Symmetries, Lagrangians and conserved vectors for forms of the complex-valued Klein-Gordon Equation

TL;DR

This paper investigates the complex-valued Klein-Gordon Equation using Lie symmetry analysis, identifying symmetries and conserved quantities for models with power-law nonlinearity in two and three dimensions.

Contribution

It introduces a symmetry-based analysis of the complex Klein-Gordon Equation, deriving conserved vectors and Lagrangians for models with power-law nonlinearities.

Findings

Identified Lie point symmetries for the 2D and 3D models.

Constructed Lagrangians and conserved vectors for the 2D model.

Extended symmetry analysis to complex-valued equations with nonlinearities.

Abstract

We analyse the complex-valued Klein-Gordon Equation from an integrability perspective by the implementation of the Lie Theory of Continuous Groups, where this equation is governed by power-law nonlinearity. We write the equations in terms of real dependent variables and examine both the two-dimensional and three-dimensional models. In the case of the two-dimensional model the analysis is extended by the determination of Noether point symmetries and conserved vectors via the construction of a Lagrangian.