Recent ly, we have been able to solve this problem in the affirmative. I saw a phrase "the Klein-Gordon flow behaves like the wave flow at high frequency and the Schrödinger flow at low frequency. Next, Nonlaopon et al. Some of the modern methods of integrability are variational iteration method, homo-topy analysis method, semi-inverse variational principle, exponential function method, (G /G) method, Riccati equation method, Fan’s F-expansion method and so on. The solution we obtained is unique. Authors: James Colliander, Justin Holmer and Nikolaos Tzirakis Journal: Trans. of a particle of rest mass min our units where c= ¯h= 1. In the second method, we translate the behavior in momentum space into position space by means of Fourier transform. The objections to the Klein-Gordon equation are three-fold 1. Solution to klein-gordon type equation. Fourier,Grenoble 61,6(2011)2463-2506 GLOBAL EXISTENCE FOR COUPLED KLEIN-GORDON EQUATIONS WITH DIFFERENT SPEEDS byPierreGERMAIN(*) Abstract. 4 Fourier transform pairs If f(x) is very narrow, then its Fourier transform A(k) is a very broad function and vice versa. In the last decades Iterative methods have been used for solving fractional differential equations. Jafari and 1 M. 2 - Derives that spin operators are the generators of rotation Spin, Helicity and the Dirac Equation by Thomson (Class Notes). This relativistic wave equation is now most commonly known as the Klein-Gordon equation. The Solution of Nonlinear Klein - Gordon Equation using Reduced Differential Transform Method Ita B. In this paper, we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schro¨dinger equation. ) The Klein{Gordon equation can be easily generalized to motions in an external eld by. We focus mainly on the one-dimensional Klein-Gordon equation, which is associated with the hyperbola. This Klein–Gordon problem has a long history. 81) Suppose one is interested in obtaining solutions to the inhomogeneous Klein-Gordon (KG) equation i. Besides plane-wave solutions of definite momentum, we can have solutions which are the Fourier transform of which depend only on momentum:. Let m>0 and V : R !R a real. Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition Cunha, Patrícia L. work, is that the finite Fourier sine transform method can be applied to other partial differential equation, due to the efficiency in the application to get the possible results. The Klein-Gordon equation reads: $$(\partial ^2 + m^2)\phi(x) = 0. Then we use this to derive dispersion estimates for solutions of the associated Schr odinger and Klein{Gordon equations. The /1+1 Klein-Gordon and Dirac equations are converted into finite-dimensional matrix equations by using a projection method. full record;. 1 ) without the restriction of a compact support on the initial data which was assumed in the previous work [ 1 ]. Nonlinear Partial Differential Equations for Scientists and Engineers, Third. This banner text can have markup. 2 1Physical and Theoretical Chemistry Group, Department of Pure and Applied Chemistry, University of Calabar, Calabar, NIGERIA 2Department of Mathematics and Statistics, Ngoundere University, CAMEROON. The method of [4] was based on a new L2–L∞ estimate for the inhomogenous Klein-Gordon equation which gave existence of global solutions of. Therefore the Fourier transform of Φ is approximately equal to ˜ A(p) (3. Recall ﬁrst the classical results for the Euclidean Laplace operator on Rn, n≥ 2. ] "Large-degree asymptotics of rational Painlevé-II functions. Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition Cunha, Patrícia L. It is second-order in space and time and manifestly Lorentz-covariant. We prove a sharp bilinear estimate for the one dimensional Klein{Gordon equation. Its solution can be written as a convolution integral (using Laplace transforms) [2,3] or expressed as an infinite series (obtained from a Fourier transform inversion) [1]. NOTE ON WEIGHTED STRICHARTZ ESTIMATES FOR KLEIN-GORDON EQUATIONS WITH POTENTIAL By Hideo KUBO and Sandra LUCENTE Abstract. An interesting nonlinear3 version of the wave equation is the Korteweg-de Vries equation u. First, we do inverse Fourier transformation of the scalar field. In the Klein Gordon equation we get particular solution, so we also obtain particular solution of linear two channel dissipation model. THROUGH KLEIN-GORDON THEORY JONATHAN EMBERTON Abstract. 2 Homotopy Perturbation Method16 3. This transform was sugg. Wavelet transform of a function, as Fourier transform, is powerful tool for analyzing the components of stationary phenomena.$$ If I let $$\ph. We compare these methods and discuss the convergence of them. In this work, we completely solve these two problems. Sidorenko St. Klein Gordon Equation F eynman Stuec k elburg In terpretation Dirac Equation An ti-particles F ermion spin Co v arian t notation Massless fermions Learning Outcomes Be able to deriv e K G equation and explain ph ysical meaning of -v e E solutions. In order to prove this theorem, some elements of linear fractional theory and ergodic. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different machines for a problem size of 512 3. Using Fourier transforms, the field from a point source can be computed. In this paper, Elzaki transform and variational iteration method were applied to solve non-homogeneous Klein-Gordon equation. We provide an introduction to quantum eld theory by observing the methods required to quantize the classical form of Klein-Gordon theory. Wavelet-based Integral Representation of Solutions of Wave and Klein-Gordon Equations Maria V. One can think that the main problem of the Klein-Gordon equation is that it is quadratic: if it wouldn't be, we could expect negative energies to vanish and we could get a correct expression for a covariant quantum mechanic equation. 81) Suppose one is interested in obtaining solutions to the inhomogeneous Klein- Gordon (KG) equation i. Wiener algebra of functions with integrable Fourier transforms. For a fixed , this is an ordinary differential equation in time. These methods are based on applying fourth-order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The proof involves an unlikely combination of ve trigono-metric identities. and modiﬁed fractional diﬀerential transform method applied successfully for ﬁnding the approximate analytical solution of the nonlinear fractional Klein-Gordon equa-tion. Recall ﬁrst the classical results for the Euclidean Laplace operator on Rn, n≥ 2. [math]\phi(t,\vec{x})=\int \frac{d^{3}\vec{p}}{(2\pi)^{3}} \phi(t,\vec{p. Propagator for the Klein-Gordon Equation. In this paper, we applied homotopy perturbation Sumudu transform method (HPSTM) to obtain the analytical exact and approximate solutions for the fractional Klein-Gordon equation with time-fractional derivatives of the. For newcomers to quantum field theory, the Klein Gordon Field is a solution to Klein Gordon’s equation: where as usual, is the d’Alembertian operator and m is the mass of the scalar field (for now, if we consider a real Klein Gordon field, this is just the pass of the particle excitation mode of the field). Let m>0 and V : R !R a real. In principle, the Klein-Gordon equation could be solved by separation of variables and an applica- tion of an appropriate Fourier transform. This site uses cookies. The discrete Fourier transform of u: Z2 → C is deﬁned by the formula bu(θ) = X x∈Z2 u(x)eiθx,θ∈ T2:= R2/2πZ2. 24) where the index is used to label different solutions of identical mass. 3 can actually be derived from 2 by using a Fourier transform. Fourier uniqueness sets and the Klein-Gordon equation H˚akan Hedenmalm Abstract. Also we use the method of normal forms of 7 by which we transform the quadratic nonlinearity into a cubic one with a nonlocal operator. A Heisenberg uniqueness pair (HUP) is a pair (Γ, Λ), where Γ is a curve in the plane and Λ is a set in the plane, with the following property: any bounded Borel measure µ in the plane supported on Γ, which is absolutely continuous with respect to arc length, and whose Fourier transform bµ vanishes on Λ, must automatically be the zero measure. I will try to explain, without going into too much detail, how one can consider a non-linear wave equation as a dynamical system and what it brings to the study of its solutions. Marangell, and R. It is a quantized version of the relativistic energy–momentum relation. Its inverse Fourier transform is called the "sampling function" or "filtering function. On page 22 of this text, they show how we can delineate all the different solutions to the Klen-Gordon equation by fourier transforming the ϕ(x,t) into ∫ ϕ(p,t) exp(ipx) d 3 p/(2π) 3 This then apparently pulls a p 2 out of it using the ∇ 2, but I don't see how. They can be written (using) as a Fourier transform of, (4. Through this proposed strategy, the nonlinear term is substituted by associating Adomian polynomials obtained by utilization of a multistep approach. 10) where c+ and c_ are arbitrary constants. Theclassical(integerorder)Klein– Gordon equation was developed on the basis of homo-geneousspaceandtime. Historically, same as Dirac equation, Klein-Gordon equation was derived by using correspondence principle Eqs. In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential of the form = − −,where g is a magnitude scaling constant, i. The Klein-Gordon equation, the Hilbert transform, and dynamics of Gauss-type maps Joint work with Alfonso Montes-Rodr guez June 1-4, 2015. Then we use this to derive dispersion estimates for solutions of the associated Schro¨dinger and Klein-Gordon equations. The proof involves an unlikely combination of five trigonometric identities. In this paper, we propose an efficient numerical scheme for the space-fractional Klein-Gordon-Schrödinger (SFKGS) equations. "Spectral and modulational stability of periodic wavetrains for the nonlinear Klein-Gordon equation," Journal of Differential Equations 257, pages 4632-4703, 2014. The Klein-Gordon equation reads:$$ (\partial ^2 + m^2)\phi(x) = 0. For the model problems, we present the corresponding algebraic equations of the differential transform in the standard DTM and the projected DTM at. This paper extends the homotopy perturbation Sumudu transform method (HPSTM) to solve linear and nonlinear fractional Klein-Gordon equations. Highly localized explicit solutions to the wave and Klein–Gordon–Fock multidimensional linear equations are presented. NASA Astrophysics Data System (ADS) Pérenne, Nicolas; Renouard, Dominique P. , St Petersburg, Russia New Trends and Directions in Harmonic Analysis, Approximation Theory, and Image Analysis Inzell, Germany, 2007. Global Existence and Dispersion of Solutions to Nonlinear Klein-Gordon Equations with Potential / The Normal Forms Transform; The Fourier Transform and Bilinear. We will only consider the free-particle Klein-Gordon propagator in this section. Joint with Rowan Killip, Monica Visan. Having developed the nonrelativistic propagator, we now turn to the development of the relativistic propagators. 2 Waves in an Elastic Medium 142 Longitudinal Waves. For the classical conic sections, the Fourier transform of the measure solves a second order partial diﬀeential equation. The Fourier transform of the position space propagators can be thought of as propagators in momentum space. relativistic particle: the Klein-Gordon equation (@ @ + m2) (x) = 0 (2. For the model problems, we present the corresponding algebraic equations of the differential transform in the standard DTM and the projected DTM at. 1 with initial conditions u x,0 g 1 x ,u t x,0 g 2 x , 3. The Fourier transform of radial functions are radial. Skip navigation. 3 Klein-Gordon Equation. These methods are based on applying fourth-order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. We provide an introduction to quantum eld theory by observing the methods required to quantize the classical form of Klein-Gordon theory. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The nonlinear models of real- -life problems are still difficult to solve either numerically or theoretically. Zimmermann:. the complete set of the Maxwell-Klein-Gordon-equations. * 1-particle Hilbert space: The completion of the (Klein-Gordon) inner product space of smooth, rapidly falling positive-frequency solutions of the Klein-Gordon equation; It is isomorphic to L 2 (positive mass shell in Fourier transform space). However, we present a solution of the Dirichlet boundary value problem posed on the half line using a different approach. The /1+1 Klein-Gordon and Dirac equations are converted into finite-dimensional matrix equations by using a projection method. Integral Transforms and Spectral Theory in Analysis and Geometry May 20-23, Carlton, Nahariya, Israel RESEARCH WORKSHOP OF THE ISRAEL SCIENCE FOUNDATION. (2005) The tanh and the sine–cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation. Our main result is that each nite energy solution converges as t! 1 to the set of onlinear eigenfunctions" (x)e i!t. In principle, the Klein-Gordon equation could be solved by separation of variables and an applica- tion of an appropriate Fourier transform. 1 Practical use of the Fourier transform The Fourier transform is beneﬁcial in differential equations because it can transform them into equations which are easier to solve. Overall, the Klein-Gordon equation appears to be a good relativistic replacement for the non-relativistic Schr¨odinger equation at the ﬁrst sight, but it completely fails to give the conventional probability interpretation of a single-particle wave function. De Broglie also arrived at the same equation in 1928. the nonlinear Klein-Gordon equation and a theoretical analysis was performed. Fourier Transform. Roohani 1 Department of Mathematics, Islamic Azad University, Ghaemshahr. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. What does it mean to claim that. Ziolkowski. Beatrice Pelloni. 1) for small ﬁnal conditions ϕ 0, ϕ˙ 0, with ˆϕ 0, ϕˆ˙ 0 ∈ C∞ 0 (IR 3), where fˆis the Fourier transform of f. The Fourier transform and the convolution are used to solve the problem. The general solution to the Klein-Gordon equation has either positive or negative energy. The aim of this lecture note is to explain how to derive sharp a priori estimates which enable us to prove a global in time existence of solutions to semilinear wave equation and non-linear Klein-Gordon equation. process and single iteration gives the exact solution. Nonlinear Klein-Gordon solitary waves - or solitons in a loose sense - in n+1 dimensions, driven by very general external fields which must only satisfy continuity - together with regularity conditions at the boundaries of the system, obey a quite simple equation of motion. The function represented by relation (8) satisfies the Klein Gordon Equation. We continue our in our derivation of the Fourier transform by expanding the interval to infinity. Batrasev, M. Watch Queue Queue. , 262 (2012), 4986-5020. Odlyzko has done extensive computations of the Fourier transform of the nontrivial zeros of the Zeta function, and they seem to form a one-dimensional quasicrystal. They showed that the results for the two. of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. When you susbsitute this back into the klein gordon equation you get down two factors of p, i. HESAMEDDINI2, A. The solutions are actually constructed in the spatial and temporal Fourier transform domain. A Direct Proof of Global Existence for the Dirac-Klein-Gordon Equations in One Space Dimension Yung-fu Fang1 Abstract. We prove a sharp bilinear estimate for the one dimensional Klein{Gordon equation. For the classical conic sections, the Fourier transform of the measure solves a second order partial diﬀeential equation. $Then we know the initial. This expansion is used to estimate the Lp norms | W{-, t)\v as / —*- oo. For the model problems, we present the corresponding algebraic equations of the differential transform in the standard DTM and the projected DTM at. If you like you can view the Klein-Gordon equation as a \toy model" for the Maxwell equations. 7 The Sine-Gordon Equation, Soliton and Anti-Soliton Solutions 470 11. The objections to the Klein-Gordon equation are three-fold 1. Fourier analysis on the lattice Rn h,α (subsection 2. A second order semi-implicit Fourier pseu-dospectral discretizationscheme used by Donninger and Schlag [2] and. Let m>0 and V : R !R a real. The solution we obtained is unique. Master of Science in Applied Mathematics. First, the semi-discrete scheme is given by using Fourier spectral method in spatial direction, and conservativeness and convergence of the semi-discrete scheme are discussed. The scattering problem for this equation belongs to the borderline between the short range case and the long range one. The case of the semi-axis$\R_+$as well as the holomorphic counterpart remained open. In other words, the Klein-Gordon equation, if. A nonlocal square root of the Klein-Gordon equation is proposed. It is a good relativistic equation for a massive scalar field. Be able to deriv e Dirac eqn and algebra for i; Be able to solv e the free Dirac equation and in. Suppose we just apply the phase transformation to the Dirac equation of the same form that we use with the Klein-Gordon Lagrangian. Perel, Mikhail S. For the classical conic sections, the Fourier transform of the measure solves a second order partial diﬀeential equation. Transverse Waves. process and single iteration gives the exact solution. Received: January 18, 2011 AMS Subject Classification: 46F10 Key Words and Phrases. The Klein-Gordon Propagator Dan Styer, Oberlin College Physics Department, Oberlin, Ohio 44074 [email protected] To illustrate the reliability of the method, some examples are presented. 2) Compare the results from the Natural transform decomposition method with other methods that are already in existence for solving fractional differential equations. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. Finally we propose the use of the obtained equations for the Fourier„s transform of the. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity. The Feynman propagator is aGreen's function. The value of is depicted through the constant “a” 2. AB - We prove a sharp bilinear estimate for the one-dimensional Klein-Gordon equation. The equation is given as. The Klein-Gordon equation reads:$$(\partial ^2 + m^2)\phi(x) = 0. choosing µ= 0 (the temporal component of Aµ) we get ∂iF 0i+qJ0 = 0, (7) which is, upon deﬁning the electric ﬁeld as Ei= F0i, nothing but the Maxwell equation. This nonlocal equation is a special relativistic equation for a scalar field of first order in the time derivative. Using Fourier transforms, the field from a point source can be computed. In Section 5 we extend the results to the discrete Klein-Gordon equation. The triply periodic domain allows for Fourier in all three directions, and we can as such create one instance of this base class using Basis() with family Fourier for each direction. In these notes, I shall show that the propagator (1) is a Green’s function of the Klein– Gordon equation, and then I shall explain why there are many diﬀerent Green’s functions and which particular Green’s function happens to be the Feynman propagator. 364 (2012), 1571-1631. In this paper, Elzaki transform and variational iteration method were applied to solve non-homogeneous Klein-Gordon equation. linear Schr odinger equation and the nonlinear Klein-Gordon equation. Theclassical(integerorder)Klein– Gordon equation was developed on the basis of homo-geneousspaceandtime. In the inhomogeneous case, the screened Poisson equation is very similar to the inhomogeneous Helmholtz equation, the only difference being the sign within the brackets. Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm Inverse scattering for the Dirac equation. The main idea is that £[u(x,t)] can be considered as a function of x in solving process by Laplace transform, and this can be dealt with a linear equation. I am learning about adv quantum and field theory and i have run across something unfamiliar mathematically. The aim of this lecture note is to explain how to derive sharp a priori estimates which enable us to prove a global in time existence of solutions to semilinear wave equation and non-linear Klein-Gordon equation. Wavelets Numerical Methods for Solving Differential Equations By Yousef Mustafa Yousef Ahmed Bsharat Supervisor Dr. Some Klein-Gordon equation notes. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different machines for a problem size of 512^3. An interesting nonlinear3 version of the wave equation is the Korteweg-de Vries equation u. 3 Klein-Gordon Equation In non-relativistic mechanics, the energy for a free particle is E= p2 2m: (35) To get quantum mechanics, we make the following substitutions:. 1 Dirac Klein-Gordon Equations The majority of the thesis is concerned with looking at regularity results of the coupled Dirac Klein-Gordon equations. Now L= L(φ,∂φ/∂x µ). Internal waves and rectification in a linearly stratified fluid. In the homogeneous case (f=0), the screened Poisson equation is the same as the time-independent Klein-Gordon equation. Mittala & Rachna Bhatiaa a Department of Mathematics, IIT Roorkee, Roorkee 247667, Uttarakhand India Accepted author version posted online: 29 Sep 2014. To get the dimensions to work out, m is not a mass, but a constant with dimensions of length-1. of the Klein-Gordon equation the corresponding phase function is nondegenerate in any ﬁnite region of the Fourier space, but is degenerate at inﬁnity, which makes it diﬃcult to deduce uniform bounds. The purpose of this paper is to consider the time-splitting Fourier spectral (TSFS) method to solve the fractional coupled Klein-Gordon-Schrödinger (K-G-S) equations. Klein–Gordon equation is the gateway to relativistic quantummechanics. These methods are based on applying fourth-order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. Retarded and advanced field operators are obtained as solutions, whose frequency parts satisfy a complex inhomogeneous K-G equation. Klein-Gordon-Schr¨odinger Equations in Two Space Dimensions By Akihiro Shimomura Abstract. Skip navigation. Scattering for the cubic Klein-Gordon equation in two space dimensions. invariant, we are led to the Klein-Gordon equation, (∂2 + k2 c)ψ =0 where λ = λ c 2π =! mc denotes the Compton wavelength. 3) The propagator in momentum space can now be written as. The function represented by relation (8) satisfies the Klein Gordon Equation. Laplace transform, which literally transforms the orig-inal diﬀerential equation into an elementary algebraic equation. Introduction There is much interest in applications of the square-root Klein-Gordon oper-ator, p m2 r 2, to problems in quantum mechanics. Sidorenko St. Particularly, we will start with studying the Dispersion Relations, then we will solve analytically the Signalling Problem and the Cauchy Problem for this equation using integral transform methods (Laplace and Fourier). Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In non-relativistic mechanics, the energy for a free particle is E= p2. " Remember: totally classical! No factors of hbar. The non-linear Fourier transform. Supposing that the initial data is in the non-weighted modulation space, M0 p;q (R n), using spec-i ed nonhomogenuous functions they were able to show that both the nonlinear Schr odinger equation and the nonlinear Klein-Gordon equation have solutions in a. Note that the eigenvalue λ(q) is a function of the continuous parameter q in the Mathieu ODEs. For the model problems, we present the corresponding algebraic equations of the differential transform in the standard DTM and the projected DTM at. Khusnutdinovaa,b,∗, D. fr/hal-01078696v2 Submitted on 4 Jan 2016 HAL is a multi-disciplinary open access archive for the deposit and. Wiener algebra of functions with integrable Fourier transforms. [email protected] The Klein-Gordon Propagator Dan Styer, Oberlin College Physics Department, Oberlin, Ohio 44074 [email protected] Heisenberg uniqueness pairs and the Klein-Gordon equation 3 In this case, we see from the example of the line that it is necessary for (Γ,Λ) to be a Heisenberg uniqueness pair that π 1 (Λ) be dense. Since being ˝almost constant˝, its Fourier transform is a sequence of delta type when β → 0. The method of [4] was based on a new L2–L∞ estimate for the inhomogenous Klein-Gordon equation which gave existence of global solutions of. Wavelet analysis is a recently developed mathematical tool in applied mathematics. It is not a simple problem to introduce these transforms in the non- archimedean case, see [3 - 7]. Home; web; books; video; audio; software; images; Toggle navigation. To get the dimensions to work out, m is not a mass, but a constant with dimensions of length-1. (Here, x= (t;x) is the coordinate 4-vector and mis the mass of the particle. UROOSA2,x Abstract. ysis for standing waves of a perturbed Klein-Gordon equation, Discrete Contin. In the context of a hyperbola 0, the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets 3of a collection of solutions to the Klein-Gordon equation. Fourier Transforms - Solving the Wave Equation This problem is designed to make sure that you understand how to apply the Fourier transform to di erential equations in general, which we will need later in the course. The Cubic Nonlinear Schrodinger Equation; The Two- and Three-Dimensional Navier-Stokes Equations; The Klein-Gordon Equation; Gray Scott; Maxwell's Equations; Incompressible Magnetohydrodynamics; GPU programs for Fourier pseudospectral simulations of the Navier-Stokes, Cubic Nonlinear Schrodinger and sine Gordon equations; Visualization with. Now L= L(φ,∂φ/∂x µ). Forced Motion of the Elastically Braced String. Introduction There is much interest in applications of the square-root Klein-Gordon oper-ator, p m2 r 2, to problems in quantum mechanics. work, is that the finite Fourier sine transform method can be applied to other partial differential equation, due to the efficiency in the application to get the possible results. Overall, the Klein-Gordon equation appears to be a good relativistic replacement for the non-relativistic Schr¨odinger equation at the ﬁrst sight, but it completely fails to give the conventional probability interpretation of a single-particle wave function. The plane waves (10) are basic solutions and the field (9) is constructed by. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Wave equation—D’Alembert’s solution First as a revision of the method of Fourier transform we consider the one-dimensional (or 1+1 including time) homogeneous wave equation. The Klein--Gordon and sine-Gordon equations are a two nonlinear hyperbolic partial differential equations that model problems in classical and quantum mechanics, solitons, and condensed matter physics. These methods are based on applying fourth-order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. The approximate analytical solution of the equation is calculated in the. Peeter Joot peeter. Fourier space. 81) Suppose one is interested in obtaining solutions to the inhomogeneous Klein-Gordon (KG) equation i. The Dirac delta function provides the most extreme example of this property. As we can see now, the "apparent" problems were manifestations of the new physics. UROOSA2,x Abstract. invariant, we are led to the Klein-Gordon equation, (∂2 + k2 c)ψ =0 where λ = λ c 2π =! mc denotes the Compton wavelength. The Fourier transform of the position space propagators can be thought of as propagators in momentum space. Klein-Gordon equation with Coulomb potential - Hypergeometric functions and numerical solutions Klein-Gordon equation - interaction with electromagnetic field Klein-Gordon equation with scalar 1/r potential. We associate to its restriction to a spacelike wedge a scattering from the past light cone to the future light cone, which is then shown to be (essentially) the Hankel transform of order zero. I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave Wave equation solution using Fourier Transform | Physics Forums. 2) is given by. We can write solutions of (8) as Eˆ(z,!)=E+(!)eik(!)z +E(!)eik(!)z, (10). In Physics there is an equation similar to the Di usion equation called the Wave equation @2C @t 2 = v2 @2C @x: (1). Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In this paper we study the Klein--Gordon equation, and show that making use of the standard canonical commutation relations, in some specific way which is fully consistent from the mathematical point of view, leads to new types of canonical commutation relations and, for this reason, to existence of new types of particles called the virtual.$ Then we know the initial. , in the presence of a source. Theclassical(integerorder)Klein– Gordon equation was developed on the basis of homo-geneousspaceandtime. [email protected] In the last decades Iterative methods have been used for solving fractional differential equations. SMOOTHING PROPERTIES OF EVOLUTION EQUATIONS VIA CANONICAL TRANSFORMS AND COMPARISON PRINCIPLE MICHAEL RUZHANSKY AND MITSURU SUGIMOTO Abstract. Download with Google. Finally we propose the use of the obtained equations for the Fourier„s transform of the. The value of is depicted through the constant “a” 2. For newcomers to quantum field theory, the Klein Gordon Field is a solution to Klein Gordon’s equation: where as usual, is the d’Alembertian operator and m is the mass of the scalar field (for now, if we consider a real Klein Gordon field, this is just the pass of the particle excitation mode of the field). Journal of Physics A: Mathematical and Theoretical, Volume 45, Number 39. dispersive equations - the Schrödinger, wave and Klein-Gordon equations - associated to an asymptotically ﬂat metric, and with (or without) an obstacle. Because each of those satisfies the Klein-Gordon equation. I use techniques from Fourier analysis to study partial differential equations. In order to numerically verify this approximation we want to present numerical methods in this chapter which enable us to obtain a good numerical solution of both the Gross-Pitaevskii equation (3. , University of California, Santa Cruz October 18, 2015. Phrased diﬀerently, we show that e πiαnt,e βn/t, n ∈ Z, span a weak-star dense subspace in L∞(R) if and only if αβ≤ 1. This paper explores the approximate analytical solution of nonlinear Klein-Gordon equations (NKGE) by using Multistep Modified Reduced Differential Transform Method (MMRDTM). In order to prove this theorem, some elements of linear fractional theory and ergodic theory are needed, such as the Birkhoﬀ. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. Equations (1) through (3) address matter which, classically, behave the same there should exist a common equation describing all cases. The collective behavior of electrons is modeled by a Klein-Gordon equation, which is nonlinearly coupled with the electromagnetic wave through the Maxwell and Poisson equations. Quantization 5 5. Plan Nacional I+D+I (Ministerio de Educación y Ciencia) Repositorio de producción científica de la Universidad de Sevilla Heisenberg uniqueness pairs and the Klein-Gordon equation. The Klein-Gordon equation is the linear partial differential equation which is the equation of motion of a free scalar field of possibly non-vanishing mass m m on some (possibly curved) spacetime (Lorentzian manifold): it is the relativistic wave equation with inhomogenety the mass m 2 m^2. The equation is given as. Introduction There is much interest in applications of the square-root Klein-Gordon oper-ator, p m2 r 2, to problems in quantum mechanics. Besieris The Bradley Department of Electrical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 Richard W. 2 The Sine-Gordon equation 2. This course is recommended only for students who have achieved a strong performance in Mathematics as well as Physics in Part IB, or an equivalent qualification. 0378v1 [math. Let us consider the Klein--Gordon equation. 6 The Klein-Gordon Equation and the Whitham Averaged Variational Principle 467 11. Then, we study the Fourier transform of the diamond Klein-Gordon kernel and also the Fourier transform of their convolution. Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm Inverse scattering for the Dirac equation. Heisenberg uniqueness pairs and the Klein-Gordon equation 3 In this case, we see from the example of the line that it is necessary for (Γ,Λ) to be a Heisenberg uniqueness pair that π 1 (Λ) be dense. Nonlinear Klein-Gordon and Schrodinger Equations by the Projected¨ Differential Transform Method Younghae Do1 and Bongsoo Jang2 1 Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea 2 Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 689-798, Republic of Korea. KHALID 1, M. Numerical solution of nonlinear system of Klein-Gordon equations by cubic B-spline collocation method R. In this work, we completely solve these two problems. Let m>0 and V : R !R a real. Please try again later. Global Existence and Dispersion of Solutions to Nonlinear Klein-Gordon Equations with Potential / The Normal Forms Transform; The Fourier Transform and Bilinear. Authors: James Colliander, Justin Holmer and Nikolaos Tzirakis Journal: Trans. When you susbsitute this back into the klein gordon equation you get down two factors of p, i. We multiply both sides of equation 1. It is this parameter dependence that complicates the analysis of Mathieu functions and makes them among the most difficult special functions used in physics. The Fourier transform of a measure supported on a hyperbola solves the one-dimensional Klein-Gordon equation, so the theorem supplies very thin uniqueness sets for a class of solutions to this equation. 1 Laboratory of Numerical Analysis, Kibernetics and Applications, University Marien NGOUABI, Brazzaville, Congo. The Dirac delta function provides the most extreme example of this property. I'm studying Quantum Field Theory and the first example being given in the textbook is the massless Klein Gordon field whose equation is just the wave Wave equation solution using Fourier Transform | Physics Forums. Subcritical and supercritical Klein-Gordon-Maxwell equations without Ambrosetti-Rabinowitz condition Cunha, Patrícia L. Separation of Variables—idea is to reduce a PDE of N variables to N ODEs. Using Fourier transforms, the field from a point source can be computed. The proposed method based on applying the Laplace transform to nonlinear partial differential equation and replacing the nonlinear terms by the Adomian polynomials. operator for the Klein-Gordon equation Dw = m2u, D = A-82/8t2, A the M-dimensional Laplacian, perturbed by a linear potential V= V(x) to " = m2u+Vu, determines the scatterer V. We prove weighted $L^2$ estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. • In section 3 we ﬁnd some explicit representations underlying to the.