# Nonlinear Klein Gordon Equation

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. We establish the linear profile decomposition, then by using the solution of the mass-critical nonlinear Schrodinger equation to approximate the large scale profile, we can prove the scattering result by the concentration-compactness/rigidity method. equations occurring in applied mathematics is that associated with the name of Klein–Gordon. Goncharovskii, I. Abstract | PDF (389 KB) (2014) A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime. title = "Scattering threshold for the focusing nonlinear Klein-Gordon equation", abstract = "We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig and Merle for the H1 critical wave and Schr{\"o}dinger equations. 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). In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein Gordon equation with quadratic and cubic nonlinearity. As far as we know, this is the ﬁrst blow up result for the Klein-Gordon equations with arbitrarily high initial energy on the whole space Rn. Distribution Solution of Some Nonlinear PDEs Related to the Elastic Bessel Klein-Gordon Wave Operator This paper studies the existence and uniqueness of the solution to the boundary value problem for the nonlinear partial differential equation. Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations. 2 þ u2 ¼ 0, t Z0, uðx,0Þ ¼ 1 þ sinðxÞ: ð19Þ @t a @x We construct a Homotopy substituting (a = 0, b= 1 and Example 3. The nonlinear Klein-Gordon equation (KGE) models many nonlinear phenomena. Global Existence and Dispersion of Solutions to Nonlinear Klein-Gordon Equations with Potential /. Breather solutions in nonlinear Klein-Gordon equations on the necklace graph Existence theorem: Let k be an odd integer. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein–Gordon equation can be used as numerical algorithm. A non-local version of these equations are properly described by the fractional version of them. nonlinear Klein-Gordon equation which appears in nonlinear optics, plasma physics, and fluid me- chanics. Biswas⁄ and R. This chapter deals with the Nonlinear Klein-Gordon Equation (NKG). Dispersive partial differential equation. (2017) Arbitrarily high-order time-stepping schemes based on the operator spectrum theory for high-dimensional nonlinear Klein–Gordon equations. for a "cosmic superfluid". Nonlinear Partial Differential Equations for Scientists and Engineers, Third. We analyze three finite difference approximations of the nonlinear Klein--Gordon equation and show that they are directly related to symplectic mappings. The results reveal that the method is very effective, simple, and convenient. / Shatah, Jalal. 1) is the Euler-Lagrange equation with respect to the La-grangian density L NLKG. The time fractional derivative is described in the Caputo sense. In the last decades Iterative methods have been used for solving fractional differential equations. 2) by Fourier pseudospectral method, and then solving the ordin-ary diﬀerential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between diﬀerent time intervals or applying Crank–Nicolson/leap-frog for linear/nonlinear. BELLOR´IN ,L. The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear generalization of the celebrated telegraph equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schrodinger equation, and the power-law diffusion (porousmedia) equation. 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. Hylomorphic solitons in the nonlinear Klein-Gordon equation J. Introduction One of the fundamental partial differential equations is the nonlinear Klein- Gordon equation 2. : Variational Iteration Method for Solving Coupled Schrödinger-Klein-Gordon Equation errors is avoided. Then the functions. Distribution Solution of Some Nonlinear PDEs Related to the Elastic Bessel Klein-Gordon Wave Operator This paper studies the existence and uniqueness of the solution to the boundary value problem for the nonlinear partial differential equation. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation, Journal of Computational Physics 28 (1978) 271-278. The nonlinear Schrodinger equation actually had nothing to do with quantum mechanics. The Klein-Gordon (KG) equation is an extension of the linear wave equation with additional linear and/or nonlinear terms. nonlinear Klein-Gordon equation which appears in nonlinear optics, plasma physics, and fluid me- chanics. Differential Transformation Method The differential transformation method is applied mainly to solve initial value problems. 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. Exact solutions of sine Gordon and multiple sine Gordon equations are constructed in terms of solutions of a linear base equation, the Klein Gordon equation and also in terms of nonlinear base equations where the nonlinearity is polynomial in the dependent variable. Suppose w =w(x,t) is a solution of the nonlinear Klein–Gordon equation. (2014) A note on the instability of a focusing nonlinear damped wave equation. A simple method for generating the exact solutions of the nonlinear Klein-Gordon equation is proposed. Al-Luhaibi Abstract In this paper, Sumudu decomposition method is applied to solve various forms of linear and nonlinear Klein-Gordon equations. Ding and L. w1=w(±x+C1,±t+C2), w2=w(xcoshﬂ +tsinhﬂ, tcoshﬂ +xsinhﬂ), where C1, C2, and ﬂ are arbitrary constants, are also solutions of the equation (the plus or minus signs in w1are chosen arbitrarily). maybe try Lane-Emden (or Lane-Emden-Fowler) (actually i not sure of exact name either) or even try 'semilinear elliptic' and you will get a million hits $\endgroup$ - Math604 Nov 3 '18 at 14:33. Motet-Bachelot [20], W. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov. Note: Citations are based on reference standards. A simple nonlinear partial differential equation suitable for study by an undergraduate in applied science is discussed. The local energy also decays. This site uses cookies. The Nonlinear Klein-Gordon Equation and Average Variational Principle. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equation is recovered correctly from the lattice Boltzmann equation. Breather solutions in nonlinear Klein-Gordon equations on the necklace graph Existence theorem: Let k be an odd integer. We solve the global Cauchy problem for large data with minimum regularity. Below the energy of the ground state, we show that the energy space can be split into two complements sets leading to two opposite. The equation has attracted much at-. Recently Jafari [34] used ADM and DJM to solve time–fractional Klein-Gordon type equation. To illustrate the reliability of the method, some examples are presented. In this paper, we study the convergence of solutions in the limit from the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation. 2) by Fourier pseudospectral method, and then solving the ordin-ary diﬀerential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between diﬀerent time intervals or applying Crank–Nicolson/leap-frog for linear/nonlinear. The Klein–Gordon equation plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [2]. The solutions obtained depend on two arbitrary functions and are in the form of running waves. In paper ]8[ the nonlinear one-dimensional Klein-Gordon equation was solved with the help of the. The average variational principle is used to derive the nonlinear Klein-Gordon equation. A Nonlinear Klein-Gordon Equation: American Journal of Physics: Vol 37, No 1. However, to prove asymptotic stability of solutions of non-linear equations, one. We recall that (1. The modified simple equation (MSE) method is especially effective and highly proficient mathematical instrument to obtaining exact traveling wave solutions to NLEEs arising in science, engineering and mathematical physics. Jalal Shatah. For nonzero mass, the short-time behavior of the field variable has an inverse-sine spectrum rather than an exponential one. We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction, and under the same regularity assumptions on the initial data required for the integration of the corresponding limit system. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. The convergence and stability of the method are proved. 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. A wavelet method for a class of space-time fractional Klein–Gordon equations with constant coefficients is proposed, by combining the Haar wavelet and operational matrix together and efficaciously dispersing the coefficients. Keywords: Variational iterative method, Linear and Nonlinear Klein-Gordon equations, Lagrange multiplier 1 Introduction. Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations. This paper focuses on the nonlinear Klein-Gordon equation with dual power law nonlinearity. Essential steps in the. Math, 22 (2016) 209-227. Numerical illustrations that include non‐linear Klein-Gordon equations and non‐linear partial differential equations are investigated to show the pertinent. d'Avenia, Pietro; Pisani, Lorenzo. Lindblad and A. Existing literature on the so-called NKG family and other NKG-type equations are first discussed, including interrelations, wherever possible, and existing particular solutions. Numerical solution of the nonlinear Klein–Gordon equation Numerical solution of the nonlinear Klein–Gordon equation Rashidinia, J. In this paper, we study the convergence of solutions in the limit from the Klein–Gordon–Zakharov system to the nonlinear Schrödinger equation. Research Papers and Preprints. Context Differential geometry. It has application, for example, to relativistic fields (waves). 2, 274-290 Efﬁcient high-order ﬁnite difference methods for nonlinear Klein-Gordon equations. We compare the properties of four explicit finite difference schemes used to integrate the nonlinear Klein-Gordon equation @f"t"[email protected]"x"x+f(@f)=0. Journal of Computational Physics 340 , 243-275. 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. Under suitable smoothness and decay assumptions on the potential and a genericity assumption on the nonlinearity, we prove that all small energy solutions are asymptotically free. Wavelet Galerkin Method for the Solution of Nonlinear Klein-Gordon Equations By Using B-Spline Wavelets K. The technique is a combined form of the. Keywords-Nonlinear optics, Plasma physics, Fluid mechanics, Decomposition method. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. The Klein–Gordon equation plays a significant role in many scientific applications such as solid state physics, nonlinear optics and quantum field theory [2]. ; Tsutsumi, Y. The interaction of solitons with external potentials in nonlinear Klein-Gordon field theory is investigated using an improved model. The nonlinear Klein–Gordon equation appears in many types of nonlinearities. (NKG) reduces to the Klein-Gordon equation. 4) r7!r ieA (1. Let us consider the Klein--Gordon equation. 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. Comparisons are made between the Adomian. 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. However, the arithmetic mean (AM) method has not been studied on the Klein-Gordon equation. The Legendre collocation method is used to reduce. (solid) and the damped nonlinear Klein-Gordon equation with ﬂ = 0 : 0001 (dashed), ﬂ = 0 : 001 (dash-dotted) and ﬂ = 0 : 005, nonlinear term G 0 ( u ) = u 7 , initial data ` ( r ) = h ( r ) and ˆ ( r ) = h 0 ( r ) + h ( r ) =r ,. 1969-01-01 00:00:00 A simple nonlinear partial differential equation suitable for study by an undergraduate in applied science is discussed. Antony Vijesh [ + - ] Author and Article Information. (NKG) reduces to the Klein-Gordon equation. We note that our proof is very simple. By utilizing the properties of interpolation on the element, high accuracy analysis and derivative delivery techniques with respect to time t instead of the traditional Ritz projection operator, which is an indispensable tool in the traditional finite element analysis, the. yDicle University, Department of Mathematics, 21280 Diyarbak‹r, Turkey 315. 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. Sine-Gordon Equation. 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. We consider the initial value problem for systems of nonlinear Klein–Gordon equations with quadratic nonlinearities. In: Communications on Pure and Applied Mathematics, Vol. Biswas⁄ and R. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. nonlinear Klein-Gordon equations with nonlinear damping terms, Dynamical Syst, 2013, 28(2): 287-298. L]~TTER]~ AL NUOVO CIMENTO VOL. GUERRERO AND J. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov. Harish Kumar and V. With the Taylor and Chapman-Enskog expansion, the nonlinear Klein-Gordon equa-. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Learn new and interesting things. An application of one of the solutions for the (2 + 1) – dimensional sine-Gordon equation is proposed. A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. (NKG) reduces to the Klein-Gordon equation. Breather solutions in nonlinear Klein-Gordon equations on the necklace graph Existence theorem: Let k be an odd integer. It would be very useful to carry out a systematic study on J-GL-C method with general indexes (θ,ϑ>−1). (with Xuwen Chen ) On the rigorous derivation of the 2D cubic nonlinear Schroedinger equation from 3D quantum many-body dynamics, Archive for Rational Mechanics & Analysis 210 (2013) pp. , Hokkaido Mathematical Journal, 2008 New distributional travelling waves for the nonlinear Klein-Gordon equation Paiva, A. Lindblad and A. GUERRERO AND J. We present an approximate formula of the fractional derivative. However, the arithmetic mean (AM) method has not been studied on the Klein-Gordon equation. Bellazzini, V. We study the asymptotic behavior of the semilinear Klein-Gordon equation with nonlinearity of fractional order. Math, 22 (2016) 209-227. Nonlinear Klein–Gordon equation. Our computations show that (i) the solutions remain bounded as t → ∞, (ii) the amplitude decreases as p increases, and (iii) the number of oscillations increases as p increases. Shirokov Research output : Contribution to journal › Article. A simple nonlinear partial differential equation suitable for study by an undergraduate in applied science is discussed. The short-time evolution of a class of nonlinear Klein-Gordon systems is studied. A numerical method based on collocation points is developed to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. fr/hal-01078696v2 Submitted on 4 Jan 2016 HAL is a multi-disciplinary open access archive for the deposit and. Vazquez, Numerical solution of a nonlinear Klein–Gordon equation, Journal of Computational Physics 28 (1978) 271–278. 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. Pasquali 1Universit a dell'Aquila 2Universit a degli studi di Milano. Introduction One of the fundamental partial differential equations is the nonlinear Klein– Gordon equation 2. In this paper, we investigate two types of nonlocal soliton equations with the parity-time (PT) symmetry, namely, a two dimensional nonlocal nonlinear Schrödinger (NLS) equation and a coupled nonlocal Klein-Gordon equation. 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. of the nonlinear Klein-Gordon equation (1. , 419 (2014), 1214-1234. 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. Klein Gordon is related to a wave equation. Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations. Communications in Theoretical Physics, Vol. BELLOR´IN ,L. of the nonlinear Klein-Gordon equation (1. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation can be used as numerical algorithm. This equation is associated with the relativistic relationship between the energy £ and the momentum p of a particle £ 2 = p2c2 + m2c4, where m is the mass of the particle and c is the speed of light. Finally, solutions with spherical symmetry, of nonlinear Klein Gordon equations are given. (NKG) reduces to the Klein-Gordon equation. Shirokov Research output : Contribution to journal › Article. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. The phase portraits of the solutions are also given. Wazwaz used ADM to solve boundary value problem [30], Kadomtsev-Petviashvili equation [31], partial differential equations [32] and diffusion equations [33]. 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. View Notes - NakanishiEnergy scattering for 2D nonlinear Klein-Gordon equation in the critical case Kenji Nakanishi (Kyoto Univ. It is proved that most small-amplitude finite-gap solutions of the SG equation, which satisfy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are “Lipschitz manifolds with holes”. CONCLUSION AND DISCUSSION In this paper, the procedure introduced by Nakamura is probably applicable to a class of nonlinear evolution equation. ; Ortigueira, M. To analyse the system’s dynamics, a transformation is used which uncouples the equations into a linear differential-difference equation and a discrete nonlinear Klein–Gordon system with second-neighbour interactions. Available online. In the field of physics, the nonlinear Klein-Gordon equation plays an important role especially in the applications of quantum mechanics and condensed matter physics [1,2]. 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. Further, exact solutions of. 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. Courant Institute. It is proved that most small-amplitude finite-gap solutions of the SG equation, which satisfy either Dirichlet or Neumann boundary conditions, persist in the NKG equation and jointly form partial central manifolds, which are “Lipschitz manifolds with holes”. Feng, Philosophical Transactions A, v. (2004), and then Shang (2010) and Yusufoglu and Bekir (2008) gave further result by using the ideas of the tanh method and the general integral method. 3, 405-460. In the last part of this chapter, we show that some relativistic effects such as the space contraction, the time dilation, the Einstein equation, are consequences of the Poincarè invariance of NKG. In Section 4, we prove the convergence of the time-scaled modulated nonlinear Klein–. Our proof is based on the concavity method which was ﬁrst introduced by Levine [8], [9]. 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. 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. In this paper, we study a nonlinear Klein-Gordon equation coupled with a Maxwell equation. In paper ]8[ the nonlinear one-dimensional Klein-Gordon equation was solved with the help of the. Interesting non-linear generalization of both Schrödinger’s and Klein–Gordon’s equations have been recently advanced by Tsallis, Rego-Monteiro and Tsallis (NRT) in Nobre et al. A high-order compact scheme for the nonlinear fractional Klein-Gordon equation. It is used to model many different phenomena, including the propagation of dislocations in crystals and the behaviour of elementary particles. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. View Notes - NakanishiEnergy scattering for 2D nonlinear Klein-Gordon equation in the critical case Kenji Nakanishi (Kyoto Univ. On the Fluid Dynamical Approximation to the nonlinear Klein-Gordon Equation Chi-Kun Lin1 and Kung-Chien Wu2 Department of Applied Mathematics National Chiao Tung University Hsinchu 30010, TAIWAN We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its. to this equation beyond the trivial φ= 0 (the appendix, however, discusses a special class of exact solutions, the “kinks” or “domain-walls”). In an effort to make the book more useful for a diverse readership, updated modern examples of applications are chosen from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Three limits are determined when the light speed c →∞and → 0. 2 þ u2 ¼ 0, t Z0, uðx,0Þ ¼ 1 þ sinðxÞ: ð19Þ @t a @x We construct a Homotopy substituting (a = 0, b= 1 and Example 3. We study the nonrelativistic limit of the Cauchy problem for the nonlinear Klein-Gordon equation and prove that any finite energy solution converges to the corresponding solution of the nonlinear Schrödinger equation in the energy space, after the infinite oscillation in time is removed. 1) blows up in a ﬁnite time. Several physical realizations are outlined and the construction of an analog model is described. The Klein-Gordon Equation is invariant under a Lie group generated by three vector fields. -Expansion Method and Travelling Wave Solutions to Nonlinear Klein-Gordon Equation 227 ( 𝜔 2 − 2 𝑘 2 ) 𝑢 ′′ + 𝑢 + 𝑢 3 = 0 (8) where 𝜁 0 is an arbitrary constant, 𝑘 , 𝜔 are constants to be determined later. The Klein-Gordon equation plays an important role in mathematical physics. The interaction of solitons with external potentials in nonlinear Klein-Gordon field theory is investigated using an improved model. We generalize the Ginzburg-Landau equation in such a way that the topological defects supported by this equation present long-range interaction both. 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. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. 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. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones. 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. First, we developed the classical field theory corresponding to the non-linear q-Klein-Gordon equation, improving upon the work of Rego-Monteiro and Nobre. Introduction One of the fundamental partial differential equations is the nonlinear Klein- Gordon equation 2. We prove the existence of the scattering operator in in the neighborhood of the origin for the nonlinear Klein–Gordon equation with a power nonlinearity where μ ∈ C , n=1,2. UNSTABLE GROUND STATE OF NONLINEAR KLEIN-GORDON EQUATIONS BY JALAL SHATAH Abstract. 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. In this research, the hybrid method of differential transform and finite difference method is applied to solve the nonlinear Klein -Gordon problem. 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). 1), it is better to start from (1. (2004), and then Shang (2010) and Yusufoglu and Bekir (2008) gave further result by using the ideas of the tanh method and the general integral method. We present an e ective algorithm to solve the Linear and Nonlinear Klein-Gordon equation, which is based on the Perturbation Iteration Transform Method (PITM). native method for solving the linear and nonlinear Klein Gordon equations. By the aid of a suitable generalization of the weighted Sobolev spaces we define the weighted Sobolev spaces on the upper branch of the unit hyperboloid. Learn new and interesting things. The nonlinear Schrödinger equation. The solutions obtained depend on two arbitrary functions and are in the form of running waves. JOURNAL OF COMPUTATIONAL PHYSICS 124, 442448 (1996) ARTICLE NO. Special emphasis is given to various methods of solutions of these equations. equations occurring in applied mathematics is that associated with the name of Klein–Gordon. 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. MNIM for solving linear and nonlinear Klein-Gordon equations 2985 The exact solution of ( 16) is u(x,t)=xt (a) NIM Equation (16) is equivalent to the following integral form u = xt+ x2t4 12 + t 0 t 0 (u xx − u2)dtdt Set u0 = xt + x 2t4 12 and N(u)= t 0 t 0 (u xx − u2)dtdt. In paper ]8[ the nonlinear one-dimensional Klein-Gordon equation was solved with the help of the. By utilizing the properties of interpolation on the element, high accuracy analysis and derivative delivery techniques with respect to time t instead of the traditional Ritz projection operator, which is an indispensable tool in the traditional finite element analysis, the. The method is demonstrated by several physical models of Klein Gordon equations. The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear generalization of the celebrated telegraph equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schrodinger equation, and the power-law diffusion (porousmedia) equation. Its name is a wordplay on its similar form to the Klein-Gordon equation. Polynomial scaling and wavelet functions are rarely used in the contexts of numerical computation [2, 3]. It has application, for example, to relativistic fields (waves). 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. Jalal Shatah. 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. 1) Rego-Monteiro and Nobre use the higher-order Lagrangian procedures of Bollini and Giambiagi [4,5], while we have used the conventional Lagrangian treatment. 1969-01-01 00:00:00 A simple nonlinear partial differential equation suitable for study by an undergraduate in applied science is discussed. Ding and L. (with Xuwen Chen ) On the rigorous derivation of the 2D cubic nonlinear Schroedinger equation from 3D quantum many-body dynamics, Archive for Rational Mechanics & Analysis 210 (2013) pp. 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. NASA Astrophysics Data System (ADS) Bengochea, G. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. Nonlinear Analysis: Modelling and Control, 2015, Vol. The nonlinear Schrödinger equation. JOURNAL OF COMPUTATIONAL PHYSICS 124, 442448 (1996) ARTICLE NO. Chebyshev Wavelet Quasilinearization Scheme for Coupled Nonlinear Sine-Gordon Equations K. Solitons and periodic line waves as exact solutions of these two nonlocal. This approach leads us to consider the equation "2u tt ". In this paper, numerical solution of non‐linear Klein–Gordon equations with power law non‐linearities are obtained by the new application of He's variational iteration method. The case of Maxwell-Schrödinger type coupled equations is also considered. On the contrary, if W has a nonlinear suitable component the wave packets do not disperse, hence one can ﬁnd solitons. non-linear Klein–Gordon equations ([17], [18]) required more precise information about the decay of solutions of the corresponding linearized equations, and this is what stimulated our investigations in [1]–[3]. Strauss, L. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. A numerical method based on collocation points was developed in ]7[ to solve the nonlinear Klein-Gordon equations by using the Taylor matrix method. The primary application is to. Normal forms and quadratic nonlinear Klein‐Gordon equations. Numerical illustrations that include non‐linear Klein–Gordon equations and non‐linear partial differential equations are investigated to show the pertinent. Google Scholar [15] L. Several physical realizations are outlined and the construction of an analog. Key words: nonlinear Klein-Gordon equation, nonlinear Schr¨odinger equation, wellposed-ness, nonrelativistic limit. 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). Differential Equations & Applications Volume 4, Number 3 (2012), 445–457 doi:10. Interaction among nonresonant waves of the nonlinear Klein–Gordon equation in ordinary (three-dimensional) space is investigated, by an asymptotic perturbation method, based on Fourier expansion and spatio-temporal rescaling. 909--954, pdf. 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. Equations & Applications Volume 3, Number 4 (2011), 527–553 INTERNAL DEGREES OF FREEDOM IN PERTURBED NONLINEAR KLEIN–GORDON EQUATIONS L. 7, 3019-3092. The convergence and stability of the method are proved. Available online. ] "Large-degree asymptotics of rational Painlevé-II functions. The Klein-Gordon Equation is invariant under a Lie group generated by three vector fields. Further, exact solutions of. The dissipative effects explored here are described by an evolution equation that can be regarded as a nonlinear generalization of the celebrated telegraph equation, unifying within one single theoretical framework the nonlinear Klein-Gordon equation, a nonlinear Schrodinger equation, and the power-law diffusion (porousmedia) equation. An application of one of the solutions for the (2 + 1) – dimensional sine-Gordon equation is proposed. The Klein–Gordon equation was first considered as a quantum wave equation by Schrödinger in his search for an equation describing de Broglie waves. Klein-Gordon equation with a special type of nonlocal nonlinearity in commutative homogeneous spaces with invariant metric A. We present an approximate formula of the fractional derivative. Further, exact solutions of. It consists of both original articles and survey papers covering a wide scope of research topics in partial differential equations and their applications. Then the functions. We compare these methods and discuss the convergence of them. The nonlinear Klein-Gordon equation (KGE) models many nonlinear phenomena. The rest of the thesis is organized as follows: In chapter 2, we derive the hydrodynamic structure of the modulated non-linear Klein-Gordon equation and discuss their relation to the compressible. 1), it is better to start from (1. In this paper, the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities are studied. 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. The bifurcation analysis is subsequently carried out that gives an additional set of solutions. This video is unavailable. Background This work is based on the equations of electromag-. Numerical illustrations that include non‐linear Klein-Gordon equations and non‐linear partial differential equations are investigated to show the pertinent. On the Fluid Dynamical Approximation to the nonlinear Klein-Gordon Equation Chi-Kun Lin1 and Kung-Chien Wu2 Department of Applied Mathematics National Chiao Tung University Hsinchu 30010, TAIWAN We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its. A NUMERICAL METHOD FOR COMPUTING RADIALLY SYMMETRIC SOLUTIONS OF A DISSIPATIVE NONLINEAR MODIFIED KLEIN-GORDON EQUATION A Thesis Submitted to the Graduate Faculty of the University of New Orleans in partial fulﬂllment of the requirements for the degree of Master of Science in The Department of Physics by Jorge Eduardo Mac¶‡as D¶‡az. Finally, the numerical experiments, including a nonlinear Schrödinger equation, a sine-Gordon equation, a nonlinear Klein-Gordon equation, and the well-known Fermi-Pasta-Ulam problem, are implemented in comparison with the corresponding symplectic RK methods and the prominent numerical results definitely coincide with the theories and. Some comments will be made about the relativistic and non relativistic Klein-Gordon equations. In the first part we construct a model which describes solitary waves of the nonlinear Klein-Gordon equation interacting with the electromagnetic field. BELLOR´IN ,L. The solutions are localized and oscillatory and so termed oscillons. Nonlinear Analysis: Modelling and Control, 2015, Vol. The resonance generates a solitary packets of waves. 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. In this paper, we propose a scheme for numerical approximation of solutions of the one-dimensional nonlinear KGE. A technique to approximate the solutions of nonlinear Klein–Gordon equation and Klein–Gordon-Schrödinger equations is presented separately. , Differential and Integral Equations, 2017. Further, exact solutions of nonlinear generalizations of the Schrodinger equation and of additional nonlinear general- izations of the Klein Gordon equation are constructed in terms of solutions of linear base equations. Small data scattering for nonlinear Schrödinger equations, nonlinear wave equations, nonlinear Klein-Gordon equations with power type nonlinearities is studied in the scheme of Sobolev spaces on the whole space $\mathbb{R}^n$ with. Ibrahim, M. Nevertheless, it is not suit for the nonlinear Klein-Gordon equation. In this paper, the Homotopy perturbation method has been successively applied for finding approximate analytical solutions of the fractional nonlinear Klein-Gordon equation can be used as numerical algorithm. 2) by Fourier pseudospectral method, and then solving the ordin-ary diﬀerential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between diﬀerent time intervals or applying Crank–Nicolson/leap-frog for linear/nonlinear. Introduction Nonlinear systems of. ) This is joint work with S. 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. KHALID 1, M. It was in short that Strichartz-type space-time estimates for the Schr¨odinger equation with the L2 scaling follow from the same estimates for the Klein-Gordon equation with H1 initial data. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein-Gordon equation with quadratic and cubic nonlinearity. nonlinear Klein–Gordon equation and discuss their relation to the compressible and incompressible Euler equations. This site uses cookies. MNIM for solving linear and nonlinear Klein-Gordon equations 2985 The exact solution of ( 16) is u(x,t)=xt (a) NIM Equation (16) is equivalent to the following integral form u = xt+ x2t4 12 + t 0 t 0 (u xx − u2)dtdt Set u0 = xt + x 2t4 12 and N(u)= t 0 t 0 (u xx − u2)dtdt. Exact Solutions > Nonlinear Partial Differential Equations > Second-Order Hyperbolic Partial Differential Equations > Nonlinear Klein-Gordon Equation 7. To analyse the system's dynamics, a transformation is used which uncouples the equations into a linear differential-difference equation and a discrete nonlinear Klein-Gordon system with second-neighbour interactions. In this paper we examine the Klein-Gordon equation, using the Laplace-Adomian de- composition method and He-Laplace method to get the exact solution. A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. In Section 4, we prove the convergence of the time-scaled modulated nonlinear Klein-. Recently Jafari [34] used ADM and DJM to solve time–fractional Klein-Gordon type equation.