Some Exact Solutions for a Klein Gordon Equation

Journal of Physics Communications, 2019

Some exact solutions of KdV-Burgers-Kuramoto (KBK) equation are derived by the anzas and tanh methods. Also, the most general Lie point symmetry group of the KBK equation are presented using the basic Lie symmetry method. As well as, the non-classical and weak symmetries of this equation, as well as the corresponding similarity reductions, are investigated. Finally, the classical and non-classical symmetries of KBK and KdV-Burgers (KB) equations are compared.

THIRD INTERNATIONAL CONFERENCE OF MATHEMATICAL SCIENCES (ICMS 2019), 2019

DKP equation describes spin-0 and spin-1 relativistic particles. It is well known [4-5] that the DKP equation is equivalent to the Klein-Gordon and Proca equations, in [7] the equivalence to the KG equation is established via the matrix S and the reduction formula LSZ. In a different manner, we give an explicit relation between the DKP and the KG equations for both the spin-0 particle in (1 + 3) dimensions and spin-1 particle in (1 + 1) dimensions. From the DKP equation in its explicit form, we get another system generated by the KG equation, which gives us the equivalence between the DKP equation and the KG equation. Using this equivalence, the Volkov-like solution of the DKP equation for the spin-0 particle in the field of an electromagnetic plane wave, is calculated.

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.

Computational Methods for Differential Equations, 2019

In this paper, the Lie approximate symmetry analysis is applied to investigate new exact solutions of the perturbed nonlinear Klein-Gordon equation. The nonlinear Klein-Gordon equation is used to model many nonlinear phenomena. The tanh-coth method, is employed to solve some of the obtained reduced ordinary differential equations. As a result, we construct new analytical solutions with small parameter which is effectively obtained by the proposed method.

WSEAS TRANSACTIONS ON MATHEMATICS, 2021

We investigate a case of the generalized Korteweg – De Vries Burgers equation. Our aim is to demonstrate the need for the application of further methods in addition to using Lie Symmetries. The solution is found through differential topological manifolds. We apply Lie’s theory to take the PDE to an ODE. However, this ODE is of third order and not easily solvable. It is through differentiable topological manifolds that we are able to arrive at a solution

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.

Physics Letters A, 2006

In this work we discuss some exactly solvable Klein-Gordon equations. We basically discuss the existence of classes of potentials with different nonrelativistic limits, but which shares the intermediate effective Schroedinger differential equation. We comment about the possible use of relativistic exact solutions as approximations for nonrelativistic inexact potentials.

Few-Body Systems, 2012

We present the exact solution of the Klein-Gordon with Hylleraas Potential using the Nikiforov-Uvarov method. We obtain explicitly the bound state energy eigenvalues and the corresponding eigen function are also obtained and expressed in terms of Jacobi Polynomials.

Open Physics

For a generalized KdV-Burgers-Kuramoto equation we have studied conservation laws by using the multiplier method, and investigated its first-level and second-level potential systems. Furthermore, the Lie point symmetries of the equation and the Lie point symmetries associated with the conserved vectors are determined. We obtain travelling wave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and antikinks.

Physics Letters A, 2007

Using the shape invariance property we obtain exact solutions of the (1 + 1) dimensional Klein-Gordon equation for certain types of scalar and vector potentials. We also discuss the possibility of obtaining real energy spectrum with non-Hermitian interaction within this framework.