Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations

TL;DR

This paper extends Lie symmetry analysis to a class of (1+1)-dimensional nonlinear Klein-Gordon equations, providing a complete classification of their symmetries using algebraic methods and realizations of Lie algebras on the line.

Contribution

It introduces a new approach leveraging classical Lie theorems and the structure of the equivalence group to simplify and enhance symmetry classification of these equations.

Findings

Complete group classification of the class of equations.

Identification of invariant symmetry extension cases.

Simplified proof using classical Lie algebra realizations.

Abstract

Essentially generalizing Lie's results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.