Group invariant transformations for the Klein–Gordon equation in three dimensional flat spaces

Abstract

We perform the complete symmetry classification of the Klein-Gordon equation in maximal symmetric spacetimes. The central idea is to find all possible potential functions V (t, x, y) that admit Lie and Noether symmetries. This is done by using the relation between the symmetry vectors of the differential equations and the elements of the conformal algebra of the underlying geometry. For some of the potentials, we use the admitted Lie algebras to determine corresponding invariant solutions to the Klein-Gordon equation. An integral part of this analysis is the problem of the classification of Lie and Noether point symmetries of the wave equation.

Group Analysis for Klein-Gordon Equation via Their Symmetries

“…m 2 c 2 f (x, y, t), u (5) = f (x cos ε − y sin ε, y cos ε + x sin ε, t), u (6) = f (x cosh cε − tc sinh cε, y, t cosh cε − (x/c) sinh cε), u (7) = f (x, y cosh cε − tc sinh cε, t cosh cε − (y/c) sinh ε), u (α) = f (x, y, t) + εα(x, y, t) where ε is any real number and α(x, y, t) is any other solution to two dimensional KG equation for a free particle with mass m. At the end the most general solution that we can obtain from a given solution u = f (x, y, t), by group transformations is in the form given below…”

“…Gupta and Sharma [4] obtained the exact travelling wave solutions for the KG equation with cubic nonlinearity by using First Integral Method. Other researcher also applied the different approaches to obtain the invariant solution of KG equation [7,10].…”

Lie symmetry analysis and similarity solutions for the Camassa-Choi equations

“…The theory of Lie symmetries of differential equations is the standard technique for the computation of solutions and describes the algebra for nonlinear differential equations. The novelty of the Lie symmetries is that invariant transformations can be found in order to simplify the given differential equation [3][4][5][6][7][8][9]. There is a plethora of applications in the Lie symmetries in the fluid dynamics with important results which have been used to understand the physical properties of the models.…”

Lie Symmetry Analysis and Similarity Solutions for the Jimbo-Miwa Equation and Generalisations

“…Then, by using normal coordinates, the solution of the differential equation can be written in terms of the invariant functions for the Lie symmetry vector and in that way to reduce the number of independent variables in case of PDEs, or the order of a ordinary differential equation (ODE) [8,16,26]. Applications of Lie symmetries can be found for instance in [18,20,22,24,25,27,34,35,43] and references therein.…”