Linearizability of Systems of Ordinary Differential Equations Obtained by Complex Symmetry Analysis

TL;DR

This paper classifies and analyzes the symmetry structures of systems of two second-order ordinary differential equations that can be linearized, focusing on those derived from complex scalar equations and their canonical forms.

Contribution

It introduces a reduced optimal canonical form for systems from scalar complex equations, linking them to specific symmetry algebra dimensions and classifying their linearizability.

Findings

Identified five equivalence classes of linearizable systems.

Established symmetry algebra dimensions for these classes.

Provided examples illustrating the classification and symmetry structures.

Abstract

Five equivalence classes had been found for systems of two second-order ordinary differential equations, transformable to linear equations (linearizable systems) by a change of variables. An "optimal (or simplest) canonical form" of linear systems had been established to obtain the symmetry structure, namely with 5, 6, 7, 8 and 15 dimensional Lie algebras. For those systems that arise from a scalar complex second-order ordinary differential equation, treated as a pair of real ordinary differential equations, a "reduced optimal canonical form" is obtained. This form yields three of the five equivalence classes of linearizable systems of two dimensions. We show that there exist 6, 7 and 15-dimensional algebras for these systems and illustrate our results with examples.