Linearization from Complex Lie Point Transformations

Abstract

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimension d, with d ≤ 4. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue in IR 3 of the linearizability criteria in IR 2 .

Linearization criteria for two-dimensional systems of third-order ordinary differential equations by complex approach

“…Linearizability of such systems implies linearizability of the complex scalar equation constructed from it; the converse is not true; linearizability of the complex scalar equation does not imply linearizability of the system [3]. If the scalar ODE is linearizable and the resulting system is also linearizable, this procedure is called complex linearization.…”

Noether-Like Operators and First Integrals for Generalized Systems of Lane-Emden Equations

“…arbitrary and non-linear, as given in Case 1. Equations (7) and (12), after some straightforward calculations, show that ς 1 = 1, ς 2 = 0, χ 1 = χ 2 = 0, and A 1 , A 2 are constants. Therefore, we have a single Noether-like operator X = ∂ ∂x .…”

“…For instance, the use of the complex variable technique to discuss linearization of systems of two second-order ODEs and PDEs has been presented in [6]. The procedure of converting a system of two second-order ODEs admitting Lie algebra of dimension d (d ≤ 4) into linearizable form with the help of complex Lie point symmetries of the base equation was given in [7]. Using semi-invariants, Mahomed et al [8] studied systems of two linear hyperbolic PDEs when they arise from a complex scalar ODE.…”

Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods

“…In this section, we reconsider a class of two-dimensional, systems of second order ODEs that is solved using complex methods [13]. There it was shown that, for this class of systems, dimensions of the Lie point symmetry algebra remain less than 5, while the base complex equations in most of the cases possess an eight-dimensional Lie and a five-dimensional Noether algebra.…”

“…The explicit use of complex functions of complex or real variables is demonstrated in [7][8][9][10] where solvability of systems of DEs is achieved through Noether symmetries and corresponding first integrals. Furthermore, by employing complex symmetry procedures: the energy stored in the field of a coupled harmonic oscillator was studied in [11] and linearizability of systems of two second order ODEs was addressed in [12,13]. The complex procedure, indeed, has been extended to higher dimensional systems of second order ODEs [14] and two-dimensional, systems of third order ODEs [15].…”