Dimensional Klein-Gordon Equation Research Articles

[formula omitted]-dimensional Klein–Gordon equation in presence of deformed generalized Deng–Fan with Yukawa potential class: Approximate bound state solutions in relativistic and non-relativistic regimes

In this paper, we examine the bound state solutions of the D-dimensional Klein–Gordon equation for a deformed Deng–Fan potential in generalized form along with Yukawa potential class. The supersymmetric quantum mechanics method and Nikiforov–Uvarov method are employed, utilizing a proper correspondence to the centrifugal potential term. Remarkably, in both methods, we obtain the same analytical expressions for normalized wave functions and energy eigenvalues, expressed in closed form by hypergeometric functions and Jacobi polynomials, for all l and n quantum states. Additionally, we explore the thermodynamic quantities (internal energy, partition function, specific heat, entropy and free energy) in case of both relativistic and non-relativistic regimes for the aforementioned potential. Finally, we demonstrate energy variation against various potential parameters for a few diatomic molecules pictorially.

New Exact Traveling Wave Solutions of the Non-linear (2+1)-dimensional Klein-gordon Equation

In this article, we discuss the nonlinear (2+1)-dimensional Klein-Gordon equation with an additional term. The functional variable method is used to construct exact solutions of the quadratic and cubic nonlinear (2+1)-dimensional Klein-Gordon equation. The exact solutions of these equations including soliton and periodic wave solutions are obtained. The advantage of the used method beyond other existing methods is that it provides more new exact solutions. Some selected solutions of the equations are presented graphically by Matlab program. This method is efficient and it can be successfully used to obtain another nonlinear wave equations in mathematical physics and engineering.

EXPLICIT OPTICAL DROMIONS WITH KERR LAW HAVING FRACTIONAL TEMPORAL EVOLUTION

In this work, we derived the (2+1)-dimensional Schrödinger equation from the (2+1)-dimensional Klein–Gordon equation. We also obtained the fractional order form of this equation at the same time so as to discover the connection between them. For the (2+1)-dimensional Klein–Gordon equation, symmetries and conservation laws are pres ented. For different gauge constraint, from the perspective of conservation laws, the corresponding symmetries are obtained. After that, based on the fractional complex transform, soliton solutions of the time fractional (2+1)-dimensional Schrödinger equation are displayed. Some figures are showed behaviors of soliton solutions. It is important to discover the relationships between these equations and to obtain their explicit solutions. These solutions will perhaps provide a theoretical basis for the explanation of complex nonlinear phenomena. From the results of this paper, it is clear that the Lie symmetry method is a particularly important tool for dealing with differential equations.

Blow-up Dynamics of Higher Dimensional Klein-Gordon Equation with Nonminimal Coupling in Subcritical Case

The aim of this present work is to study the blow-up dynamics and lifespanestimate for solution to higher dimensional Klein-Gordon Equation in subcritical case, in which1 < p < psc. We construct the equation of motion from the Lagrangian of Klein-Gordon withnon-minimal coupling, where the coupling interaction of the scalar field is proportional to thescalar curvature of the spacetime. The equation of motion has the form like nonlinear dampedwave equation with mass. The novelty of this work is the time dependent of nonlinear term. Weuse test function method to proof the lifespan estimate.

No-short scalar hair theorem for spinning acoustic black holes in a photon-fluid model

It has recently been revealed that spinning black holes of the photon-fluid model can support acoustic `clouds', stationary density fluctuations whose spatially regular radial eigenfunctions are determined by the $(2+1)$-dimensional Klein-Gordon equation of an effective massive scalar field. Motivated by this intriguing observation, we use {\it analytical} techniques in order to prove a no-short hair theorem for the composed acoustic-black-hole-scalar-clouds configurations. In particular, it is proved that the effective lengths of the stationary bound-state co-rotating acoustic scalar clouds are bounded from below by the series of inequalities $r_{\text{hair}}>{{1+\sqrt{5}}\over{2}}\cdot r_{\text{H}}>r_{\text{null}}$, where $r_{\text{H}}$ and $r_{\text{null}}$ are respectively the horizon radius of the supporting black hole and the radius of the co-rotating null circular geodesic that characterizes the acoustic spinning black-hole spacetime.

Generalized symmetries and conservation laws of (1 + 1)-dimensional Klein–Gordon equation

Using advantages of nonstandard computational techniques based on the light-cone variables, we explicitly find the algebra of generalized symmetries of the (1 + 1)-dimensional Klein–Gordon equation. This allows us to describe this algebra in terms of the universal enveloping algebra of the essential Lie invariance algebra of the Klein–Gordon equation. Then, we single out variational symmetries of the corresponding Lagrangian and compute the space of local conservation laws of this equation, which turns out to be generated, up to the action of generalized symmetries, by a single first-order conservation law. Moreover, for every conservation law, we find a conserved current of minimal order contained in this conservation law.

The RBF partition of unity method for solving the Klein-Gordon equation

In this paper, a localized radial basis function (RBF) method is applied to obtain a global approximation of the solution of two dimensional Klein-Gordon equation on a given bounded domain. We use the RBF partition of unity (RBF-PU) method which is based on partitioning the original domain to several patches and using the RBF approximation on each local domain. Low computational cost and well conditioned final linear system are the main advantages of this method comparing with the original (global) RBF techniques. Numerical experiments show that the given problem could be solved successfully with a reasonable accuracy.

Generalized symmetries, conservation laws and Hamiltonian structures of an isothermal no-slip drift flux model

We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein–Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation laws. We also find broad families of local recursion operators and a nonlocal recursion operator, and construct an infinite family of Hamiltonian structures involving an arbitrary function of a single argument. For each of the constructed Hamiltonian operators, we obtain the associated algebra of Hamiltonian symmetries.

Photon velocity, power spectrum in Unruh effect with modified dispersion relation

In this paper we propose a new form of generalized uncertainty principle which involves both a linear and a quadratic term in the momentum. From this we have obtained the corresponding modified dispersion relation which is compared with the corresponding relation in rainbow gravity. The new form of the generalized uncertainty principle reduces to the known forms in appropriate limits. We then calculate the modified velocity of photons and we find that it is energy-dependent, allowing therefore for a superluminal propagation. We then derive the ()-dimensional Klein-Gordon equation taking into account the effects of the modified dispersion relation. The positive frequency mode solution of this equation is then used to calculate the power spectrum arising due to the Unruh effect. The result shows that the power spectrum depends on the energy of the particle owing its origin to the presence of the generalized uncertainty principle. Our results capture the effects of both the simplest form as well as the linear form of the generalized uncertainty principle and also points out an error in the result of the power spectrum up to first order in the generalized uncertainty principle parameter existing in the literature.

Perturbation solution for a solitary wave of the nonlinear higher dimensional disturbed Klein-Gordon equation

Perturbation solution for a solitary wave of the nonlinear higher dimensional disturbed Klein-Gordon equation

Exact wave solutions to the (2+1)-dimensional Klein-Gordon equation with special types of nonlinearity

Exact wave solutions to the (2+1)-dimensional Klein-Gordon equation with special types of nonlinearity

From the nonrelativistic Morse potential to a unified treatment of a large class of bound‐state solutions of a modified D ‐dimensional Klein–Gordon equation

From the nonrelativistic Morse potential to a unified treatment of a large class of bound‐state solutions of a modified <i>D</i> ‐dimensional Klein–Gordon equation

A note on the Gaussons of some new logarithmic evolution equations

In this paper, we study the Gaussian solitary waves for some nonlinear evolution equations with logarithmic nonlinearities. These studied logarithmic evolution equations are the generalized logarithmic BBM equations, the logarithmic (2+1)-dimensional KP-like equations, the logarithmic (3+1)-dimensional KP-like equations, the generalized logarithmic (2+1)-dimensional Klein-Gordon equations and the generalized logarithmic (3+1)-dimensional Klein-Gordon equations. We not only prove that they possess Gaussons: solitary wave solutions of Gaussian shape but also derive the relationships among the parameters.

New solutions of the [formula omitted]-dimensional Klein–Gordon equation via mapping onto the nonrelativistic one-dimensional Morse potential

New exact analytical bound-state solutions of the D-dimensional Klein–Gordon equation for a large set of couplings and potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of irrational equations at the worst. Several analytical results found in the literature, including the so-called Klein–Gordon oscillator, are obtained as particular cases of this unified approach.

Direct meshless local Petrov–Galerkin method for the two-dimensional Klein–Gordon equation

In this paper we apply the direct meshless local Petrov–Galerkin (DMLPG) method to solve the two dimensional Klein–Gordon equations in both strong and weak forms. Low computational cost is the main property of this method compared with the original MLPG technique. The reason lies behind the approach of generalized moving least squares approximation where the discretized functionals, obtained from the PDE problem, are directly approximated from nodal values. This shifts the integration over polynomials rather than the MLS shape functions, leading to an extremely faster scheme. We will see that this method can successfully solve the problem with a reasonable accuracy.

Pseudo-spectral fourier method as applied to finding localized spherical soliton solutions of (3 + 1)-dimensional Klein–Gordon equations

Nonlinear Klein–Gordon equations with fractional power and logarithmic potentials and with a variation in the φ4 potential are found for which the existence of long-lived stable spherically symmetric solutions in the form of pulsons is numerically established. Their mean oscillation amplitude and the frequency of the fast oscillation mode do not vary in the course of the numerical simulation. It is shown that the stability of these pulsons is explained by the presence of a potential well.

On the nonlinear (3 + 1)-dimensional Klein–Gordon equation allowing oscillating localized solutions

Certain nonlinear scalar Klein–Gordon equations have been specified for which the existence of long-lived (t ~ 1000) stable spherically symmetric solutions in the form of pulsons has been numerically revealed. Their average amplitude of oscillations and the frequency of the fast oscillation mode do not change during the entire time of calculation. It has been shown that the wave solutions of the Klein–Gordon equation with zero mass hold for these equations.

Relativistic solutions for diatomic molecules subject to pseudoharmonic oscillator in arbitrary dimensions

The exact solutions of the N-dimensional Klein—Gordon equation in the presence of an exactly solvable potential of V(r) = De(r/re − re/r)2 type have been obtained. The N dimensional Klein—Gordon equation has been reduced to a first-order differential equation via Laplace transformation. The exact bound state energy eigenvalues and corresponding wave functions for CH, H2, and HCl molecules interacting with pseudoharmonic oscillator potential in the arbitrary N dimensions have been determined. Bound state eigenfunctions used in applications related to molecular spectroscopy are obtained in terms of confluent hypergeometric functions.

Chebyshev-Tau method for the linear Klein-Gordon equation

In this paper, we study a numerical method based on polynomial approximation, using the shifted Chebyshev polynomial, to construct the approximate solutions of the one dimensional linear Klein-Gordon equation with constant coefficients. Also, we give general forms of the operational matrices of integral and derivative. We solve two illustrative examples to test this method known as the shifted Chebyshev Tau method. By using this method, the problem is reduced to a set of linear algebraic equations by the operational matrices of integral and derivative. Then, solving the systems, we obtain the exact solutions to these problems. It is shown that the method produces accurate results. Key words: Chebyshev polynomials, Tau method, shifted Chebyshev series, Klein-Gordon equation.

The Klein-Gordon equation with ring-shaped potentials: Asymptotic iteration method

This study presents the solutions of three dimensional Klein-Gordon equation for the spherically and non-spherically harmonic oscillatory ring-shaped potentials within the framework of asymptotic iteration method. Using the method of variable separation, this study obtains the radial and angular equations. And then the bound states energy eigenvalues and corresponding eigenfunctions are obtained analytically.