In this note, we use an extension of the trial equation method also called the direct integral method for partial differential equations with nonconstant coefficients to derive exact solutions in the form of nonlinear waves. The resulting spincorrected kg equation is a second order differential. Exact solution of the klein gordon equation for the. Nov 01, 2014 we obtain exact solutions of both angular and radial parts of the kleingordon equation in this spacetime, which are given in terms of the confluent heun functions. New solitary wave solutions for the conformable kleingordon. Exact solutions of the klein gordon equation and hawking radiation in a kerrnewman background h.
We present new exact solutions in the form of solitary waves for the conformable klein gordon equation with quintic nonlinearity. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations odes by introducing symmetry variables and then seeking exact solutions for more tractable. Separation of variables and exact solution of the klein. Jan 04, 2012 exact solutions of the kleingordon equation by modi. Analytical solutions of the kleingordon equation for rosen. Klein gordon kg equation is a basic relativistic wave equation that is well known to describe the motion of spin zero particles. It can be solved by means of inverse scattering method 1. Kapitza spring 2018 second quantization of the kleingordon.
However, the problem of exact solution of the kge for a number of special potential has been of great interests in the recent years. Exact solutions for the generalized klein gordon equation via a transformation and expfunction method and. A bidirectional traveling plane wave representation of. We present an algorithm for constructing the firstorder symmetry algebra and describe its structure in terms of lie algebra extensions. International journal of theoretical physics, 2008. New explicit and exact solutions of the klein gordon zakharov equations in this section, we shall apply the method developed in section 2 to the klein gordon zakharov equations 12 and give it a series of explicit exact solutions. Lijuan, du xianyun use the new auxiliary equations to obtain a diversity of exact periodic wave solutions in terms of the jacobi elliptic functions for two forms of nonlinear klein gordon equation, and get solitary wave solutions while limit. Using the asymptotic iteration and wave function ansatz method, we present exact solutions of the klein gordon equation for the quarkantiquark interaction and harmonic oscillator potential in the case of the positiondependent mass.
Finding exact solutions to this equations provides importan information about the. Exact solutions of the klein gordon equation in the presence of a dyon, magnetic flux and scalar potential in the specetime of gravitational defects. Solution of the klein gordon equation the covariant generalization of the klein gordon equation in curved spacetime has the form 12 g r. Kge is of high importance for mixed scalar and vector potentials. A generalization of the sine gordon equation, sometimes referred to as a multiple sine gordon equation io, is. A systematic method for constructing particular solutions of the nonlinear kleingordon and liouville equations in four. We use functional variable method which converts a conformable pde to a secondorder ordinary differential equation through a traveling wave transformation. New exact traveling wave solutions for the kleingordon. Exact solutions of the kleingordon equation with position.
Approximate solution of nonlinear fractional kleingordon. The solutions obtained include solitons and periodic solutions. Exact elliptic solution for nonlinear kleingordon equation. Based on the idea of the infinite series method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear evolution equations. The ptsymmetry for these potentials is also considered. Pdf on exact solutions of kleingordon equations using.
Pdf we present the exact solution of the klein gordon with hylleraas potential using the nikiforovuvarov method. It is secondorder in space and time and manifestly lorentzcovariant. Exact solutions of the kleingordon equation for the rosen. Aug 07, 2006 exact solutions of the klein gordon equation in the presence of a dyon, magnetic flux and scalar potential in the spacetime of gravitational defects. Exact solutions of the kleingordon equation with a. With the taylor and chapmanenskog expansion, the nonlinear klein gordon equation is recovered correctly from the lattice boltzmann equation. Our results reveal that the new approximate solutions obtained via opim are more accurate and impressive than many other techniques in literature. The problem of finding exact solutions of the klein gordon equation for a number of special potentials has been a line of great interest in recent years, some authors, by using different methods. Exact solutions of the symmetric regularized long wave. The kleingordon equation in the spacetime representation 9 the quantity q2 is represented by the differential operator dalembert operator from the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field. We show that auxiliary equation method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Algebraic approach to the problem has also been studied. Exact solutions of the massdependent kleingordon equation.
We consider for the kleingordon equation in gravitational field of massive point source in general relativity. This article examines the klein gordon kg equation with cubic nonlinearity by an ansatz method. Numerical solution of the kleingordon equation in an. When the parameters are assigned special values, solitary wave solutions are originated from the exact solutions. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. The energy eigenvalues equation and the corresponding wave functions are obtain explicitly. The klein gordon equation plays a significant role in mathematical physics and many scientific applications such as solidstate physics, nonlinear optics, and quantum field theory 1. Solutions of the sine gordon equation equivalent to eq.
Pdf exact solution of the klein gordon equation in the. By this approach it was possible to find several exact invariant solutions for the klein gordon equation uxx utt ku. The general solutions found, could be used for future explorations on the study for. The exact energy eigenvalues and wavefunctions are derived analytically by using the nikiforov and uvarov method.
A new class of exact traveling wave solutions to the klein. Pdf exact solutions of the kleingordon equation with hylleraas. Thus, in general, we will have to resort to an approximate procedure. New exact traveling wave solutions for the nonlinear klein. Novel approaches to the solution of differential, integral and integrodifferential. The relativistic problem of spinless particles with positiondependent mass subject to kinklike potentials tanh. Exact solutions of the kleingordon equation in the. Traveling wave, exact solutions and auxiliary equation. Shape invariance approach to exact solutions of the klein. Ikhdair department of physics, near east university, nicosia, north cyprus, mersin 10, turkey. Exact travelling wave solutions of the coupled kleingordon. Exact solutions for the nonlinear kleingordon and liouville.
New solitary wave solutions for the conformable klein. We obtain explicitly the bound state energy eigenvalues and the corresponding eigen function for swave. Boundstate solutions of the kleingordon equation for the. Analytical solutions of the klein gordon equation for the rosenmorse potential 1 with equal scalar and vector potentials are studied in this paper. In this paper, we employ an entirely dierent approach, namely, the travelling wave variable approach along with the simplest equation method to obtain exact solutions of the kgz equations a andb. Gordon, duffin kemmer petiau and dirac equations for mixed vector and scalar. Its solutions include a quantum scalar or pseudoscalar field, a field whose. Since the exact solution cannot be obtained explicitly for arbitrary timedependence of the eld, we discuss the asymptotic behavior of the solutions with the help of the relativistic. An application discussed to illustrate the effectiveness and the performance of the proposed method, which successively provided for finding the solutions of the nonlinear klein gordon equation. It has root coming from the study of theoretical physics with importance in quantum mechanics. To obtain the exactly normalized bound state wave function and energy expressions we have used the asymptotic iteration method aim 23. The wave functions obtained are expressed in terms of jacobi polynomials.
By using symbolic computation, we apply auxiliary equation method to construct exact solutions of nonlinear klein gordon equation. Sep 23, 2019 in addition, the solutions of these wave equations are highly applicable in chemical physics and highenergy physics at higher spatial dimensions. Pdf exact solutions of the kleingordon equation in the. The plane waves 10 are basic solutions and the field 9 is constructed by.
Pdf exact solutions of the kleingordon equation with. The exact bound state wavefunctions and energy equations of klein gordon and dirac equations are given with equal scalar and vector potential sr vr vr2 v0 tanh2 rd. We present the exact solution of the klein gordon with hylleraas potential using the nikiforovuvarov method. Akbar2 1d ep ar tm nofm hic s,c gu v ye dt lb 2d ep ar tm nofa li dm hc s,u v y r j b g. In recent years, many studies have been carried out to explore the relativistic energy eigenvalues and the corresponding wave functions of the klein gordon and the dirac equation. This method not very well known and used is of great importance in the scientific community.
Some new exact traveling wave solutions to the kg equation have been extracted in the simplest form through this ansatz. Pdf exact solutions of the kleingordon equation for the scarf. Pdf we present the exact solution of the kleingordon with hylleraas potential using the nikiforovuvarov method. The lie symmetry method allows to find invariant solutions under certain groups of transformations for differential equations. Finding the exact solution for most of these equations is not an easy task, thus analytical and numerical methods must be used. It was found that the asymptotic iteration method provides the closedforms for the energy eigenvalues as well as the eigenfunctions. Numerical solution of nonlinear kleingordon equation. A new decomposition of exact solutions to the scalar wave equation into bidirectional, forward and backward, traveling plane wave solutions is described. We give the solutions of some special cases of the poschl teller doubleringshaped coulomb potential in sect.
The klein gordon equation with quadratic nonlinearity in the interval 11. Exact solutions of the zoomeron and kleingordonzakharov. Using the dalembert operator and separation of variables, we work on the complicated differential equation governing the radial component. Exact solutions of the kleingordon equation for spherically. We solve the klein gordon and dirac equations in an open cosmological universe with a partially horn topology in the presence of a time dependent magnetic eld. Exact solutions of the kleingordon equation in the kerr. Exact solutions of nonlinear generalizations of the klein. Boundstate solutions of the kleingordon equation with q. For example, the exact solutions of the klein gordon equation for an hydrogen atom and for a harmonic oscillator in 3d represent two typical examples 1. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the hawking radiation of charged massive scalar particles. Furthermore the kg equation has also been considered in the context. Exact solutions for the generalized kleingordon equation via a. The klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. We report the quasi exact analytical solutions by working on a.
We present an algorithm for constructing the firstorder symmetry algebra and describe its structure in terms of lie. In this ansatz a nonlinear ordinary differential equation has been used as an auxiliary equation whose solutions are identified. The lie symmetry approach along with the simplest equation and expfunction methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the klein gordon zakharov equations. Pdf we obtain the exact analytical solution of the klein gordon equation for the exponential vector and scalar potentials by using the. Exact solutions of nonlinear kleingordon equation with non. Exact travelling wave solutions of the coupled klein. These equations appear in the study of relativistic and quantum physics. The relativistic klein gordon equation with equal scalar and vector spherically asymmetrical singular oscillators is solved using the asymptotic iteration method. Pdf in this paper we present the exact solutions of the onedimensional klein gordon equation for the scarftype potential with equal scalar. Gordon equation of the symmetric generalized woods. Exact solutions of the kleingordon equation in the presence. Therefore, a system that can be described by a complex solution to the kleingordon equation also be described by a system of two independent particles with equal mass that have real solutions to the klein klein gordon equation. Published 7 august 2006 2006 iop publishing ltd classical and quantum gravity, volume 23, number 17.
The exact solutions of the onedimensional kleingordon equation for the rosenmorse type potential with equal scalar and vector potentials are presented. We shall now use the above results to obtain solutions of kleingordon equation. It is a quantized version of the relativistic energymomentum relation. Pdf exact solutions of coupled nonlinear kleingordon equations. Some exact solutions for a klein gordon equation scielo colombia. Let us now consider stationary kleingordon equation of the form. The general solutions found, could be used for future explorations on. Numerical solution of nonlinear kleingordon equation using. Solutions to common problems with probability interpretation and inde. Solutions of the klein gordon equation with generalized. For example, the exact solutions of the kleingordon equation for an hydrogen atom and for a harmonic oscillator in 3d represent two typical examples 1. In relativistic quantum mechanics, the particle in teraction should be described by either the klein. We obtain periodic wave and solitary wave solutions including particularly kinkprofile and bellprofile type.
Using this method, wave functions and corresponding exact energy equation are obtained for the swave bound state. Klein gordon, equation, nonlinear, exact solutions created date. The method is applied on some test examples, and the numerical results have been compared with the analytical solutions or the numerical solutions reported in previous studies. The resulting representation is a natural basis for synthesizing pulse solutions that can be tailored to give directed energy transfer in space. The work confirms the significant features of the employed method and shows the variety of the obtained solutions. Nonlinear klein gordon equation exact solutions keywords. The obtained results include new soliton and periodic solutions. Exact solutions of the kleingordon equation by modi. Thus, the exact solutions of the coupled klein gordon equation can be expressed as. Exact solutions of coupled nonlinear kleingordon equations e. Exact solutions of nonlinear kleingordon equation with. Research article boundstate solutions of the kleingordon. We solve the klein gordon equation with a new anharmonic oscillator potential and present the exact solutions. Exact solutions of klein gordon equation with exponential scalar and vector potentials.
Apr 01, 2015 the first integral method was used to construct exact solutions of the zoomeron and kleingordonzakharov equations. On exact solutions of klein gordon equations using the semi analytic iterative method december 2020 project. Pdf exact solutions of kleingordon equation with exponential. In nuclear and high energy physics the study of exact solution of the klein gordon equation kge is of high importance for mixed scalar and vector potentials.
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