and Qαβ. The main result states that the solution to the 2D coupled system exists globally provided that the initial data is small in some weighted Sobolev space, which does not necessarily have compact support, and we also study the asymptotic behavior of the solution. In particular, we show the optimal time decay and scattering of the solution.The major difficulties lie in the slow decay nature of the wave and the Klein-Gordon components in two space dimensions, in addition, extra difficulties arise due to the presence of the null form Q0 which is not of divergence form and is not compatible with the Klein-Gordon equations. To overcome the difficulties, a new observation for the structure of the null form Q0 is required.

PDM Klein–Gordon particles in Som–Raychaudhuri cosmic string spacetime with space-like dislocation and a magnetic field: Spacetime associated degeneracies, vorticity-energy, and charge-energy correlations

We explicitly study and report the effects of the gravitational fields on the spectroscopic structure of Klein–Gordon (KG) particles in Som–Raychaudhuri cosmic string spacetime with space-like dislocation and a magnetic field. We extend this study to include the effects of the gravitational fields on some position-dependent mass (PDM) KG-particles. As effects of the gravitational fields, we report new spacetime associated degeneracies, vorticity-energy correlations and charge-energy correlations. Such degeneracies and correlations are observed in the spectroscopic structure of both constant mass KG-particles and PDM KG-particles at hand.

Blow-up solutions of damped Klein–Gordon equation on the Heisenberg group

We prove the blow-up of solutions of the semilinear damped Klein–Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper Ruzhansky and Tokmagambetov (J Differ Equ 265(10):5212–5236, 2018), where the global in time well-posedness was proved for the small energy solutions.

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Exact analytical solution of the one-dimensional time-dependent radiative transfer equation with linear scattering

The radiative transfer equation (RTE) is a cornerstone for describing the propagation of electromagnetic radiation in a medium, with applications spanning atmospheric science, astrophysics, remote sensing, and biomedical optics. Despite its importance, an exact analytical solution to the RTE has remained elusive, necessitating the use of numerical approximations such as Monte Carlo, discrete ordinate, and spherical harmonics methods. In this paper, we present an exact solution to the one-dimensional time-dependent RTE. We delve into the moments of the photon distribution, providing a clear view of the transition to the diffusion regime. This analysis offers a deeper understanding of light propagation in the medium. Furthermore, we demonstrate that the one-dimensional RTE is equivalent to the Klein-Gordon equation with an imaginary mass term determined by the inverse reduced scattering length. Contrary to naive expectations of superluminal solutions, we find that our solution is strictly causal under appropriate boundary conditions, determined by the light transport problem. We validate the found solution using Monte Carlo simulations and benchmark the performance of the latter. Our analysis reveals that even for highly forward scattering, dozens of random light scatterings are required for an accurate estimate, underscoring the complexity of the problem. Moreover, we propose a method for faster convergence by adjusting the parameters of Monte Carlo sampling. We show that a Monte Carlo method sampling photon scatterings with input parameters (μs,g), where μs is the inverse scattering length and g is the scattering anisotropy parameter, is equivalent to that with (μs(1−g)/2,−1). This equivalence leads to a significantly faster convergence to the exact solution, offering a substantial improvement of the Monte Carlo method for the one-dimensional RTE. Our findings not only contribute to the theoretical understanding of the RTE but also have potential implications for improving the numerical methods used to approximate it.

A reverse Ozawa–Rogers estimate

We provide a reverse bilinear estimate for the one-dimensional Klein–Gordon equation that complements a result of Ozawa and Rogers. The proof relies on the cosine formula for frequency vectors adapted to the Klein–Gordon equation.

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

In this paper, we propose an effective numerical method using two-dimensional Shifted fractional-order Gegenbauer Multi-wavelets to find the approximate solutions of the time-fractional distributed order non-linear partial differential equations. The method is applied to numerically solve the fractional distributed order non-linear Klein–Gordon equation. We derive an exact formula for the Riemann-Liouville fractional integral operator for the Shifted fractional Gegenbauer Multi-wavelets. Applying function approximations obtained by this method turns the considered equation into a system of algebraic equations. Error estimation and convergence analysis of the method are also studied. Some numerical examples are included to show and check the effectiveness of the proposed method.

Scattering for the Mass-Critical Nonlinear Klein–Gordon Equations in Three and Higher Dimensions

In this paper we consider the mass-critical nonlinear Klein–Gordon equations in three and higher dimensions. We prove the dichotomy between scattering and blow-up below the ground state energy in the focusing case, and the energy scattering in the defocusing case. We use the concentration-compactness/rigidity method developed by C. E. Kenig and F. Merle. The main novelty from the work of R. Killip, B. Stovall, and M. Visan (Trans. Amer. Math. Soc. 364, 2012) is to approximate the large scale (low-frequency) profile by the solution of the mass-critical nonlinear Schrödinger equation when the nonlinearity is not algebraic.

Covariant formalism and Noether symmetries on scalar-field cosmologies

In this work, we investigate a cosmological model in the context of the covariant formalism proposed by Canuto et al. where the notion of atomic and gravitational units is invoked. We consider a minimally coupled scalar field as gravitational source, including a self-interacting potential. It is possible to determine the symmetries associated to the correspondent field theory using the Noether procedure, through the Lie derivative construction. This permits to determine the scale function [Formula: see text], present in the covariant formalism, that maps the physics between descriptions in atomic and gravitational units. In gravitational units it is verified that the Klein–Gordon field behaves as a dark constituent. On the other hand, in atomic units what emerges is an initially accelerated universe, with a later incursion into a decelerated one, mirroring the usual matter/radiation age.

Quantum description of free electron laser radiation and nonlinear amplitude equations

A relativistic quantum mechanical model to describe the quantum free electron laser dynamics has been developed. Neglecting the spin of electrons in the impacting beam, this model is based on the Klein–Gordon equation coupled to the Poisson equation for the space-charge potential and the wave equation for the transverse components of the radiation field. Furthermore, a system of coupled nonlinear envelope equations for the slowly varying amplitudes of the electron beam distribution and the radiation field has been derived. The fundamental system of basic equations has been cast into a suitable hydrodynamic formulation. In the framework of the hydrodynamic representation, a new dispersion relation has been derived and analyzed in both the quantum and the quasi-classical regimes, where the space-charge oscillations of the electron beam are taken into account.

L$^{\infty}$ Decay Estimate for the Klein-Gordon Equation in the Anti-de Sitter Model of the Universe

We consider the inital value problem for the Klein-Gordon equation with non-zero initial data in anti-de Sitter spacetime. $L^{\infty}$ decay estimate is derived for the solutions to the linear Klein-Gordon equations in the anti-de Sitter spacetime without source term.

Nonlinear stability of self-gravitating massive fields. A wave-Klein–Gordon model

In recent years, significant progress has been made in understanding the global evolution of self-gravitating massive matter in the small-perturbative regime near Minkowski spacetime. To investigate the interaction between a Klein–Gordon equation and Einstein’s field equations, we developed a new approach called the Euclidean-hyperboloidal foliation method. This method involves constructing a spacetime foliation that is well-suited for deriving precise decay estimates for wave and Klein–Gordon equations in curved spacetime. In this article, we provide an overview of our method and present a complete proof for a wave-Klein–Gordon model that captures some of the key challenges associated with the Einstein-matter system.

Generalised tanh-shaped hyperbolic potential: Klein–Gordon equation's bound state solution

The development of potential theory heightens the understanding of fundamental interactions in quantum systems. In this paper, the bound state solution of the modified radial Klein–Gordon equation is presented for generalised tanh-shaped hyperbolic potential from the Nikiforov–Uvarov method. The resulting energy eigenvalues and corresponding radial wave functions are expressed in terms of the Jacobi polynomials for arbitrary l states. It is also demonstrated that energy eigenvalues strongly correlate with potential parameters for quantum states. Considering particular cases, the generalised tanh-shaped hyperbolic potential and its derived energy eigenvalues exhibit good agreement with the reported findings. Furthermore, the rovibrational energies are calculated for three representative diatomic molecules, namely H2, HCl and O2. The lowest excitation energies are in perfect agreement with experimental results. Overall, the potential model is displayed to be a viable candidate for concurrently prescribing numerous quantum systems.

Unstable Ground State and Blow Up Result of Nonlocal Klein–Gordon Equations

Unstable Ground State and Blow Up Result of Nonlocal Klein–Gordon Equations

On the solution behavior of a nonlinear time-fractional Klein–Gordon equation: Theoretical study and numerical validation

This work is devoted to the behavior analysis of solutions to a time-fractional Klein–Gordon equation in a multidimensional bounded domain. To get a more flexible representation, the diffusive formalism is adopted to deal with well-posedness issues. In case of negative initial energy, we discuss some conditions related to the solution blow-up in finite time. Prior to the blow-up phenomenon occurrence, we show that the solution component u holds under some conditions an exponential growth in Lp+1 norm. The validity of the theoretical findings is illustrated through the development of a finite difference approach for the one dimension problem, where the convergence of the proposed numerical method is also addressed.

Two dimensional wave-Klein–Gordon equations with a below-critical nonlinearity

In this paper we investigate the small data global existence and pointwise decay of solutions to two systems of coupled wave-Klein–Gordon equations in two spatial dimensions. In particular, we consider critical (in the sense of time decay) semilinear nonlinearities for the wave equation and below-critical semilinear nonlinearities for the Klein–Gordon equation, a situation that has not been studied before in the context of coupled wave and Klein–Gordon equations. An interesting feature of our two systems is that the below-critical nonlinearity causes the Klein–Gordon field to lose its linear behaviour close to the light cone, even though it enjoys optimal time decay.