Analysis of non-linear Klein–Gordon equations using Lie symmetry

Journal of Mechanics of Continua and Mathematical Sciences , 2019

In this article, we investigate the traveling wave solutions to the Klein-Gordon equation in (2+1)-dimension with special types of nonlinearity. The types include quadratic, cubic and polynomial nonlinearity. The Klein-Gordon equation assumes a significant role in numerous types of scientific investigation such as in quantum field theory, nonlinear optics, nuclear physics, magnetic field etc. To investigate the aimed traveling wave solutions, we execute the (G'?G)-expansion method. The attained solutions are in the form of hyperbolic, trigonometric and rational functions. The results acknowledged that the applied method is very efficient and suitable for discovering differential equations with various types of nonlinearity considered in optics and quantum field theory. The solutions of the Klein-Gordon equation with quadratic, cubic, and polynomials nonlinearity play a significant role in many scientific measures notably optics and quantum field theory.

Communications in Theoretical Physics, 2018

We obtain solutions of the nonlinear Klein-Gordon equation using a novel algebraic operational method combined with the Adomian polynomial expansion of nonlinear functions. Our operational method does not use any integral transforms nor integration processes. We illustrate the application of our method by solving several examples and present numerical results that show the accuracy of the truncated series approximations to the solutions.

Pramana, 2013

In this letter, the Lie point symmetries of a class of Gordon-type wave equations that arise in the Milne space-time are presented and analysed. Using the Lie point symmetries, it is showed how to reduce Gordon-type wave equations using the method of invariants, and to obtain exact solutions corresponding to some boundary values. The Noether point symmetries and conservation laws are obtained for the Klein-Gordon equation in one case. Finally, the existence of higherorder variational symmetries of a projection of the Klein-Gordon equation is investigated using the multiplier approach.

2002

In this paper we prove the existence of infinitely many radially symmetric standing waves in equilibrium with their own electro-magnetic field. The interaction is described by means of the minimal coupling rule; on the other hand the Lagrangian density for the electro-magnetic field is the second order approximation of the Born-Infeld Lagrangian density. * Mathematics Subject Classifications: 58E15, 35J50, 35Q40, 35Q60.

2006

The s-wave Klein-Gordon equation for the bound states is separated in two parts to see clearly the relativistic contributions to the solution in the non-relativistic limit. The reliability of the model is discussed with the specifically chosen two examples.

Nonlinear Dynamics, 2013

This paper obtains the conservation laws of the Klein-Gordon equation with power law and log law nonlinearities. The multiplier approach with Lie symmetry analysis is employed to obtain the conserved densities. The 1-soliton solutions are subsequently used to compute the conserved quantities from the conserved densities. Later the perturbation terms are added and the conservation laws of the perturbed Klein-Gordon equation are studied.

In this paper we present a complete classification of nonlinear differential equations of the form u xy = f (u, u x , u y) reduced to the Klein-Gordon equation v xy = F (v) by differential substitutions v = ϕ(u, u x , u y), such that ϕ ux ϕ uy = 0.

Applicable Analysis, 2003

In this work, we study the multiplicity of solutions for a stationary nonhomogeneous problem associated to the nonlinear one-dimensional Klein-Gordon Equation. We prove that the existence of positive solutions is equivalent to the solvability of a scalar equation 2F(M) = 1, where F is a real function depending on V. Moreover, we prove some existence and multiplicity results for the

International Journal of Theoretical Physics, 2009

A model with interparticle inharmonic interaction under the 4 external potential is studied. Making use of the special relation of relevant parameters, in the continuous limit, it was possible to obtain various non-classical soliton solutions, specifically, compact and peak-like solutions. The solutions undergo a jump on their first derivatives at some points of the space-time manifold. These analytic solutions were obtained by considering strong restrictions on their velocities and on the jump conditions. The energy concentrated in these solutions shows in several cases a discrete spectrum.

2007

In this paper we study existence and orbital stability for solitary waves of the nonlinear Klein-Gordon equation. The energy of these solutions travels as a localized packet, hence they are a particular type of solitons. In particular we are interested in sufficient conditions on the potential for the existence of solitons. Our proof is based on the study of the