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﻿ Nonlinear Klein Gordon Equation

# Nonlinear Klein Gordon Equation

Scattering and Blow-Up for the Focusing Nonlinear Klein-Gordon Equation with Complex Valued Data Below the Standing Waves Speaker: Takahisa Inui, Kyoto University Location: Warren Weaver Hall 1302. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. Singular solitons and bifurcation analysis of quadratic nonlinear Klein-Gordon equation Ming Song1,2, Zhengrong Liu1, Essaid Zerrad3 and Anjan Biswas4,5,∗ 1Department of Mathematics, South China University of Technology, Guangzhou-510640, CHINA 2Department of Mathematics, Yuxi Normal University, Yuxi-653100, CHINA. Considering a nonlinear evolution equation con-sists of a linear part Lu, nonlinear part Nu, and a free term g(= g(x,t)) represented as Lu. JONES, ROBERT MARANGELL, PETER D. This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon. Lindblad and J. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f, f(u) behaves as a power u 1+4 / n near zero. 2? Nonlinear Partial Differential Equations > Second-Order Hyperbolic Partial Differential Equations > Nonlinear Klein-Gordon Equation 7. First, we provide an alternative proof of the stability of multibreathers in a chain with nearest neighbor interactions only. Many nonlinear Klein-Gordon equations have been studied numerically, and in a few cases, analytical solutions have been found. The equation, as well as several solution techniques, were known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions ("kink" and "antikink") with the collisional properties of solitons (Perring and Skyrme 1962. We recall that (1. In this paper, we want to understand the dynamics of large, quantized energy. Keywords Double Laplace transform, nverse double Laplace transform, I ative method, Iter Nonlinear Klein-Gordon equation 1. [1] Scattering and the Levandosky-Strauss conjecture for fourth order nonlinear wave equations, J. A Variational Iteration Method for Solving the Nonlinear Klein-Gordon Equation B. 7, 3019-3092. We used the variational method to study three different equations in this family. Introduction. In particular, any global solution is bounded. In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one- dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. 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. Wang, Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA Journal of Applied Mathematics, 74 (2009), 392-415. AMS subject classi cation: 65M06, 65M20 Keywords: Scalar auxiliary variable approach, linearly implicit scheme, energy-preserving scheme, exponential integrator, Klein-Gordon. Let us consider the Klein--Gordon equation $u_{tt} - y_{xx} + b\, u + g(u) = f(x,t),$. In 1926, Oskar Klein and Walter Gordon independently proposed one of the nonlinear wave equations emerging from the relativistic motion of electrons. The phase portraits of the solutions are also given. ear Klein-Gordon equation is the defocusing nonlinear Schr odinger equation. Bonanno, and E. nonlinear Klein-Gordon equation boundedness of space-time means. U(1)-invariant nonlinear Klein–Gordon equation in one dimension, with the non- linearity concentrated at a single point: each ﬁnite energy solution converges as t →±∞to the set of all “nonlinear eigenfunctions” of the form ψ(x)e − iωt. In the second part we study the electrostatic case. The hyperbolic function method and Riccati equation method are employed to solve some of the obtained reduced ordinary differential equations. Wang, Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy, IMA Journal of Applied Mathematics, 74 (2009), 392-415. Differential Equations 10 (2013), no. nonlinear Klein-Gordon equation: u tt αu xx βu γun f x,t ,x∈ a,b , 3. Let us consider the Klein--Gordon equation. In this paper, we implement the spectral collocation method with the help of the Legendre poly-nomials for solving the non-linear Fractional (Caputo sense) Klein-Gordon Equation (FKGE). Existence Of Solutions To Nonlinear th-Order Coupled Klein-Gordon Equations With Nonlinear Sources And Memory Terms Khaled Zenniry, Amar Guesmiaz Received 5 December 2014 Abstract In this article, we consider a system of th-order derivatives of the dependent variables of coupled Klein-Gordon equations to improve recent results obtained. A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods. method PDTM , for solving nonlinear partial diﬀerential equations. Math is a program designed to calculate roots of nonlinear equation, roots of system of linear equations, differentiate, integrate, approximate using LMS, matrix calculation using numerical methods for calculating results, estimated errors and much more. Communications in Theoretical Physics, Vol. KUKSIN Abstract. DEEBA AND S. The nonrelativistic-semiclassical limit, ~ !0;c= ~ !1for some >0, of the modulated defocusing cubic nonlinear Klein-Gordon equation is the classical wave map for the limit wave function and typical linear wave equa-tion for the associated phase function. 2010-02-15 00:00:00 A numerical method is developed to solve the nonlinear one-dimensional Klein-Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. Nonlinear Klein-Gordon Equation Necat Polata, Hatice Taskesenb aDepartment of Mathematics, Dicle University, 21280, Diyarbakir bDepartment of Statistics, Yuzuncu Yil University, 65080, Van Abstract. Hylomorphic solitons in the nonlinear Klein-Gordon equation J. In both equations the Laplacian appears on the right hand side. Nonlinearity 18 (2005), No. Math is a program designed to calculate roots of nonlinear equation, roots of system of linear equations, differentiate, integrate, approximate using LMS, matrix calculation using numerical methods for calculating results, estimated errors and much more. Special emphasis is given to various methods of solutions of these equations. Viewed 96 times 1. In the book "Quantum Field theory and the Standard Model" by Matthew Schwartz, the author states: In quantum field theory, we generally work in the Heisenberg picture, where all time dependence. We solve the problem for both Dirichlet and Neumann boundary conditions. Presented at the 19th annual Hudson River Undergraduate Mathematics Conference. }, abstractNote = {We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in R{sup n} of the Klein-Gordon type. Malhi, to appear in Mathematische Annalen, 2018. Baoxiang, On existence and scattering for critical and subcritical non-linear Klein-Gordon equations in Hs, Nonlinear Anal. We consider the scattering problem for the nonlinear Klein-Gordon Equation with long range nonlinearity in one dimension. linear ﬁelds on the Schwarzschild geometry : the non linear Dirac equation, Yang-Mills ﬁelds and non linear Klein-Gordon ﬁelds were studied respectively by A. We used the variational method to study three different equations in this family. In: Communications on Pure and Applied Mathematics, Vol. The generalized nonlinear Klien-Gordon equation plays an important role in quantum mechanics. Since then, this equation has been known as the Klein{Gordon equation. Then the equation becomes, Solution of the equation can be obtained more simple by reducing the two variables (x,t) become one new variable z that containing mixed x and t with certain velocity. This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon. This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon. Global Attractor for Nonlinear Oscillator Coupled to Klein-Gordon Field 109 (iii) First results on the asymptotics of type (1. We observe that ADM is a powerful method to solve linear and nonlinear Klein-Gordon partial diﬀerential equations. In the Appendices Nonlinear Klein Gordon equations with solitary (travelling) wave solutions are reviewed and solved for particular cases. We discuss relations between the nonlinear Klein-Gordon equation and the nonlinear Schr¨odinger equation in view of the global wellposedness in the energy space and L2. The Klein–Gordon equation frequently arises in various fields of scientific applications such as solid state physics, nonlinear optics, quantum field theory and relativistic quantum mechanics. / Shatah, Jalal. 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. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional. Some exact solutions including soliton solutions and rational and elliptic function solutions to the Klein-Gordon-Zakharov equation with power law nonlinearity in (1？+？2) dimensions are obtained by this. Differential Equations & Applications Volume 4, Number 3 (2012), 445-457 doi:10. Klainerman, S 1985, ' Global existence of small amplitude solutions to nonlinear klein‐gordon equations in four space‐time dimensions ', Communications on Pure and Applied Mathematics, vol. Pis¸kin, Uniform decay and blow-up of solutions for coupled nonlinear Klein-Gordon equations with nonlinear damping terms, Math Methods in the Applied Sci, DOI: 10. For nonzero mass, the short-time behavior of the field variable has an inverse-sine spectrum rather than an exponential one. For general nonlinear Klein-Gordon equations with dissipation we show that any ﬁnite energy radial solution either blows up in ﬁnite time or asymptotically approaches a stationary solution in H1 L2. View Notes - NakanishiEnergy scattering for 2D nonlinear Klein-Gordon equation in the critical case Kenji Nakanishi (Kyoto Univ. Several physical realizations are outlined and the construction of an analog. pdf), Text File (. UNSTABLE GROUND STATE OF NONLINEAR KLEIN-GORDON EQUATIONS BY JALAL SHATAH Abstract. We present exact solutions to the nonlinear Klein–Gordon equation in (3 + 1) dimensions. LONG TIME DYNAMICS FOR DAMPED KLEIN-GORDON EQUATIONS N. 1) into an equivalent system, which inherits a quadratization en-ergy, by utilizing the idea of the SAV approach. Sine-Gordon Equation. strongly nonlinear evolution equations. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. BELLOR´IN ,L. 0071 A Decomposition Method for Solving the Nonlinear KleinGordon Equation E. Marcati , S. Japan, 58 (2006), 379-400. The first one to be studied here was the linear equation, Utt - Uzz + U = 0, where U is a real Klein-Gordon field. In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one-dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. d 2N/; (1-1) where fWR!R is a given function. We consider a finite element method for the nonhomogeneous second order wave equation which is formulated in terms of continuous approximation functions in both space and time, thereby giving a unified treatment of the spatial and temporal discretizations. 2, 345{35 27. Show more. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. NASA Astrophysics Data System (ADS) Bengochea, G. Kharibegashvili and B. In this paper we give sufficient conditions for the stability of the standing waves of least energy for nonlinear Klein-Gordon equations. SCHLAG Abstract. The short-time evolution of a class of nonlinear Klein-Gordon systems is studied. We prove the existence of infinitely many pairs (psi, E), where psi is a solitary wave for the nonlinear Klein-Gordon equation and. method PDTM , for solving nonlinear partial diﬀerential equations. Hayashi, Critical nonlinear Schrödinger equations with data in homogeneous weighted $\mathbf{L}^{2}$ spaces, J. Berikelashvili and O. The Klein-Gordon equation plays an important role in mathematical physics. KUKSIN Abstract. BOBENKO and SERGEJ B. nonlinear partial differential equations included some well-known fractional Klein-Gordon equations. Shu [28] and the author [21]. Due to the Klein-Gordon equation's enormous popularity, the SGE was named so as a wordplay. We recall that (1. Keywords Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing. nonlinear Klein–Gordon equation with Dirichlet boundary condi-tion on a bounded domain, which shows the overﬂow solution as expected. In this paper, we apply two decomposition methods, the Adomian decomposition method (ADM) and a well-established iterative method, to solve time-fractional Klein-Gordon type equation. Thus, the main purpose of this paper is to present the rst step towards a general theory. DEEBA AND S. We study the initial value problem for the quadratic nonlinear Klein-Gordon equation v tt + v − v xx = λv 2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v 0 (x), v t (0, x) = v 1 (x), x ∈ R, where v 0 and v 1 are real-valued functions, λ ∈ R. GONZALEZ´ Abstract. (1) arises in many scientiﬁc applications such as solid state physics, nonlinear optics and quantum ﬁeld theory. The name of the transient version of the sine-Gordon equation is a play on words on the similar form of the Klein – Gordon equation. The solutions are localized and oscillatory and so termed oscillons. Equations and Applications 19 (2012), no. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equa-. 2 and we prove the existence of hylomorphic solitons. The equation has attracted much attention in studying solitons ([20], [21]) and condensed matter physics, in investigating. 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. of the nonlinear Klein-Gordon equation (1. In a suitable approximation it is shown that for the linear potential which is of interest in quark interactions, their confinement for example, we get solutions that mimic the Harmonic oscillator energy levels, surprisingly. Abstract We study the initial value problem for the quadratic nonlinear Klein-Gordon equation v tt + v - v xx = λv 2, t ∈ R, x ∈ R, with initial conditions v(0, x) = v 0 (x), v t (0, x) = v 1 (x), x ∈ R, where v 0 and v 1 are real-valued functions, λ ∈ R. On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on R^3 (with J. Its name is a wordplay on its similar form to the Klein-Gordon equation. We prove the existence of infinitely many pairs (psi, E), where psi is a solitary wave for the nonlinear Klein-Gordon equation and. Comparing the methodology with some other known techniques shows that the present approach. Scattering and Blow-Up for the Focusing Nonlinear Klein-Gordon Equation with Complex Valued Data Below the Standing Waves Speaker: Takahisa Inui, Kyoto University Location: Warren Weaver Hall 1302. No complete scattering theory has to this day been developed for such equations ; the required Strichartz estimates are still. A Variational Iteration Method for Solving the Nonlinear Klein-Gordon Equation B. Nonlinear Klein-Gordon equation. Many nonlinear Klein-Gordon equations have been studied numerically, and in a few cases, analytical solutions have been found. The first one to be studied here was the linear equation, Utt - Uzz + U = 0, where U is a real Klein-Gordon field. Second-Order Nonlinear Hyperbolic Partial Differential Equations - Exact Solutions Keywords hyperbolic, nonlinear, partial, differential, equations, Klein--Gordon, wave, sine-Gordon, second-order, equation, exact solutions. Pasquali 1Universit a dell'Aquila 2Universit a degli studi di Milano. This modiﬁcation is called the He-multiple-scales method 2. Exact solutions of nonlinear evolution equations (NLEEs) are very crucial to realize the obscurity of many physical phenomena in mathematical science. Marangell, and R. wave equation (where the relevant nonlinear object is the stationary function W whichsolves 1+4W+ W =(d 2) = 0). A numerical method based on collocation points is developed to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. Klainerman, S 1985, ' Global existence of small amplitude solutions to nonlinear klein‐gordon equations in four space‐time dimensions ', Communications on Pure and Applied Mathematics, vol. Math is a program designed to calculate roots of nonlinear equation, roots of system of linear equations, differentiate, integrate, approximate using LMS, matrix calculation using numerical methods for calculating results, estimated errors and much more. 38 (1985), 631-641. Batiha Department of Mathematics and Basic Sciences, Faculty of Science Philadelphia University, Amman, Jordan Abstract: In this paper, variational iteration method (VIM) is presented as an alternative method for solving the nonlinear Klein-Gordon equation. This chapter deals with the theory and applications of nonlinear Klein-Gordon (KG) and Sine-Gordon (SG) equations. In this paper, in order to extend the lattice Boltzmann method to deal with more nonlinear equations, a one- dimensional (1D) lattice Boltzmann scheme with an amending function for the nonlinear Klein-Gordon equation is proposed. This equation is the. A numerical method based on collocation points was developed in ]7[ to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. Keywords Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing. The equation is analyzed to determine nonlinear pulse and periodic waveforms, and the stability of these waveforms is investigated using a recently developed "averaged Lagrangian" technique. of the nonlinear Klein-Gordon equation (1. JONES, ROBERT MARANGELL, PETER D. The Klein‐Gordon (KG) equation is an extension of the linear wave equation with additional linear and/or nonlinear terms. We consider a modified Klein-Gordon equation that arises at ultra high energies. UROOSA2,x Abstract. The above-mentioned equations are decomposed into a system of partial differential. For general nonlinear Klein-Gordon equations with dissipation we show that any ﬁnite energy radial solution either blows up in ﬁnite time or asymptotically approaches a stationary solution in H1 L2. The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space H s = H s (R n) with s ≥ n/2. This is a demonstration of how the Python module shenfun can be used to solve the time-dependent, nonlinear Klein-Gordon equation, in a triply periodic domain. This chapter deals with the Nonlinear Klein-Gordon Equation (NKG). Search for more papers by this author. Alagesan, T, Chung, Y & Kaliyaperumal, N 2003, ' Painlevé analysis of N-coupled nonlinear Klein-Gordon equations ' Journal of the Physical Society of Japan, vol. , 419 (2014), 1214-1234. Then the functions. The main theme of this paper is the construction of symmetric and arbitrarily high-order time-stepping numerical methods for the nonlinear Klein{Gordon equa-tion and the analysis of their. In the compact case, we establish existence of traveling wave solutions via energy minimization methods and prove that at least compact isotropic manifolds have genuinely traveling waves. The method of solution presented here first consists of reducing nonlinear partial differential equations to ordinary differential equations (ODE's) by introducing symmetry variables and then seeking exact solutions for more tractable. Read "An integral equation formalism for solving the nonlinear Klein-Gordon equation, Applied Mathematics and Computation" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. nonlinear partial differential equations included some well-known fractional Klein-Gordon equations. New York J. 2, 345{35 27. The phase portraits of the solutions are also given. Bilinear Estimates and applications to global well-posedness for the Dirac-Klein-Gordon equation, J. of the nonlinear Klein-Gordon equation (1. Taking the nonlinear Klein-Gordon equation for example, we derive the concrete energy-preserving scheme and demonstrate its high e ciency through numerical experiments. Motet-Bachelot [20], W. Mikael Mortensen (mikaem at math. [1] Scattering and the Levandosky-Strauss conjecture for fourth order nonlinear wave equations, J. Maleknejad1, M. The nonlinear coupled Klein-Gordon equation was first studied by Alagesan et al. Nosrati Sahlan 2, and A. On the almost sure global well-posedness of energy sub-critical nonlinear wave equations on R^3 (with J. The Klein-Gordon equation is the ﬁrst relativistic equation in quantum. We compare these methods and discuss the convergence of them. Ask Question Asked 4 years, 6 months ago. A Novel Method to Solve Nonlinear Klein-Gordon Equation Arising in Quantum Field Theory Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods. linear ﬁelds on the Schwarzschild geometry : the non linear Dirac equation, Yang-Mills ﬁelds and non linear Klein-Gordon ﬁelds were studied respectively by A. Both Klein and Fock used Kaluza and Klein's method [6]. The Klein–Gordon equation frequently arises in various fields of scientific applications such as solid state physics, nonlinear optics, quantum field theory and relativistic quantum mechanics. Baoxiang, On existence and scattering for critical and subcritical non-linear Klein-Gordon equations in Hs, Nonlinear Anal. The Klein-Gordon equation plays an important role in mathematical physics. persistence of regularity for nonlinear Klein-Gordon equation. In this work, we have applied two variables -expansion method to the Burgers-Huxley and the Klein-Gordon nonlinear evolution equations to check the effectiveness of this method and to obtain further new form exact traveling wave solutions. Kharibegashvili and B. Considering a nonlinear evolution equation con-sists of a linear part Lu, nonlinear part Nu, and a free term g(= g(x,t)) represented as Lu. The initial value problem for the nonlinear Klein-Gordon equation with various cubic nonlinearities depending on v,v t,v x,v xx,v tx and having a suitable nonresonance structure was studied in 4-6 , where small solutions were found in the neighborhood of the free solutions when the initial data are small and regular and decay rapidly at. Special emphasis is given to various methods of solutions of these equations. In the Appendices Nonlinear Klein Gordon equations with solitary (travelling) wave solutions are reviewed and solved for particular cases. We use the. This is due to the fact that the solution becomes highly-oscillatory in time in this regime which causes the breakdown of classical integration schemes. nonlinear Klein-Gordon equation. The linear Klein-Gordon equation occurs as a modification of the linear Schrödinger equation that is consistent with special relativity, see for example Landau or Grenier. Abstract— Reduced differential transform method (RDTM) is implemented for solving the linear and nonlinear Klein Gordon equations. Courant Institute. For nonzero mass, the short-time behavior of the field variable has an inverse-sine spectrum rather than an exponential one. The homotopy perturbation method (HPM) has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation which can be used as a numerical algorithm []. The name of the transient version of the sine-Gordon equation is a play on words on the similar form of the Klein - Gordon equation. Sinibaldi Communicated by Giovanni Gallavotti, received April 17, 2009. whether it is adapt to other coupled bilinear equation of this type or not is still need further study. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement 14 : 30 -- 15 : 20: Jun-ichi Segata (Tohoku University) Modified scattering for the Klein-Gordon equation with the critical nonlinearity in two and three dimensions 15 : 40 -- 16 : 30: Makoto Nakamura (Yamagata University). The Klein‐Gordon (KG) equation is an extension of the linear wave equation with additional linear and/or nonlinear terms. The Klein-Gordon equation plays an important role in mathematical physics. 3 Numerical treatment of coupled nonlinear hyperbolic Klein-Gordon equations 249 Legendre, Chebyshev and Gegenbauer polynomials [44–46]. Nosrati Sahlan 2, and A. the global behaviour of solutions to the radial cubic non-linear Klein-Gordon equation in three dimensions in the energy space H1 ⇥L2. , Differential and Integral Equations, 2017. 1), it is better to start from (1. Some results on the nonlinear Klein-Gordon-Maxwell equations Alessio Pomponio Dipartimento di Matematica, Politecnico di Bari, Italy Granada, Spain, 2011. The method is applied to some test examples and the numerical results are compared with the exact solutions. Keywords Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing. Advances in Difference Equations Solving the fractional nonlinear Klein-Gordon equation by means of the homotopy analysis method Muhammet Kurulay In this paper, the homotopy analysis method is applied to obtain the solution of nonlinear fractional partial differential equations. of the modulated cubic nonlinear Klein‐Gordon equation is the classical wave map for the limit wave function and the typical linear wave equation for the associated phase function. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrodinger equations. Therefore, we extend this method to solve the time-fractional Klein-Gordon-type equation. We describe the proof of scattering for certain global solutions with energies less than that of the ground state, Q. We construct new analytical solutions with a small parameter which is effectively obtained by the proposed method. CONCLUSION AND DISCUSSION In this paper, the procedure introduced by Nakamura is probably applicable to a class of nonlinear evolution equation. Keywords Double Laplace transform, nverse double Laplace transform, I ative method, Iter Nonlinear Klein-Gordon equation 1. It is used to model many different phenomena, including the propagation of dislocations in crystals and the behaviour of elementary particles. Author of [27] investigated three different ﬁnite differ-ence approximations of the initial nonlinear Klein-Gordon equa-tion, showed they are directly related to symplectic mappings and. New York J. We solve the problem for both Dirichlet and Neumann boundary conditions. The solutions are related to soliton solutions of the sine-Gordon equation. nonlinear Klein-Gordon equation. Math is also capable of drawing standard functions using graph dialog window to help with. for the nonlinear Klein–Gordon equation (NLKG): uR †uCu Df0. Several physical realizations are outlined and the construction of an analog. Two of the schemes, the Perring--Skyrme and Ablowitz--Kruskal--Ladik, are long established, and the third is a new, higher order accurate scheme. Theory, Methods &. 2, 345{35 27. u-bordeaux1. Several physical realizations are outlined and the construction of an analog. Conclusions The main objective of this work is to obtain a solution for linear and non lin-ear Klein-Gordon partial diﬀerential equations. Bonanno, and E. Wang Time quasi-periodic solutions to the nonlinear Klein-Gordon equations For NLKG, which is a second-order in time equation, Multiscale Analysis is, so far, theonlymethod that works. Bournaveas and T. The Klein-Gordon equation frequently arises in various fields of scientific applications such as solid state physics, nonlinear optics, quantum field theory and relativistic quantum mechanics. Fig:4:3 : The exact solution of linear inhomogeneous Klein-Gordon equation. We discuss relations between the nonlinear Klein-Gordon equation and the nonlinear Schr¨odinger equation in view of the global wellposedness in the energy space and L2. For the defocusing nonlinear Schr odinger equation, the semiclassical limit for initial data with Sobolev regularity in short time has been studied by Gre-nier [5]. Bilinear Estimates and applications to global well-posedness for the Dirac-Klein-Gordon equation, J. Vortex rings, spirals, and other solutions are demonstrated. 3042 (in press). The Klein-Gordon equation is the name given to the equation of motion of a quantum. nonlinear Klein-Gordon equations with nonlinear damping terms, Dynamical Syst, 2013, 28(2): 287-298. Its name is a wordplay on its similar form to the Klein-Gordon equation. The equation, as well as several solution techniques, were known in the 19th century, but the equation grew greatly in importance when it was realized that it led to solutions ("kink" and "antikink") with the collisional properties of solitons (Perring and Skyrme 1962. In particular the Payne-Sattinger dichotomy for solutions with energy less than that of the ground state, the Bates and Jones approach to the construction of Invariant Manifolds for semilinear partial di↵erential equations and the recent. Differential Equations & Applications Volume 4, Number 3 (2012), 445-457 doi:10. for some semi-linear elliptic equations in R u. whether it is adapt to other coupled bilinear equation of this type or not is still need further study. A numerical method is developed to solve the nonlinear one-dimensional Klein–Gordon equation by using the cubic B-spline collocation method on the uniform mesh points. This class includes the Q-balls which are spherically symmetric solutions of the nonlinear Klein-Gordon equation (NKG) as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equation and in gauge theories. We analyze three finite difference approximations of the nonlinear Klein--Gordon equation and show that they are directly related to symplectic mappings. Hydrodynamic limits of the nonlinear Klein-Gordon equation. Search for more papers by this author. Hylomorphic solitons in the nonlinear Klein-Gordon equation J. the Local fractional variational iteration method (LFVIM) [45], wave equations on the Cantor sets (WECSs) [3], local fractional Laplace equation (LFLE) [42] and Newtonian mechanics (NM) on fractals subset of real-line [17]. Available online. Introduction. Lindblad and A. The nonrelativistic-semiclassical limit, ~ !0;c= ~ !1for some >0, of the modulated defocusing cubic nonlinear Klein-Gordon equation is the classical wave map for the limit wave function and typical linear wave equa-tion for the associated phase function. For nonzero mass, the short-time behavior of the field variable has an inverse-sine spectrum rather than an exponential one. 2018-05-01. Math is a program designed to calculate roots of nonlinear equation, roots of system of linear equations, differentiate, integrate, approximate using LMS, matrix calculation using numerical methods for calculating results, estimated errors and much more. The Klein–Gordon equation frequently arises in various fields of scientific applications such as solid state physics, nonlinear optics, quantum field theory and relativistic quantum mechanics. title = "The method of lines and the nonlinear Klein-Gordon equation",. We present exact solutions to the nonlinear Klein–Gordon equation in (3 + 1) dimensions. [CrossRef] [MathSciNet] [Google Scholar] C. We now briefly describe how this paper is arranged. 34 (2005) p. It would be very useful to carry out a systematic study on J-GL-C method with general indexes (θ,ϑ>−1). Conclusions The main objective of this work is to obtain a solution for linear and non lin-ear Klein-Gordon partial diﬀerential equations. 4, 1841-1857. We study the existence of cylindrically symmetric electro-magneto-static solitary waves for a system of a nonlinear Klein-Gordon equation coupled with Maxwell's equations in presence of a positive. For the defocusing nonlinear Schr odinger equation, the semiclassical limit for initial data with Sobolev regularity in short time has been studied by Gre-nier [5]. Jokhadze and S. The initial value problem for the nonlinear Klein-Gordon equation with various cubic nonlinearities depending on v,v t,v x,v xx,v tx and having a suitable nonresonance structure was studied in 4-6 , where small solutions were found in the neighborhood of the free solutions when the initial data are small and regular and decay rapidly at. Lindblad and J. In this paper, we want to understand the dynamics of large, quantized energy. Multi-solitons for nonlinear Klein-Gordon equations - Volume 2 - RAPHAËL CÔTE, CLAUDIO MUÑOZ Skip to main content Accessibility help We use cookies to distinguish you from other users and to provide you with a better experience on our websites. The FRDTM has several advantages for dealing directly with suggested equations; it needs a few iterations to get high accuracy, it is very simple for obtaining analytical-approximate solutions in rapidly convergent formulas, it. over a region. The cubic nonlinear Klein-Gordon equation is a wave equation important for many scientific applications such as solid state physics, nonlinear optics and quantum field theory. Using the method of normal forms of Shatah ["Normal forms and quadratic. 1), it is better to start from (1. Solve a coupled stationary nonlinear sine-Gordon equation. This is a survey on recent results on global attractors of the Hamilton nonlinear wave equations in an in nite space. A numerical method based on collocation points was developed in ]7[ to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. In this paper, we want to understand the dynamics of large, quantized energy. Since then, this equation has been known as the Klein{Gordon equation. The nonrelativistic-semiclassical limit, ~ !0;c= ~ !1for some >0, of the modulated defocusing cubic nonlinear Klein-Gordon equation is the classical wave map for the limit wave function and typical linear wave equa-tion for the associated phase function. Special emphasis is given to various methods of solutions of these equations. (2005) gave arguments for the existence of exact travelling wave solutions of a perturbed sine Gordon equation on the real line or on the circle and classified them [7 ]. UROOSA2,x Abstract. Yue Liu, Masahito Ohta, Grozdena Todorova, Strong instability of solitary waves for nonlinear Klein–Gordon equations and generalized Boussinesq equations Gilles Lebeau, Perte de régularité pour les équations d’ondes sur-critiques. Segata, Modified scattering for the quadratic nonlinear Klein-Gordon equation in two space dimensions, preprint, 2016. U(1)-invariant nonlinear Klein–Gordon equation in one dimension, with the non- linearity concentrated at a single point: each ﬁnite energy solution converges as t →±∞to the set of all “nonlinear eigenfunctions” of the form ψ(x)e − iωt. The Green function method combined with integral transforms is employed to solve the linear Klein-Gordon equation. We propose and analyze a multiscale time integrator Fourier pseudospectral (MTI-FP) method for solving the Klein--Gordon (KG) equation with a dimensionless parameter $0<\varepsilon\leq1$ which is i. Aug 23, 2019. whether it is adapt to other coupled bilinear equation of this type or not is still need further study. Berikelashvili and O. Lindblad and J. , Nonlinearity 19 (2006), no. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave's frequency or the wave's speed of propagation and on the. V (X, T) = In This Numerical Research, We Examine The Klein-Gordon And Sine-Gordon Equations N(X)= ᴪ ᴪ Using The Numerical Method Of Nonlinear We Use The Approximation Of Rbfs In The Methods And Compare The Obtained Results. This class includes the Q-balls which are spherically symmetric solutions of the nonlinear Klein-Gordon equation (NKG) as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equation and in gauge theories. In the last decades Iterative methods have been used for solving fractional differential equations. No complete scattering theory has to this day been developed for such equations ; the required Strichartz estimates are still. The name of the transient version of the sine-Gordon equation is a play on words on the similar form of the Klein - Gordon equation. The homotopy perturbation method (HPM) has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation which can be used as a numerical algorithm [ ]. Wavelet Galerkin Method for the Solution of Nonlinear Klein-Gordon Equations By Using B-Spline Wavelets K. SPECTRAL AND MODULATIONAL STABILITY OF PERIODIC WAVETRAINS FOR THE NONLINEAR KLEIN-GORDON EQUATION CHRISTOPHER K. In this paper we prove the instability of the ground state, i. Equations with no damping terms ( = = 0) have been considered by many authors; see [3,6,9,12,13,15,. Abstract The Klein{Gordon equation with nonlinear potential features in a large number of applications, yet its computation represents a major challenge. Hylomorphic solitons in the nonlinear Klein-Gordon equation J.