Coupled Klein-Gordon Equations Research Articles

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

An invariance constrained deep learning network for partial differential equation discovery

The discovery of partial differential equations (PDEs) from datasets has attracted increased attention. However, the discovery of governing equations from sparse data with high noise is still very challenging due to the difficulty of derivatives computation and the disturbance of noise. Moreover, the selection principles for the candidate library to meet physical laws need to be further studied. The invariance is one of the fundamental laws for governing equations. In this study, we propose an invariance constrained deep learning network (ICNet) for the discovery of PDEs. Considering that temporal and spatial translation invariance (Galilean invariance) is a fundamental property of physical laws, we filter the candidates that cannot meet the requirement of the Galilean transformations. Subsequently, we embedded the fixed and possible terms into the loss function of neural network, significantly countering the effect of sparse data with high noise. Then, by filtering out redundant terms without fixing learnable parameters during the training process, the governing equations discovered by the ICNet method can effectively approximate the real governing equations. We select the one-dimensional (1D) Kuramoto–Sivashinsky equation, the two-dimensional (2D) Burgers equation, the equation of two-dimensional channel flow over an obstacle, and the equation of three-dimensional (3D) intracranial aneurysm as examples to verify the superiority of the ICNet for fluid mechanics. Furthermore, we extend similar invariance methods to the discovery of wave equation (Lorentz Invariance) and verify it through single and coupled Klein–Gordon equation. The results show that the ICNet method with physical constraints exhibits excellent performance in governing equations discovery from sparse and noisy data.

Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes

We study linear scalar perturbations of slowly accelerating Kerr-Newman-anti-de Sitter black holes using the method of isomonodromic deformations. The conformally coupled Klein-Gordon equation separates into two second-order ordinary differential equations with five singularities. Nevertheless, the angular equation can be transformed into a Heun equation, for which we provide an asymptotic expansion for the angular eigenvalues in the small acceleration and rotation limit. In the radial case, we recast the boundary value problem in terms of a set of initial conditions for the isomonodromic tau function of Fuchsian systems with five regular singular points. For the sake of illustration, we compute the quasi-normal modes frequencies.

Novel waves structures for two nonlinear partial differential equations arising in the nonlinear optics via Sardar-subequation method

In this paper, our aim is to construct the novel wave structures for two non-linear evolution equations which are rising in non-linear optics, mathematical biological models, fluid dynamics, waves theory, mechanics, quantum mechanics, and many more. We employed an efficient analytical technique namely, the Sardar-subequation method to build the wave solutions for the modified Benjamin-Bona-Mahony equation and the Coupled Klein–Gordon equations. We have built the numerous type of soliton wave structures of modified Benjamin-Bona-Mahony equation and the Coupled Klein–Gordon equations via the Sardar-subequation method. Acquired results reveal the dynamics behavior of waves structures including the bright, singular, dark, and periodic singular solitons solutions. To illustrate the behavior of these solutions some selected solutions are sketched in two-, and three-dimensional graphs. On the basis of these results, our technique is suitable, up-to-date, and powerful. The obtained solutions are very efficacious and influential in non-linear optics, mathematical biology, mechanics, fluid mechanics, plasma physics, and many more. This study will assist to predict some new hypothesis and theories in the field of mathematical physics.

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations

Unconditional Superconvergence Analysis of Energy Conserving Finite Element Methods for the Nonlinear Coupled Klein-Gordon Equations

The zero mass problem for Klein-Gordon equations: quadratic null interactions

Abstract We study in $\mathbb {R}^{3+1}$ a system of nonlinearly coupled Klein-Gordon equations under the null condition, with (possibly vanishing) mass varying in the interval $[0, 1]$ . Our goal is three-fold, which extends the results in the earlier work of [5, 3]: 1) we want to establish the global well-posedness result to the system that is uniform in terms of the mass parameter (i.e., the smallness of the initial data is independent of the mass parameter); 2) we want to obtain a unified pointwise decay result for the solution to the system, in the sense that the solution decays more like a wave component (independent of the mass parameter) in a certain range of time, while the solution decays as a Klein-Gordon component with a factor depending on the mass parameter in the other part of the time range; 3) the solution to the Klein-Gordon system converges to the solution to the corresponding wave system in a certain sense when the mass parameter goes to 0. In order to achieve these goals, we will rely on both the flat and hyperboloidal foliation of the spacetime and prove a mass-independent $L^2$ –type energy estimate for the Klein-Gordon equations with possibly vanishing mass. In addition, the case of the Klein-Gordon equations with certain restricted large data is discussed.

Variational iteration method for solving coupled nonlinear system of Klein-Gordon equations

In this paper, variational iteration technique is applied to solve nonlinear coupled Klein-Gordon equations (CNLKGE). The equations arise when dealing with particle physics. The numerical solutions of CNLKGE have been compared to the exact solutions and presented graphically. The results of the proposed method are in good agreement with the exact solutions provided by [1]. The suggested algorithm is efficient and easier to employ compared to other methods like the perturbation method.

What can we learn from the conformal noninvariance of the Klein–Gordon equation?

It is well known that the Klein–Gordon equation in curved spacetime is conformally noninvariant, both with and without a mass term. We show that such a noninvariance provides nontrivial physical insights at different levels, first within the fully relativistic regime, then in the nonrelativistic regime leading to the Schrödinger equation, and then within the de Broglie–Bohm causal interpretation of quantum mechanics. The conformal noninvariance of the Klein–Gordon equation coupled to a vector potential is confronted with the conformal invariance of Maxwell’s equations in the presence of a charged current. The conformal invariance of the nonminimally coupled Klein–Gordon equation to gravity is then examined in light of the conformal invariance of Maxwell’s equations. Finally, the consequence of the noninvariance of the equation on the Aharonov–Bohm effect in curved space–time is discussed.

The studies of the linearly modified energy-preserving finite difference methods applied to solve two-dimensional nonlinear coupled wave equations

In this paper, four linearly energy-preserving finite difference methods (EP-FDMs) are designed for two-dimensional (2D) nonlinear coupled sine-Gordon equations (CSGEs) and coupled Klein-Gordon equations (CKGEs) using the invariant energy quadratization method (IEQM). The 1st EP-FDM is designed by first introducing two auxiliary functions to rewrite the original problems into the new system only including the 1st-order temporal derivatives, and then applying Crank-Nicolson (C-N) method and 2nd-order centered difference methods for the discretizations of temporal and spatial derivatives, respectively. The 2nd EP-FDM is directly devised based on the uses of 2nd-order centered difference methods to approximate 2nd-order temporal and spatial derivatives. The 1st and 2nd EP-FDMs need numerical solutions of the algebraic system with variable coefficient matrices at each time level. By modifying the 2nd EP-FDM, the 3rd EP-FDM, which is implemented by computing the system of algebraic equations with constant coefficient matrices at each time level, is developed. Finally, an energy-preserving alternating direction implicit (ADI) finite difference method (EP-ADI-FDM) is established by a combination of ADI method with the 3rd EP-FDM. By using the discrete energy method, it is shown that they are all uniquely solvable, and their solutions have a convergent rate of ${\mathscr{O}}(\varDelta t^{2}+{h^{2}_{x}}+{h^{2}_{y}})$ in H1-norm and satisfy the discrete conservative laws. Numerical results show the efficiency and accuracy of them.

Quantum field theory with the generalized uncertainty principle I: Scalar electrodynamics

Recently, the authors presented a covariant extension of the Generalized Uncertainty Principle (GUP) with a Lorentz invariant minimum length. This opens the way for constructing and exploring the observable consequences of minimum length in Relativistic Quantum Field Theories. In particular, we compute quantum gravity corrections to high energy scattering experiments, which may provide the much needed window of testing minimum length and quantum gravity theories in the laboratory. To this end, we formulate the Lagrangian of Quantum Electrodynamics for a complex scalar field from the GUP modified minimally coupled Klein–Gordon equation and write down its Feynman rules. We then calculate the Relativistic Generalized Uncertainty Principle corrections to a Quantum Electrodynamics scattering amplitude and discuss its implications.

The energy-preserving finite difference methods and their analyses for system of nonlinear wave equations in two dimensions

The coupled sine-Gordon (SG) equations and the coupled Klein-Gordon (KG) equations play an important role in scientific fields, such as nonlinear optics, solid state physics, quantum mechanics. As their energies are conservative, it is of importance to develop energy preserving finite difference method (EP-FDM) for these systems of nonlinear wave equations. However, the energy preserving finite difference methods (EP-FDMs) for one-dimensional single sine-Gordon equation and one-dimensional single Klein-Gordon equation, can not directly be generalized to solve the systems of coupled SG or coupled KG equations, and the theoretical analysis technique used for 1D single SG equation or for 1D single KG equation is not suitable for the analysis of high dimensional problems. In this paper, we develop and analyze two kinds of energy preserving FDMs for the systems of coupled SG equations or coupled KG equations in two dimensions. One proposed scheme is a two-level scheme and the other is a three-level scheme. We prove the schemes to satisfy the energy conservations in the discrete forms. By using the fixed point theorem, it is shown that they are solvable. Also, it is further proved that they have the second order convergence rate in both time and space steps. Numerical tests show the performance of the methods and confirm the theoretical findings.

Post-Newtonian corrections to Schrödinger equations in gravitational fields

In this paper we extend the WKB-like ‘non-relativistic’ expansion of the minimally coupled Klein–Gordon equation after (Kiefer and Singh 1991 Phys. Rev. D 44 1067–76; Lämmerzahl 1995 Phys. Lett. A 203 12–7; Giulini and Großardt 2012 Class. Quantum Grav. 29 215010) to arbitrary order in c−1, leading to Schrödinger equations describing a quantum particle in a general gravitational field, and compare the results with canonical quantisation of a free particle in curved spacetime, following (Wajima et al 1997 Phys. Rev. D 55 1964–70). Furthermore, using a more operator-algebraic approach, the Klein–Gordon equation and the canonical quantisation method are shown to lead to the same results for some special terms in the Hamiltonian describing a single particle in a general stationary spacetime, without any ‘non-relativistic’ expansion.

The B-spline Collocation Approach for Coupled Klein-Gordon Equation

This research presents a new approach for obtaining numerical solutions of Coupled Klein Gordon equation using the collocation method which based on cubic B-spline base functions and finite element approximation. The main advantage of the collocation method is that the structure of the method is simple and the computational cost is low. It also provides an easy and simpler procedure for solving various problems involving differential equations that model real-world phenomena . In the current research, the temporal and spatial partial derivatives are discretized with using approximate solution which is formed linear combination of B-spline basis and time dependent parameters. With the help of the idea that approximate solution satisfy the PDE at collocation points, a new numerical scheme is constructed. The newly obtained numerical scheme tested on a model problem. Numerical results are compared with exact solution with the aid of the error norms L 2 and L s presented via tables . Additionally, graphical simulations of numerical solutions are presented.

Stability Analysis of Solitary Wave Solutions for Coupled and (2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations (PDEs) play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the (2+1)-dimensional cubic Klein-Gordon (K-G) equation. The Klein-Gordon equations are relativistic version of Schrödinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which several solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions of PDEs arise in mathematical physics.

Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media

This study acquires the wave solutions of the two well-known nonlinear models, namely; the modified Benjamin–Bona–Mahony and the coupled Klein–Gordon equations. The modified Benjamin–Bona–Mahony is a nonlinear model that describes the long surface gravity waves of small amplitude and the coupled Klein–Gordon equation describes the quantized version of the relativistic energy–momentum relation. We successfully acquire some new solutions to these models such as kink-type and soliton solutions in complex hyperbolic functions form. We plot the 3D and 2D surface of the all the obtained solutions in this study. The mathematical approach used in this study is the sine-Gordon expansion method.

Bäcklund transformation and soliton–cnoidal wave interaction solution for the coupled Klein–Gordon equations

In this paper, Backlund transformation and interaction solutions of the generalized coupled Klein–Gordon (cKG) equations with the variable coefficients are studied via the residual symmetry and the consistent Riccati expansion (CRE) method. From the truncated Painleve expansion, we drive the residual symmetry which can be transformed into the local Lie point symmetry. The multiple residual symmetries are established and nth Backlund transformation in terms of determinant is presented. The cKG equations are proved to be CRE solvable by using the CRE method. More importantly, we construct exact interaction solutions between soliton and cnoidal periodic wave. It is shown that when the modulus of the Jacobi elliptic function tends to one, the interaction solutions degenerate to the special resonant soliton solutions. Moreover, these reduced resonant soliton solutions exhibit the interesting properties which differ from the usual resonant soliton solutions.

Pseudorelativistic effects on solitons in quantum semiconductor plasma

A theory for nonlinear excitations in quantum plasmas is presented for narrow-gap semiconductors by considering the combined effects of quantum and pseudorelativity. The system is governed by a coupled Klein-Gordon equation for the collective wave functions of the conduction electrons and Poisson's equation for the electrostatic potential. This gives a closed system, including the effects of charge separation, quantum tunneling, and pseudorelativity. By choosing the typical parameters of semiconductor InSb, the quasistationary soliton solution, which is a multipeaked dark soliton, is obtained numerically and shows depleted electron densities correlated with a localized potential. The dynamical simulation result shows that the dark soliton is stable and has a multipeaked profile, which is consistent with the quasistationary solution. The present model and results may be useful in understanding the nonlinear properties of semiconductor plasma on an ultrafast time scale.

Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations.

In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions–hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters.PACS numbers02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg

Modified Simple Equation Method and its Applications to Nonlinear Partial Differential Equations

In this paper, the modified simple equation method is used to construct exac t periodic and soliton solutions of some nonlinear partial differential equations. Exact solutions of the nonlinear Schr ¨ odinger equation, the Hamiltonian amplitude equation, Klein-Gordon equation in 1+2 dimension, the coupled Klein-Gordon equation, the (2 + 1)-dimensional long-wave-short-wave resonance interaction equation, the modified KdV-KP equation and the modified Benjamin-Bona-M ahony equation are successfully obtained. These solutions may be important of significance for the explanation of some practical ph ysical problems.

Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies $$\omega $$ for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $$\mathcal{R }(\omega )$$ tends to the mass of the field and the imaginary part $$\mathcal{I }(\omega )$$ tends to zero, for any angular momentum quantum number $$\ell $$ . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.