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In noncommutative phase space (NCPS), we investigate the pair creation rate for the problem of scalar and spinorial particles emerging from a vacuum under the influence of uniform external electromagnetic fields using the Schwinger formal technique. It is demonstrated that under specific conditions, where the speed of light c approaches the Fermi velocity υF and ω̄→ωc∕2 (with ωc=|e|ℬmc representing the cyclotron frequency), the behavior of the Dirac oscillator problem in a uniform electromagnetic field within NCPS is akin to the monolayer graphene problem in NCPS. In addition, the production probability per unit volume per unit time is discussed, along with the limits of parameter deformations. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of global in time small data solutions of semilinear Klein–Gordon equations in space-time with a static Schwarzschild radius in the expanding universe. [ABSTRACT FROM AUTHOR]

We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the Klein–Gordon equation having Schwartz initial data. In fact, there exist infinitely many independent such solutions. The proof is a variational argument based on the long range nature of the effective potential. We give two sets of test functions which serve to verify the hypothesis of the variational argument. One set consists of cutoff versions of the hydrogen bound states and is used to prove the existence of eigenvalues near the hydrogen spectrum. [ABSTRACT FROM AUTHOR]

We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is O (ε) with 0 < ε ≪ 1 a dimensionless parameter up to the time at O (1 / ε 2) . The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discretization at O (ε 2 τ) instead of O (τ) according to classical error estimates and at O (h m + ε 2 τ) for the full-discretization up to the time T ε = T / ε 2 with T > 0 fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the nonlinear Klein–Gordon equation with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with O (ε 2) wavelength in time and O (ε - 2) wave speed, which indicates that the temporal error is independent of ε when the time step size is chosen as O (ε 2) . Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp. [ABSTRACT FROM AUTHOR]

By the Klein-Gordon potential, we call a convolution-type integral with a kernel, which is the fundamental solution of the Klein-Gordon equation and also a solution of the Cauchy problem to the same equation. An interesting question having several important applications (in general) is what boundary condition can be imposed on the Klein-Gordon potential on the boundary of a given domain so that the Klein-Gordon equation with initial conditions complemented by this “transparent” boundary condition would have a unique solution within that domain still given by the Klein-Gordon potential. It amounts to finding the trace of the Klein-Gordon potential to the boundary of the given domain. In this article, we analyze this question and construct a novel initial boundary-value problem for the Klein-Gordon equation in characteristic coordinates.

This paper mainly presents numerical solutions to an initial-boundary value problem of the time-fractional Klein-Gordon equations. We developed a numerical scheme with the help of the finite difference methods and the predictor-corrector methods to find numerical solutions of the considered problems. The proposed scheme is based on discretizing the considered problems with respect to spatial and temporal domains. Numerical results are derived for some illustrative problems, and the outputs are compared with the exact solution in the integer order case. The solution behavior and 3D graphics of the discussed problems are demonstrated using the proposed scheme. Finally, the proposed scheme, which does not require solving large systems of linear equations, can be extended and modified to handle other classes of time-fractional PDEs. [ABSTRACT FROM AUTHOR]

In this paper, a modified method is used to approximate the solution to the time-fractional n-dimensional Navier–Stokes equation. The modified method is the Variational Iteration Transform Method, which is implemented in the equation whose fractional order derivative is described in the Caputo sense. The proposed method’s findings are presented and examined using figures. It is demonstrated that the proposed method is efficient, dependable, and simple to apply to various science and engineering applications [ABSTRACT FROM AUTHOR]

The (generalized) Derivative Nonlinear Schrödinger (DNLS) equation can be derived as an envelope equation via multiple scaling perturbation analysis from dispersive wave systems. It occurs when the cubic coefficient for the associated NLS equation vanishes for the spatial wave number of the underlying slowly modulated wave packet. It is the purpose of this paper to prove that the DNLS equation makes correct predictions about the dynamics of a Klein-Gordon model with a cubic nonlinearity. The proof is based on energy estimates and normal form transformations. New difficulties occur due to a total resonance and due to a second order resonance. [ABSTRACT FROM AUTHOR]

The impact of deformation space on the physical characteristics of diverse physics systems has been thoroughly investigated in research papers. In this work, we study the deformed Klein–Gordon equation (DKGE) in the three-dimensional relativistic non-commutative quantum space (3D-RNCQS) regime by using the improved Hua plus modified Eckart potential (IHPMEP) model. For this consideration, the DKGE in the 3D-RNCQS regime is solved using the standard perturbation theory and the well-known Bopp’s shifts method with the Greene–Aldrich approximation to the centrifugal barrier. The new relativistic energy equation and eigenfunction for the IHPMEP in the presence of deformation space-space for the heterogeneous (CO, HF, and NO) and homogeneous (N2, H2, and Li2) diatomic molecules are obtained to be sensitive to the atomic quantum numbers (j,l,s, and m), the mixed potential depths (V0,V1,V2, and V3), the inverse of the screening parameter α, and non-commutativity parameters (Φ, χ, and ζ). Analysis is performed on the non-relativistic limit of new energy spectra. By appropriately adjusting the combined potential parameters, we analyze the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IHPMEP in 3D-NCQS symmetries and obtain the new modified Eckart potential, the modified Hua potential, the modified Morse potential, and the modified Pöschl–Teller potential. Within the framework of the 3D-NRNCQS regime, the homogeneous and heterogeneous composite systems under IHPMEP models are examined. A thorough investigation is carried out into the impact of space-space deformation on the thermal parameters of the IHPMEP, including the partition function, mean energy, free energy, specific heat, and entropy. This work is of a fundamental absorbability nature and pedagogical interest in atomic and molecular physics. [ABSTRACT FROM AUTHOR]

In this study, we explore the quantum effects in the Schwarzschild spacetime for massive and massless scalar particles in the presence of an external gravitational field. The methodology involves the analytical solution of the Klein–Gordon equation for the scalar particles in the near-horizon spacetime limit, using Rindler approximation. The results show that the quantum effects differ significantly for the massive and massless cases, they possess similar characteristics near the event horizon. Bosons without mass that are distant from the event horizon undergo a reduced gravitational impact and possess a symmetric wave function that remains unchanged when they approach the event horizon. However, the symmetry is lost, indicating that the particle strives for a lower energy state and becomes erratic as it nears the event horizon. This aligns with the anticipated behavior of particles within the Schwarzschild horizon. The particles have an energy spectrum even for tremendous values of the Schwarzschild radius, which entails the validity of the Rindler approximation in the near horizon geometry. We have observed that the quantum effects of massive and massless particles differ based on the mass. Specifically, massive and massless scalars behave differently due to their geometric differences, which results in different eigenstates. Although they both tend to seek lower energy levels when they approach the event horizon, their behavior is not the same. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear Klein–Gordon equation's exact solutions are obtained through the application of an appropriate transformation based on He's semi-inverse approach. This equation considered a generalization of other famous models in applied science, such as Phi-4 equation, Duffing equation, Fisher–Kolmogorov model through population dynamics and Hodgkin–Huxley equation that characterizes the propagation of electrical signals via nervous system. The suggested approach is simple, robust, and efficient, and its application in other partial differential equations in applied science seems promising. Numerical solution of the nonlinear Klein–Gordon equation is presented using finite difference method. The method's accuracy is demonstrated by contrasting it with the exact solution that we obtained. The Von Neumann stability technique is applied to obtain the stability condition and a convergent test is presented Some 2D and 3D graphs matching to chosen solutions are simulated by taking into account appropriate values for the parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method's accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the results for the ground state of a Klein–Gordon oscillator under gravitational effects in a topologically charged Ellis–Bronnikov wormhole derived recently. We show that the authors failed to truncate a power-series and, consequently, could not obtain the desired solution to a differential equation. For this reason, their analytical expression for the ground-state eigenvalue is incorrect, a fact that affects all the conclusions drawn by the authors for the physical problem. [ABSTRACT FROM AUTHOR]

We consider wave equations with a critically singular potential ξ ⋅ σ − 2 diverging as an inverse square at a hypersurface σ = 0. Our aim is to construct counterexamples to unique continuation from σ = 0 for this equation, provided there exists a family of null geodesics trapped near σ = 0. This extends the classical geometric optics construction [3] of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of σ = 0 , by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill [12,18,19,24] and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence. [ABSTRACT FROM AUTHOR]

We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity. [ABSTRACT FROM AUTHOR]

We are interested in the global dynamics of a massive scalar field evolving under its own gravitational field and, in this paper, we study spherically symmetric solutions to Einstein's field equations coupled with a Klein-Gordon equation with quadratic potential. For the initial value problem we establish a global existence theory when initial data are prescribed on a future light cone with vertex at the center of symmetry. A suitably generalized solution in Bondi coordinates is sought which has low regularity and possibly large but finite Bondi mass. A similar result was established first by Christodoulou for massless fields. In order to deal with massive fields, we must overcome several challenges and significantly modify Christodoulou's original method. First of all, we formulate the Einstein-Klein-Gordon system in spherical symmetry as a non-local and nonlinear hyperbolic equation and, by carefully investigating the global dynamical behavior of the massive field, we establish various estimates concerning the Einstein operator, the Hawking mass, and the Bondi mass, including novel positivity and monotonicity properties. Importantly, in addition to a regularization at the center of symmetry we find it necessary to also introduce a regularization at null infinity. We also establish new energy and decay estimates for, both, regularized and generalized solutions. [ABSTRACT FROM AUTHOR]

In this paper, the time-independent Klein-Gordon equation in R 3 is treated with a decomposition of the operator Δ − γ 2 I by the Clifford algebra C l (V 3 , 3) . Some properties of integral operators associated the kind of equations and some Riemann-Hilbert boundary value problems for perturbed Dirac operators are investigated. [ABSTRACT FROM AUTHOR]

The current study is devoted to exploring the geometrical configuration of a thin‐shell in the background of symmetric teleparallel gravity. For this purpose, the well‐known cut‐and‐paste approach by matching the inner flat and outer newly calculated class of approximated black hole (BH) solutions in symmetric teleparallel gravity are considered, i.e., uncharged, charged, and anti‐de‐Sitter BHs. The dynamical analysis of thin‐shell configuration by adopting the massive and massless scalar field via Klein‐Gordon's equation of motion is discussed. The effective potential and proper time derivative of shell radius for both massive and massless scalar shells are used to discuss the collapse, expansion, and oscillatory behavior. The stable configuration of thin‐shell is observed through the linearized radial perturbation approach with a phantomlike equation of state, i.e., quintessence, dark energy, and phantom energy. It is noted that stable/unstable behavior of thin‐shell is found after the expected position of the event horizon of an exterior manifold. It is concluded that the stability of a thin‐shell is greater for f(Q)$f(\mathbb {Q})$ BH with cosmological constant as compared to the uncharged and charged f(Q)$f(\mathbb {Q})$ BHs. [ABSTRACT FROM AUTHOR]

The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave. [ABSTRACT FROM AUTHOR]

In this paper, we examine the role played by topology, and some specific boundary conditions as well, on the physics of a higher-dimensional black hole. We analyze the line element of a five-dimensional non-extremal Reissner–Nordström black hole to obtain a new family of subspaces that are types of strong retractions and deformations, and then we extend these results to higher dimensions in order to deduce the relationship between various types of transformations. We also study the scalar field perturbations in the background under consideration and obtain an analytical expression for the quasibound state frequencies by using the Vieira–Bezerra–Kokkotas approach, which uses the polynomial conditions of the general Heun functions, and then we discuss the stability of the system and present the radial eigenfunctions. Our main goal is to discuss the physical meaning of these mathematical applications in such higher-dimensional effective metric. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [ABSTRACT FROM AUTHOR]

We introduce and study a general notion of spatial localization on spacelike smooth Cauchy surfaces of quantum systems in Minkowski spacetime. The notion is constructed in terms of a coherent family of normalized POVMs, one for each said Cauchy surface. We prove that a family of POVMs of this type automatically satisfies a causality condition which generalizes Castrigiano’s one and implies it when restricting to flat spacelike Cauchy surfaces. As a consequence, no conflict with Hegerfeldt’s theorem arises. We furthermore prove that such families of POVMs do exist for massive Klein–Gordon particles, since some of them are extensions of already known spatial localization observables. These are constructed out of positive definite kernels or are defined in terms of the stress–energy tensor operator. Some further features of these structures are investigated, in particular the relation with the triple of Newton–Wigner selfadjoint operators and a modified form of Heisenberg inequality in the rest 3-spaces of Minkowski reference frames. [ABSTRACT FROM AUTHOR]

We explore superoscillations within the context of classical and quantum field theories, presenting novel solutions to Klein–Gordon's, Dirac's, Maxwell's and Einstein's equations. In particular, we illustrate a procedure of second quantization of fields and how to construct a Fock space which encompasses Superoscillating states. Furthermore, we extend the application of superoscillations to quantum tunnelings, scatterings and mixings of particles, squeezed states and potential advancements in laser interferometry, which could open new avenues for experimental tests of quantum gravity effects. By delving into the relationship among superoscillations and phenomena such as Hawking radiation, the black hole (BH) information and the Firewall paradox, we propose an alternative mechanism for information transfer across the BH event horizon. [ABSTRACT FROM AUTHOR]

In this study, we investigate the fundamental properties of (3 + 1)-D Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both 2D and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields. [ABSTRACT FROM AUTHOR]

The article presents the development of the weak Galerkin finite element method (WG-FEM) for semilinear hyperbolic problems. Semidiscrete error estimate in L 2 -norm as well as H 1 -norm have been executed for the weak Galerkin space (P k (K) , P k (∂ K) , [ P k - 1 (K) ] 2) , where k ≥ 1 is an integer. For a fully discrete scheme, we employ the Newmark scheme for temporal discretization. Finally, a few numerical results are provided to validate theoretical results. [ABSTRACT FROM AUTHOR]

Equations for two and three scalar fields, which allow localized solutions with beatings of coupled oscillators, have been presented. The amplitude of oscillations of a localized perturbation for one field decreases periodically gradually to a minimum and the amplitudes of the other scalar fields increase to a maximum; then, the reverse process occurs. In this case, all fields except for one are initially either in the state of a background solution with a small amplitude or equal to zero. Such solutions can be interesting due to analogy with neutrino oscillations. Equations of motion, where the perturbation of one of the components is obligatorily accompanied by the perturbation of the second and third components even in zeroth background state, have also been presented. It has been shown that these equations satisfy the energy conservation law. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximate analytical solutions for bound states of the l-wave Klein-Gordon equation with a position-dependent mass subjected to a hyperbolic cotangent vector potential by using the concept of the supersymmetric quantum mechanics approach. Within the framework of the proper approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the corresponding normalized wavefunctions written down in terms of the Jacobi polynomials. Furthermore, it is found that the solutions in the case of constant mass for nonzero l-values are identical to the ones obtained in the literature. Among these cases, Hulthén potential, Coulomb potential, and nonrelativistic limit are discussed [ABSTRACT FROM AUTHOR]

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown. [ABSTRACT FROM AUTHOR]

We use the semiclassical approach to solve the Klein–Gordon and Dirac equations in the presence of a time-varying electric field. Our objective is to calculate the density of particle creation in a cosmological anisotropic Bianchi- I space–time. We demonstrate that when the electric interaction is proportional to the Ricci scalar of curved space–time, the distribution of particles subjected to the electric field transforms into a thermal state. [ABSTRACT FROM AUTHOR]

We consider the solutions of the Klein–Gordon equation in the 2 + 1 -dimensional space-time which gravity is analyzed, i.e., the manifold ℝ t × ℝ + × ℝ / 2 ⁢ π ⁢ α ⁢ 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. [ABSTRACT FROM AUTHOR]

This paper examines the complex characteristics of the modified Benjamin–Bona–Mahony equation ( m BBM ) and the Klein–Gordon ( K G ) equation in the field of mathematical physics. The m BBM equation is a basic model used to describe surface water waves, especially in shallow water situations. It provides valuable information on wave propagation, stability, and the formation of solitons. The applications of this instrument are wide-ranging, including fields such as oceanography, where it plays a crucial role in comprehending wave behavior. On the other hand, the K G equation is of utmost importance in quantum field theory since it sheds light on the dynamics and interactions of scalar fields such as mesons. Within the field of particle physics, it offers substantial insights into basic concepts, acting as a fundamental basis for comprehending particle behavior. The primary goal of our work is to develop strong analytical techniques for solving these problems. In order to tackle these issues, we use two novel methodologies: the extended simple equation technique and the generalized Kudryashov method. Furthermore, we validate our results by using the extended cubic–B–spline approach for numerical computations. The work effectively solves these intricate equations, resulting in encouraging results. The presented approaches demonstrate their effectiveness, providing significant advances to mathematical physics. This work has inherent worth by presenting innovative analytical methods and perspectives, particularly designed to solve complex nonlinear equations such as the m BBM and K G equations. The finding has significant ramifications that extend across several scientific fields, offering novel approaches to tackle complex issues in mathematical physics. To summarize, our paper introduces innovative analytical techniques designed to solve nonlinear equations in the field of mathematical physics, with a particular emphasis on the m BBM and K G equations. [ABSTRACT FROM AUTHOR]

In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions.

In this study, we present the analytical solutions of Klein-Gordon equation with modified Mobius square plus Kratzer potential. The energy spectrum and wave functions are obtained via the parametric Nikiforov-Uvarov (NU) method by assuming equal scalar and vector potential. The non relativistic limit is obtained and numerical results are presented. In addition, the energy eigenvalues are obtained for special cases of this potential. Our results show that energy decreases with the screening parameter.

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. [ABSTRACT FROM AUTHOR]

A technique for obtaining an approximate breather solution of the Klein–Gordon equation is presented. A breather solution of the equation describing the propagation of nonlinear waves in a graphene-based superlattice is investigated. [ABSTRACT FROM AUTHOR]

In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical approach for solving the nonlinear fractional Klein–Gordon equation (FKGE), a model of significant importance in simulating nonlinear waves in quantum mechanics. Our method combines the Bernoulli wavelet collocation scheme with a functional integration matrix to obtain approximate solutions for the proposed model. Initially, we transform the main problem into a system of algebraic equations, which we solve using the Newton–Raphson method to extract the unknown coefficients and achieve the desired approximate solution. To theoretically validate our method, we conduct a comprehensive convergence analysis, demonstrating its uniform convergence. We perform numerical experiments on various examples with different parameters, presenting the results through tables and figures. Our findings indicate that employing more terms in the utilized techniques enhances accuracy. Furthermore, we compare our approach with existing methods from the literature, showcasing its performance in terms of computational cost, convergence rate, and solution accuracy. These examples illustrate how our techniques yield better approximate solutions for the nonlinear model at a low computational cost, as evidenced by the calculated CPU time and absolute error. Additionally, our method consistently provides better accuracy than other methods from the literature, suggesting its potential for solving more complex problems in physics and other scientific disciplines. [ABSTRACT FROM AUTHOR]

In this research article, we present an optimization algorithm aimed at finding the optimal solution for nonlinear 2-dimensional fractional optimal control problems that arise from nonlinear fractional dynamical systems governed by Caputo derivatives under Goursat–Darboux conditions. The system dynamics are described by equations such as the Klein–Gordon, convection–diffusion, and diffusion–wave equations. Our algorithm utilizes a novel class of basis functions called generalized Laguerre polynomials (GLPs), which are an extension of the traditional Laguerre polynomials. To begin, we introduce the GLPs and their properties, and we develop several new operational matrices specifically tailored for these basis functions. Next, we expand the state and control functions using the GLPs, with the coefficients and control parameters remaining unknown. This expansion allows us to transform the original problem into an algebraic system of equations. To facilitate this transformation, we employ operational matrices of Caputo derivatives, the rule of 2D Gauss–Legendre quadrature, and the method of Lagrange multipliers. [ABSTRACT FROM AUTHOR]

We consider the boundedness of the Fourier multipliers σt(ξ)=sin(tϕ(h(ξ)))ϕ(h(ξ))$\sigma _{t}(\xi) = \frac{\sin (t\phi (h(\xi)))}{\phi (h(\xi))}$ on modulation spaces, where h$h$ defined on Rd${\mathbb {R}}^{d}$ is a C∞(Rd∖0)$C^{\infty }({\mathbb {R}}^{d}\setminus \left\lbrace 0\right\rbrace)$ positive homogeneous function with degree λ>0$\lambda >0$ and ϕ:R+→R+$\phi : {\mathbb {R}}^{+} \rightarrow {\mathbb {R}}^{+}$ is a smooth function satisfying some decay conditions. We prove the boundedness of this kind of Fourier multipliers and obtain its asymptotic estimates as t$t$ goes to infinity. We remote the restriction λ>1/2$\lambda >1/2$ in Deng, Ding, and Sun's result in [Nonlinear Anal. 85 (2013), 78–92], and we consider the more general form of this multiplier. As applications, we obtain the grow‐up rate of the solutions for the Cauchy problems for the generalized wave and Klein–Gordon equations, and we obtain the local well‐posedness of nonlinear wave and Klein–Gordon equations in modulation spaces. [ABSTRACT FROM AUTHOR]

In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions. [ABSTRACT FROM AUTHOR]

In light of the significance of non-commutative quaternionic algebra in modern physics, this study proposes the existence of the Klein paradox in the quaternionic (3+1)-dimensional space-time structure. By introducing quaternionic wave function, we rewrite the Klein–Gordon equation in extended quaternionic form that includes scalar and the vector fields. Because quaternionic fields are non-commutative, the quaternionic Klein–Gordon equation provides three separate sets of the probability density and probability current density of relativistic particles. We explore the significance of these probability densities by determining the reflection and transmission coefficients for the quaternionic relativistic step potential. Furthermore, we also discuss the quaternionic version of the oscillatory, tunnelling, and Klein zones for the quaternionic step potential. The Klein paradox occurs only in the Klein zone when the impacted particle's kinetic energy is less than 0 − m 0 c 2 . Therefore, it is emphasized that for the quaternionic Klein paradox, the quaternionic reflection coefficient becomes exclusively higher than value one while the quaternionic transmission coefficient becomes lower than zero. [ABSTRACT FROM AUTHOR]

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker's Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts. [ABSTRACT FROM AUTHOR]

In this research, we utilized the auxiliary equation technique to invent the soliton results of the nonlinear third-order Klein–Fock-Gordon (KFG) equation. With the capability of explicit approach soliton results has been secured on the base of computational software. As a result, various solitary wave solutions are produced and shown in hyperbolic functions. The procedure delivers more general and wide-ranging soliton solutions separated with parameters and, for different values of constant parameters, reveals different shapes of solitons form, like dark soliton, bright soliton, combined bright and dark solitons and other types of solitons. It has been established that the method used to examine the nonlinear model is reliable, companionable, and reasonably good. Also, the method shows that it can be utilized on other types of nonlinear equations in a thorough way. [ABSTRACT FROM AUTHOR]

In this letter, we will focus on the Klein–Gordon equation with rotating axially symmetric black hole solution of the Einstein–Bumblebee theory, so called the Kerr–Bumblebee black hole, as its 3 + 1 background space-time. We start with constructing the covariant Klein–Gordon equation component by component and with the help of the ansatz of separation of variables, we successfully separate the polar part and found the exact solution in terms of Spheroidal Harmonics while the radial exact solution is discovered in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that is resulted in a complex-valued energy levels expression for a massive scalar field, where the real part is the scalar particle's energy while the imaginary part represents the quasibound stats's decay. And for a massless scalar, a pure imaginary energy levels is obtained. The quasibound states, thus, describe the decaying nature of the relativistic scalar field bound in the curved Kerr–Bumblebee space-time. We also investigate the Hawking radiation of the Kerr–Bumblebee black hole's apparent horizon via the Damour–Ruffini method by making use the obtained exact scalar's wave functions. The radiation distribution function and the Hawking temperature are successfully obtained. [ABSTRACT FROM AUTHOR]