Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2011, Article ID 171834, 17 pages
http://dx.doi.org/10.1155/2011/171834
Research Article

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

1Center for Advanced Mathematics and Physics, National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan
2School of Electrical Engineering and Computer Science, National University of Sciences and Technology, Campus H-12, Islamabad 44000, Pakistan
3Department of Mathematics, Brock University, St. Catherines, ON, Canada L2S 3A1

Received 16 April 2011; Accepted 22 August 2011

Academic Editor: Dane Quinn

Copyright © 2011 M. Safdar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, we provide a “reduced optimal canonical form.” 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.

1. Introduction

Lie used algebraic symmetry properties of differential equations to extract their solutions [14]. One method developed was to transform the equation to linear form by changing the dependent and independent variables invertibly. Such transformations are called point transformations and the transformed equations are said to be linearized. Equations that can be so transformed are said to be linearizable. Lie proved that the necessary and sufficient condition for a scalar nonlinear ordinary differential equation (ODE) to be linearizable is that it must have eight Lie point symmetries. He exploited the fact that all scalar linear second-order ODEs are equivalent under point transformations [5], that is, every linearizable scalar second-order ODE is reducible to the free particle equation. While the situation is not so simple for scalar linear ODEs of order 𝑛3, it was proved that there are three equivalence classes with 𝑛+1, 𝑛+2, or 𝑛+4 infinitesimal symmetry generators [6].

For linearization of systems of two nonlinear ODEs, we will first consider the equivalence of the corresponding linear systems under point transformations. Nonlinear systems of two second-order ODEs that are linearizable to systems of ODEs with constant coefficients, were proved to have three equivalence classes [7]. They have 7-,8-, or 15-dimensional Lie algebras. This result was extended to those nonlinear systems which are equivalent to linear systems of ODEs with constant or variable coefficients [8]. They obtained an “optimal" canonical form of the linear systems involving three parameters, whose specific choices yielded five equivalence classes, namely, with 5-,6-,7-,8-, or 15-dimensional Lie algebras.

Geometric methods were developed to transform nonlinear systems of second-order ODEs [911] to a system of the free particle equations by treating them as geodesic equations and then projecting those equations down from an 𝑚×𝑚 system to an (𝑚1)×(𝑚1) system. In this process the originally homogeneous quadratically semilinear system in 𝑚 dimensions generically becomes a nonhomogeneous, cubically semilinear system in (𝑚1) dimensions. When used for 𝑚=2 the Lie conditions for the scalar ODE are recovered precisely. The criterion for linearizability is simply that the manifold for the (projected) geodesic equations be flat. The symmetry algebra in this case is 𝑠𝑙(𝑛+2,) and hence the number of generators is 𝑛2+4𝑛+3. Thus for a system of two equations to be linearizable by this method it must have 15 generators. Separately, linearizability conditions have been derived for the equivalence of linearizable systems of two second-order ODEs and corresponding linear forms with 𝑑-dimensional Lie algebras where 𝑑=5,6,7, or 8 [12]. The linearization problem for two-dimensional systems of second-order ODEs has been addressed by constructing a general procedure to obtain invertible or noninvertible linearizing transformations [13]. A general solution scheme has been established to solve systems [14] which mainly consists of the reduction of the number of the dependent variables and linearization of the reduced systems. This method is known as sequential linearization and found to linearize those two-dimensional systems of second-order ODEs that are not linearizable via point transformations.

A scalar complex ODE involves two real functions of two real variables, yielding a system of two partial differential equations (PDEs) [15, 16]. By restricting the independent variable to be real we obtain a system of ODEs. Complex symmetry analysis (CSA) provides the symmetry algebra for systems of two ODEs with the help of the symmetry generators of the corresponding complex ODE. This is not a simple matter of doubling the generators for the scalar complex ODE. The inequivalence of these systems from the systems obtained by geometric means [11], has been proved [17]. Thus their symmetry structures are not the same. A two-dimensional system of second-order ODEs corresponds to a scalar complex second-order ODE if the coefficients of the system satisfy Cauchy-Riemann equations (CR-equations). We provide the full symmetry algebra of those systems that correspond to linearizable scalar complex ODEs. For this purpose we derive a reduced optimal canonical form for linear systems obtainable from a complex linear equation. We prove that this form provides three equivalence classes of linearizable systems of two second-order ODEs while there exist five linearizable classes [8] by real symmetry analysis. This difference arises due to the fact that in CSA we invoke equivalence of scalar second-order ODEs to obtain the reduced optimal form while in real symmetry analysis equivalence of linear systems of two ODEs was used to derive their optimal form. The nonlinear systems transformable to one of the three equivalence classes are characterized by complex transformations of the form 𝑇(𝑥,𝑢(𝑥))(𝜒(𝑥),𝑈(𝑥,𝑢)).(1.1)

Indeed, these complex transformations generate these linearizable classes of two-dimensional systems. Note that not all the complex linearizing transformations for scalar complex equations provide the corresponding real transformations for systems.

The plan of the paper is as follows. In the next section we present the preliminaries for determining the symmetry structures. The third section deals with the conditions derived for systems that can be obtained by CSA. In section four we obtain the reduced optimal canonical form for systems associated with complex linear ODEs. The theory developed to classify linearizable systems of ODEs transformable to this reduced optimal form is given in the fifth section. Applications of the theory are given in the next section. The last section summarizes and discusses the work.

2. Preliminaries

The simplest form of a second-order equation has the maximal-dimensional algebra, 𝑠𝑙(3,). To discuss the equivalence of systems of two linear second-order ODEs, we need to use the following result for the equivalence of a general system of 𝑛 linear homogeneous second-order ODEs with 2𝑛2+𝑛 arbitrary coefficients and some canonical forms that have fewer arbitrary coefficients [18]. Any system of 𝑛 second-order nonhomogeneous linear ODEs ̈̇𝐮=𝐀𝐮+𝐁𝐮+𝐜,(2.1) can be mapped invertibly to one of the following forms: ̈̇̈𝐯=𝐂𝐯,(2.2)𝐰=𝐃𝐰,(2.3) where 𝐀, 𝐁, 𝐂, 𝐃 are 𝑛×𝑛 matrix functions, 𝐮, 𝐯, 𝐰, 𝐜 are vector functions, and the dot represents differentiation relative to the independent variable 𝑡. For a system of two second-order ODEs (𝑛=2) there are a total of 10 coefficients for the system represented by (2.1). It is reducible to the first and second canonical forms, (2.2) and (2.3), respectively. Thus a system with 4 arbitrary coefficients of the form ̈𝑤1=𝑑11(𝑡)𝑤1+𝑑12(𝑡)𝑤2,̈𝑤2=𝑑21(𝑡)𝑤1+𝑑22(𝑡)𝑤2,(2.4) can be obtained by using the equivalence of (2.1) and the counterpart of the Laguerre-Forsyth second canonical form (2.3). This result demonstrates the equivalence of systems of two ODEs having 10 and 4 arbitrary coefficients, respectively. The number of arbitrary coefficients can be further reduced to three by the change of variables [8] 𝑤̃𝑦=1𝑤𝜌(𝑡),̃𝑧=2𝜌(𝑡),𝑥=𝑡𝜌2(𝑠)𝑑𝑠,(2.5) where 𝜌 satisfies 𝜌𝑑11+𝑑222𝜌=0,(2.6) to the linear system ̃𝑦=𝑑11𝑑(𝑥)̃𝑦+12(𝑥)̃𝑧,̃𝑧=𝑑21𝑑(𝑥)̃𝑦11(𝑥)̃𝑧,(2.7) where 𝑑11=𝜌3𝑑11𝑑222,𝑑12=𝜌3𝑑12,𝑑21=𝜌3𝑑21.(2.8)

This procedure of reduction of arbitrary coefficients for linearizable systems simplifies the classification problem enormously. System (2.7) is called the optimal canonical form for linear systems of two second-order ODEs, as it has the fewest arbitrary coefficients, namely, three.

3. Systems of ODEs Obtainable by CSA

Following the classical Lie procedure [19], one uses invertible point transformations 𝑋=𝑋(𝑥,𝑦,𝑧),𝑌=𝑌(𝑥,𝑦,𝑧),𝑍=𝑍(𝑥,𝑦,𝑧),(3.1) to map the general system of ODEs 𝑦=Γ1𝑥,𝑦,𝑧,𝑦,𝑧,𝑧=Γ2𝑥,𝑦,𝑧,𝑦,𝑧,(3.2) where prime denotes differentiation with respect to 𝑥, to the simplest form 𝑌=0,𝑍=0,(3.3) where the prime now denotes differentiation with respect to 𝑋 and the derivatives transform as 𝑌=𝐷𝑥(𝑌)𝐷𝑥(𝑋)=𝐹1𝑥,𝑦,𝑧,𝑦,𝑧,𝑍=𝐷𝑥(𝑍)𝐷𝑥(𝑋)=𝐹2𝑥,𝑦,𝑧,𝑦,𝑧,𝑌(3.4)=𝐷𝑥𝐹1𝐷𝑥(𝑋),𝑍=𝐷𝑥𝐹2𝐷𝑥,(𝑋)(3.5) where 𝐷𝑥 is the total derivative operator. The most general linearizable form of (3.2) obtainable from the transformations (3.1) is given by 𝑦+𝛼11𝑦3+𝛼12𝑦2𝑧+𝛼13𝑦𝑧2+𝛼14𝑧3+𝛽11𝑦2+𝛽12𝑦𝑧+𝛽13𝑧2+𝛾11𝑦+𝛾12𝑧+𝛿1𝑧=0,+𝛼21𝑦3+𝛼22𝑦2𝑧+𝛼23𝑦𝑧2+𝛼24𝑧3+𝛽21𝑦2+𝛽22𝑦𝑧+𝛽23𝑧2+𝛾21𝑦+𝛾22𝑧+𝛿2=0,(3.6) where the coefficients are functions of the dependent and independent variables, which are given in the appendix. System (3.6) is the most general candidate for two second-order ODEs that may be linearizable. On the other hand CSA deals with a class of systems of the forms (3.2) in which Γ1 and Γ2 satisfy CR-equations. The role of CR-equations in yielding the solutions of two-dimensional systems from the solutions of complex base equations has been investigated [20]. To make a complete characterization of such systems we restate the following theorem.

Theorem 3.1. A general two-dimensional system of second-order ODEs (3.2) corresponds to a complex equation 𝑢=Γ𝑥,𝑢,𝑢,(3.7) if and only if Γ1 and Γ2 satisfy the CR-equations Γ1,𝑦=Γ2,𝑧,Γ1,𝑧=Γ2,𝑦,Γ1,𝑦=Γ2,𝑧,Γ1,𝑧=Γ2,𝑦,(3.8) where Γ(𝑥,𝑢,𝑢)=Γ1(𝑥,𝑦,𝑧,𝑦,𝑧)+𝑖Γ2(𝑥,𝑦,𝑧,𝑦,𝑧).

Another candidate of linearizability of two dimensional systems obtainable from the most general form of a complex linearizable equation 𝑢+𝐸3(𝑥,𝑢)𝑢3+𝐸2(𝑥,𝑢)𝑢2+𝐸1(𝑥,𝑢)𝑢+𝐸0(𝑥,𝑢)=0,(3.9) where 𝑢 is a complex function of the real independent variable 𝑥, is also cubically semilinear, that is, a system of the form 𝑦+𝛼11𝑦33𝛼12𝑦2𝑧3𝛼11𝑦𝑧2+𝛼12𝑧3+𝛽11𝑦22𝛽12𝑦𝑧𝛽11𝑧2+𝛾11𝑦𝛾12𝑧+𝛿11𝑧=0,+𝛼12𝑦3+3𝛼11𝑦2𝑧3𝛼12𝑦𝑧2𝛼11𝑧3+𝛽12𝑦2+2𝛽11𝑦𝑧𝛽12𝑧2+𝛾12𝑦+𝛾11𝑧+𝛿12=0,(3.10) here the coefficients 𝛼1𝑖, 𝛽1𝑖, 𝛾1𝑖, and 𝛿1𝑖 for 𝑖=1,2 are functions of 𝑥, 𝑦, and 𝑧. Clearly, the coefficients 𝛼1𝑖,𝛽1𝑖,𝛾1𝑖, and 𝛿1𝑖 satisfy the CR-equations, that is, 𝛼11,𝑦=𝛼12,𝑧,𝛼12,𝑦=𝛼11,𝑧 and vice versa. It is obvious that (3.9) generates a system by decomposing the complex coefficients 𝐸𝑗, for 𝑗=0,1,2,3 into real and imaginary parts 𝐸3=𝛼11+𝑖𝛼12,𝐸2=𝛽11+𝑖𝛽12,𝐸1=𝛾11+𝑖𝛾12,𝐸0=𝛿11+𝑖𝛿12,(3.11) where all the functions are analytic. The system of the form (3.10) is called complex linearizable as it comes from a complex linearizable ODE (3.9). In order to establish correspondence between cubically semilinear forms (3.6) and (3.10), we state the following theorem.

Theorem 3.2. A system of the form (3.6) corresponds to (3.10) if and only if coefficients of both the systems satisfy the following set of equations: 𝛼111=3𝛼13=13𝛼22=𝛼24=𝛼11,𝛽211=2𝛽12=𝛽23=𝛽12,𝛼141=3𝛼121=3𝛼23=𝛼21=𝛼12,𝛽11=12𝛽22=𝛽13=𝛽11,𝛾11=𝛾22=𝛾11,𝛾21=𝛾12=𝛾12,𝛿1=𝛿11,𝛿2=𝛿12.(3.12)

Theorem 3.2 identifies those two-dimensional systems which are complex linearizable. It may be pointed out that the coefficients of (3.6) also satisfy CR-equations as a result of (3.12).

4. Reduced Optimal Canonical Forms

The simplest forms for linear systems of two second-order ODEs corresponding to complex scalar ODEs can be established by invoking the equivalence of scalar second-order linear ODEs. Consider a general linear scalar complex second-order ODE 𝑢=𝜁1(𝑥)𝑢+𝜁2(𝑥)𝑢+𝜁3(𝑥),(4.1) where prime denotes differentiation relative to 𝑥. As all the linear scalar second-order ODEs are equivalent, so (4.1) is equivalent to the following scalar second-order complex ODEs 𝑢=𝜁4(𝑥)𝑢,𝑢(4.2)=𝜁5(𝑥)𝑢,(4.3) where all the three forms (4.1), (4.2), and (4.3) are transformable to each other. Indeed these three forms are reducible to the free particle equation. These three complex scalar linear ODEs belong to the same equivalence class, that is, all have eight Lie point symmetry generators. In this paper we prove that the systems obtainable by these forms using CSA have more than one equivalence class. To extract systems of two linear ODEs from (4.2) and (4.3) we put 𝑢(𝑥)=𝑦(𝑥)+𝑖𝑧(𝑥), 𝜁4(𝑥)=𝛼1(𝑥)+𝑖𝛼2(𝑥) and 𝜁5(𝑥)=𝛼3(𝑥)+𝑖𝛼4(𝑥), to obtain two forms of systems of two linear ODEs 𝑦=𝛼1(𝑥)𝑦𝛼2(𝑥)𝑧,𝑧=𝛼2(𝑥)𝑦+𝛼1(𝑥)𝑧,𝑦(4.4)=𝛼3(𝑥)𝑦𝛼4𝑧(𝑥)𝑧,=𝛼4(𝑥)𝑦+𝛼3(𝑥)𝑧,(4.5) thus we state the following theorem.

Theorem 4.1. If a system of two second-order ODEs is linearizable via invertible complex point transformations then it can be mapped to one of the two forms (4.4) or (4.5).

Notice that here we have only two arbitrary coefficients in both the linear forms while the minimum number obtained before was three, that is, a system of the form (2.7). The reason we can reduce further is that we are dealing with the special classes of linear systems of ODEs that correspond to the scalar complex ODEs. In fact (4.5) can be reduced further by the change of variables 𝑦𝑌=𝑧𝜌(𝑥),𝑍=𝜌(𝑥),𝑋=𝑥𝜌2(𝑠)𝑑𝑠,(4.6) where 𝜌 satisfies 𝜌𝛼3𝜌=0,(4.7) to 𝑌𝑍=𝛽(𝑋)𝑍,=𝛽(𝑋)𝑌,(4.8) where 𝛽=𝜌3𝛼4, and prime denotes the differentiation with respect to 𝑋. Hence we arrive at the following result.

Theorem 4.2. Any linear system of two second-order ODEs of the form (4.5) with two arbitrary coefficients is transformable to a simplest system of two linear ODEs (4.8) with one arbitrary coefficient via real point transformations (4.6) where (4.7) holds.

The system (4.8) provides the reduced optimal canonical form associated with complex ODEs that contain a single coefficient 𝛽(𝑥), which is an arbitrary function of 𝑥. The equivalence of systems (4.4) and (4.5) can be established via invertible point transformations which we state in the form of following theorem.

Theorem 4.3. Two linear forms of the systems of two second-order ODEs (4.4) and (4.5) are equivalent via invertible point transformations 𝑦=𝑀1(𝑥)𝑦1𝑀2(𝑥)𝑦2+𝑦,𝑧=𝑀1(𝑥)𝑦2+𝑀2(𝑥)𝑦1+𝑧,(4.9) of the dependent variables only, where 𝑀1(𝑥), 𝑀2(𝑥)are two linearly independent solutions of 𝛼1𝑀1𝛼2𝑀2=2𝑀1,𝛼2𝑀1+𝛼1𝑀2=2𝑀2,(4.10) and 𝑦, 𝑧are the particular solutions of (4.4).

Proof. Differentiating the set of equations (4.9) and inserting the result in the linear form (4.4), routine calculations show that under the conditions (4.10) system (4.4) can be mapped to (4.5) where 𝛼31(𝑥)=𝑀21+𝑀22𝑀1𝛼1𝑀1𝛼2𝑀2𝑀1+𝑀2𝛼1𝑀2+𝛼2𝑀1𝑀2,𝛼4(1𝑥)=𝑀21+𝑀22𝑀1𝛼1𝑀2+𝛼2𝑀1𝑀2𝑀2𝛼1𝑀1𝛼2𝑀2𝑀1.(4.11) Thus the linear form (4.4) is reducible to (4.8).

Remark 4.4. Any nonlinear system of two second-order ODEs that is linearizable by complex methods can be mapped invertibly to a system of the form (4.8) with one coefficient which is an arbitrary function of the independent variable.

5. Symmetry Structure of Linear Systems Obtained by CSA

To use the reduced canonical form [21] for deriving the symmetry structure of linearizable systems associated with the complex scalar linearizable ODEs, we obtain a system of PDEs whose solution provides the symmetry generators for the corresponding linearizable systems of two second-order ODEs.

Theorem 5.1. Linearizable systems of two second-order ODEs reducible to the linear form (4.8) via invertible complex point transformations, have 6-, 7-, or 15-dimensional Lie point symmetry algebras.

Proof. The symmetry conditions provide the following set of PDEs for the system (4.8) 𝜉,𝑥𝑥=𝜉,𝑥𝑦=𝜉,𝑦𝑦=0=𝜂1,𝑧𝑧=𝜂2,𝑦𝑦,𝜂(5.1)1,𝑦𝑦2𝜉𝑥𝑦=𝜂1,𝑦𝑧𝜉,𝑥𝑧=𝜂2,𝑦𝑧𝜉𝑥𝑦=𝜂2,𝑧𝑧2𝜉,𝑥𝑧𝜉=0,(5.2),𝑥𝑥2𝜂1,𝑥𝑦3𝑧𝛽(𝑥)𝜉,𝑦+𝑦𝛽(𝑥)𝜉,𝑧=𝜂1,𝑥𝑧+𝑧𝛽(𝑥)𝜉,𝑧𝜉=0,(5.3),𝑥𝑥2𝜂2,𝑥𝑧+3𝑦𝛽(𝑥)𝜉,𝑧𝑧𝛽(𝑥)𝜉,𝑦=𝜂2,𝑥𝑦𝑦𝛽(𝑥)𝜉,𝑦𝜂=0,(5.4)1,𝑥𝑥+𝛽(𝑥)𝑦𝜂1,𝑧+2𝑧𝜉,𝑥𝑧𝜂1,𝑦+𝜂2+𝑧𝛽𝜂(𝑥)𝜉=0,(5.5)2,𝑥𝑥+𝛽(𝑥)𝑦𝜂2,𝑧2𝑦𝜉,𝑥𝑧𝜂2,𝑦𝜂1𝑦𝛽(𝑥)𝜉=0.(5.6) Equations (5.3)–(5.6) involve an arbitrary function of the independent variable and its first derivatives. Using (5.1) and (5.2) we have the following solution set 𝜉=𝛾1(𝑥)𝑦+𝛾2(𝑥)𝑧+𝛾3𝜂(𝑥),1=𝛾1(𝑥)𝑦2+𝛾2(𝑥)𝑦𝑧+𝛾4(𝑥)𝑦+𝛾5(𝑥)𝑧+𝛾6𝜂(𝑥),2=𝛾1(𝑥)𝑦𝑧+𝛾2(𝑥)𝑧2+𝛾7(𝑥)𝑦+𝛾8(𝑥)𝑧+𝛾9(𝑥).(5.7) Using (5.3) and (5.4), we get 𝛽(𝑥)𝛾1(𝑥)=0=𝛽(𝑥)𝛾2(𝑥).(5.8)

Now assuming 𝛽(𝑥) to be zero, nonzero constant and arbitrary function of 𝑥 will generate the following cases.

Case 1 (𝛽(𝑥)=0). The set of determining equations (5.1)–(5.6) will reduce to a trivial system of PDEs 𝜂1,𝑥𝑥=𝜂1,𝑥𝑧=𝜂1,𝑧𝑧𝜂=0,2,𝑥𝑥=𝜂2,𝑥𝑦=𝜂2,𝑦𝑦=0,2𝜉,𝑥𝑦𝜂1,𝑦𝑦=0=2𝜉,𝑥𝑧𝜂2,𝑧𝑧,𝜉,𝑥𝑧𝜂1,𝑦𝑧=0=𝜉,𝑥𝑦𝜂2,𝑦𝑧,𝜉,𝑥𝑥2𝜂1,𝑥𝑦=0=𝜉,𝑥𝑥2𝜂2,𝑥𝑧,(5.9) which can be extracted classically for the system of free particle equations. Solving it we find a 15-dimensional Lie point symmetry algebra.

Case 2 (𝛽(𝑥)0). Then (5.8) implies 𝛾1(𝑥)=𝛾2(𝑥)=0 and (5.7) reduces to 𝜉=𝛾3𝜂(𝑥),1=𝛾3(𝑥)2+𝑐3𝑦+𝑐1𝑧+𝛾6𝜂(𝑥),2=𝑐2𝛾𝑦+3(𝑥)2+𝑐4𝑧+𝛾9(𝑥).(5.10)

Here three subcases arise.

Subcase 2.1 (𝛽(𝑥)  is a non-zero constant). As equations (5.5) and (5.6) involve the derivatives of 𝛽(𝑥), which will now be zero, equations (5.3)–(5.6) and (5.10) yield a 7-dimensional Lie algebra. The explicit expressions of the symmetry generators involve trigonometric functions. But for a simple demonstration of the algorithm consider 𝛽(𝑥)=1. The solution of the set of the determining equations is 𝜉=𝐶1,𝜂1=𝐶2𝑦+𝐶4𝑒𝑥/2𝐶3𝑒𝑥/2𝑥sin2+𝐶6𝑒𝑥/2𝑥cos2+𝐶5𝑒𝑥/2cos𝑥/2+𝐶7𝜂𝑧,2=𝐶6𝑒𝑥/2+𝐶5𝑒𝑥/2𝑥sin2𝐶4𝑒𝑥/2𝑥cos2𝐶2𝑧+𝐶3𝑒𝑥/2cos𝑥/2+𝐶7𝑦.(5.11) This yields a 7-dimensional symmetry algebra.

Subcase 2.2 (𝛽(𝑥)=𝑥2,𝑥4,or (𝑥+1)4). Equations (5.3)–(5.6) and (5.10) yield a 7-dimensional Lie algebra. Thus the 7-dimensional algebras can be related with systems which have variable coefficients in their linear forms, apart from the linear forms with constant coefficients.

Subcase 2.3 (𝛽(𝑥)=𝑥1,𝑥2,𝑥2±𝐶0 or 𝑒𝑥). Using equations (5.3)–(5.6) and (5.10), we arrive at a 6-dimensional Lie point symmetry algebra. The explicit expressions involve special functions, for example, for 𝛽(𝑥)=𝑥1, 𝑥2, 𝑥2±𝐶0 we get Bessel functions. Similarly for 𝛽(𝑥)=𝑒𝑥 there are six symmetries, including the generators 𝑦𝜕𝑦𝑒𝑥𝑧𝜕𝑧, 𝑧𝜕𝑧+𝑒𝑥𝑦𝜕𝑦. The remaining four generators come from the solution of an ODE of order four.

Thus there appear only 6-, 7-, or 15-dimensional algebras for linearizable systems of two second-order ODEs transformable to (4.8) via invertible complex point transformations. We are not investigating the remaining two linear forms (4.4) and (4.5), because these are transformable to system (4.8), that is, all these forms have the same symmetry structures. Now we verify that the linear forms with Lie algebras of dimensions 6 or 7 are also obtainable from (2.7). For instance, take the coefficients in (2.7) as non-zero constants, 𝑑11(𝑥)=𝑎0, 𝑑12(𝑥)=𝑏0, and 𝑑21(𝑥)=𝑐0, where 𝑎20+𝑏0𝑐00.(5.12) The corresponding system has 7-dimensional algebra which is the same as the linear form (4.8). Moreover the 8-dimensional symmetry algebra was extracted [8] by assuming 𝑎20+𝑏0𝑐0=0.(5.13) Such linear forms cannot be obtained from (4.8). These two examples explain why a 7- dimensional algebra can be obtained from (4.8), but a linear form with an 8-dimensional algebra is not obtainable from it.

To prove these observations consider arbitrary point transformations of the form ̃𝑦=𝑎(𝑥)𝑦+𝑏(𝑥)𝑧,̃𝑧=𝑐(𝑥)𝑦+𝑑(𝑥)𝑧.(5.14)

Case a. If 𝑎(𝑥)=𝑎0,𝑏(𝑥)=𝑏0,𝑐(𝑥)=𝑐0, and 𝑑(𝑥)=𝑑0 are constants then (5.14) implies ̃𝑦=𝑎0𝑦+𝑏0𝑧,̃𝑧=𝑐0𝑦+𝑑0𝑧.(5.15) Using (2.7) and (4.5) in the above equation we find 𝑎0𝑑0𝑏0𝑐0𝑦=𝑎0𝑑0+𝑏0𝑐0𝑑11(𝑥)+𝑐0𝑑0𝑑12(𝑥)𝑎0𝑏0𝑑21𝑦+(𝑥)2𝑏0𝑑0𝑑11(𝑥)+𝑑20𝑑12(𝑥)𝑏20𝑑21𝑎(𝑥)𝑧,0𝑑0𝑏0𝑐0𝑧=𝑎0𝑑0+𝑏0𝑐0𝑑11(𝑥)+𝑐0𝑑0𝑑12(𝑥)𝑎0𝑏0𝑑21𝑧+(𝑥)2𝑎0𝑐0𝑑11(𝑥)+𝑐20𝑑12(𝑥)𝑎20𝑑21(𝑥)𝑦,(5.16) where 𝑎0𝑑0𝑏0𝑐00. Using (4.5), (5.16), and the linear independence of the 𝑑's, gives 𝑎0𝑏0=𝑐0𝑑0𝑎=0,20𝑏20=𝑐20𝑑20𝑎=0,0𝑑0+𝑏0𝑐0=𝑎0𝑐0𝑏0𝑑0=0,(5.17) which has a solution 𝑎0=𝑏0=𝑐0=𝑑0=0, which is inconsistent with (5.16) because the requirement was 𝑎0𝑑0𝑏0𝑐00.

Case b. If 𝑎(𝑥), 𝑏(𝑥), 𝑐(𝑥), and 𝑑(𝑥) are arbitrary functions of 𝑥 then ̃𝑦=𝑎(𝑥)𝑦+𝑏(𝑥)𝑧+𝑎(𝑥)𝑦+𝑏(𝑥)𝑧+2𝑎(𝑥)𝑦+2𝑏(𝑥)𝑧,̃𝑧=𝑐(𝑥)𝑦+𝑑(𝑥)𝑧+𝑐(𝑥)𝑦+𝑑(𝑥)𝑧+2𝑐(𝑥)𝑦+2𝑑(𝑥)𝑧.(5.18) Thus we obtain (𝑎𝑑𝑏𝑐)𝑦=𝑑(𝑎𝑑+𝑏𝑐)11𝑑+𝑐𝑑12𝑑𝑎𝑏21𝑎𝑑+𝑐𝑏𝑦+𝑑2𝑏𝑑11+𝑑2𝑑12𝑏2𝑑21𝑏𝑑+𝑑𝑏𝑎𝑧2𝑑𝑦+𝑏𝑧𝑐+2𝑏𝑦+𝑑𝑧,(𝑎𝑑𝑏𝑐)𝑧=𝑑2𝑎𝑐11+𝑐2𝑑12𝑎2𝑑21𝑎𝑐+𝑐𝑎𝑦+𝑑(𝑎𝑑+𝑏𝑐)11𝑑+𝑐𝑑12𝑑𝑎𝑏21𝑏𝑐+𝑑𝑎𝑎𝑧2𝑐𝑦+𝑏𝑧𝑐+2𝑎𝑦+𝑑𝑧.(5.19) Comparing the coefficients as before and using the linear independence of 𝑑's we obtain 𝑎(𝑥)=𝑏(𝑥)=𝑐(𝑥)=𝑑(𝑥)=0,(5.20) which implies that it reduces to a system of the form (5.16), which leaves us again with the same result. Thus we have the following theorem.

Theorem 5.2. The linear forms for systems of two second-order ODEs obtainable by real symmetry analysis with 5- or 8-dimensional algebras are not transformable to (4.5) by invertible point transformations.

Before presenting some illustrative applications of the theory developed we refine Theorem 5.1 by using Theorem 5.2 to make the following remark.

Remark 5.3. There are only 6-, 7-, or 15-dimensional algebras for linearizable systems obtainable by scalar complex linearizable ODEs, that is, there are no 5- or 8-dimensional Lie point symmetry algebras for such systems.

6. Applications

Consider a system of nonhomogeneous geodesic-type differential equations 𝑦+𝑦2𝑧2=Ω1𝑥,𝑦,𝑧,𝑦,𝑧,𝑧+2𝑦𝑧=Ω2𝑥,𝑦,𝑧,𝑦,𝑧,(6.1) where Ω1 and Ω1 are linear functions of the dependent variables and their derivatives. This system corresponds to a complex scalar equation 𝑢+𝑢2=Ω𝑥,𝑢,𝑢,(6.2) which is either transformable to the free particle equation or one of the linear forms (4.1)–(4.3), by means of the complex transformations 𝜒=𝜒(𝑥),𝑈(𝜒)=𝑒𝑢.(6.3) Which are further transformable to the free particle equation by utilizing another set of invertible complex point transformations. Generally, the system (6.1) is transformable to a system of the free particle equations or a linear system of the form 𝑌=Ω1𝜒,𝑌,𝑍,𝑌,𝑍Ω2𝜒,𝑌,𝑍,𝑌,𝑍,𝑍=Ω2𝜒,𝑌,𝑍,𝑌,𝑍+Ω1𝜒,𝑌,𝑍,𝑌,𝑍.(6.4) Here Ω1 and Ω2 are linear functions of the dependent variables and their derivatives, via an invertible change of variables obtainable from (6.3). The linear form (6.4) can be mapped to a maximally symmetric system if and only if there exist some invertible complex transformations of the form (6.3), otherwise these forms cannot be reduced further. This is the reason why we obtain three equivalence classes, namely, with 6-, 7-, and 15-dimensional algebras for systems corresponding to linearizable complex equations with only one equivalence class. We first consider an example of a nonlinear system that admits a 15-dimensional algebra which can be mapped to the free particle system using (6.3). Then we consider four applications to nonlinear systems of quadratically semilinear ODEs transformable to (6.4) via (6.3) that are not further reducible to the free particle system.

(1) Consider (6.1) with Ω12=𝑥𝑦,Ω22=𝑥𝑧,(6.5) it admits a 15-dimensional algebra. The real linearizing transformations 1𝜒(𝑥)=𝑥,𝑌=𝑒𝑦cos(𝑧),𝑍=𝑒𝑦sin(𝑧),(6.6) obtainable from the complex transformations (6.3) with 𝑈(𝜒)=𝑌(𝜒)+𝑖𝑍(𝜒), map the above nonlinear system to 𝑌=0,𝑍=0. Moreover, the solution of (6.5) corresponds to the solution of the corresponding complex equation 𝑢+𝑢2+2𝑥𝑢=0.(6.7)

(2) Now consider Ω1 and Ω2 to be linear functions of the first derivatives 𝑦,𝑧, that is, system (6.1) with Ω1=𝑐1𝑦𝑐2𝑧,Ω2=𝑐2𝑦+𝑐1𝑧,(6.8) which admits a 7-dimensional algebra, provided both 𝑐1 and 𝑐2 are not simultaneously zero. It is associated with the complex equation 𝑢+𝑢2𝑐𝑢=0.(6.9) Using the transformations (6.3) to generate the real transformations 𝜒(𝑥)=𝑥,𝑌=𝑒𝑦cos(𝑧),𝑍=𝑒𝑦sin(𝑧),(6.10) which map the nonlinear system to a linear system of the form (4.4), that is, 𝑌=𝑐1𝑌𝑐2𝑍,𝑍=𝑐2𝑌+𝑐1𝑍,(6.11) which also has a 7-dimensional symmetry algebra and corresponds to 𝑈𝑐𝑈=0.(6.12)

All the linear second-order ODEs are transformable to the free particle equation thus we can invertibly transform the above equation to 𝑈=0, using (𝜒(𝑥),𝑈)𝜒=𝛼+𝛽𝑒𝑐𝜒(𝑥),𝑈=𝑈,(6.13) where 𝛼, 𝛽, and 𝑐 are complex. But these complex transformations can not generate real transformations to reduce the corresponding system (6.11) to a maximally symmetric system.

(3) A system with a 6-dimensional Lie algebra is obtainable from (6.1) by introducing a linear function of 𝑥 in the above coefficients, that is, Ω1=𝑐(1+𝑥)1𝑦𝑐2𝑧,Ω2𝑐=(1+𝑥)2𝑦+𝑐1𝑧,(6.14) in (6.1), then the same transformations (6.10) converts the above system into a linear system 𝑌=𝑐(1+𝜒)1𝑌𝑐2𝑍,𝑍𝑐=(1+𝜒)2𝑌+𝑐1𝑍,(6.15) where both systems (6.14) and (6.15) are in agreement on the dimensions (i.e., six) of their symmetry algebras. Again, the above system is a special case of the linear system (4.4).

(4) If we choose Ω1=𝑐1,Ω2=𝑐2, where 𝑐𝑖(𝑖=1,2) are non-zero constants, then under the same real transformations (6.10), the nonlinear system (6.1) takes the form 𝑌=𝑐1𝑌𝑐2𝑍𝑍,=𝑐2𝑌+𝑐1𝑍.(6.16)

7. Conclusion

The classification of linearizable systems of two second-order ODEs was obtained by using the equivalence properties of systems of two linear second-order ODEs [8]. The “optimal canonical form” of the corresponding linear systems of two second-order ODEs, to which a linearizable system could be mapped, is crucial. This canonical form used invertible transformations, the invertibility of these mappings insuring that the symmetry structure is preserved. That optimal canonical form of the linear systems of two second-order ODEs led to five linearizable classes with respect to Lie point symmetry algebras with dimensions 5, 6, 7, 8, and 15.

Systems of two second-order ODEs appearing in CSA correspond to some scalar complex second-order ODE. We proved the existence of a reduced optimal canonical form for such linear systems of two ODEs. This reduced canonical form provided three equivalence classes, namely, with 6-, 7-, or 15-dimensional point symmetry algebras. Two cases are eliminated in the theory of complex symmetries: those of 5 and 8-dimensional algebras. The systems corresponding to a complex linearized scalar ODE involve one parameter which can only cover three possibilities: (a) it is zero, (b) it is a non-zero constant, and (c) it is a nonconstant function. The nonexistence of 5- and 8-dimensional algebras for the linear forms appearing due to CSA has been proved by showing that these forms are not equivalent to those provided by the real symmetry approach for systems [8] with 5 and 8 generators.

Work is in progress [20] to find complex methods of solving a class of 2-dimensional nonlinearizable systems of second-order ODEs. It is also obtainable from the linearizable scalar complex second-order ODEs, which are transformable to the free particle equation via an invertible change of the dependent and independent variables of the form 𝜒=𝜒(𝑥,𝑢),𝑈(𝜒)=𝑈(𝑥,𝑢).(7.1)

Notice that these transformations are different from (6.3). The real transformations corresponding to the complex transformations above cannot be used to linearize the real system. But the linearizability of the complex scalar equations can be used to provide solutions for the corresponding systems.

One might wonder how the procedures developed can be extended to odd dimensional systems of equations. To obtain a 2𝑛-dimensional system we can take an 𝑛-dimensional system, regard it as complex and split it. This method will not work for odd dimensions. An extension of the procedure has been developed [22] of using the splitting procedure iteratively starting with a scalar base equation. Among others, this gave a 3-dimensional system of ODEs. The procedure could be used by increasing the number of iterations or starting with a higher dimensional system and using a second iteration, to obtain any dimensional system—even or odd.

Appendix

Inserting 𝐹1(𝑥,𝑦,𝑧,𝑦,𝑧) and 𝐹2(𝑥,𝑦,𝑧,𝑦,𝑧) from (3.4) into (3.5) we obtain 𝐷𝑥(𝑋)𝐷2𝑥(𝑌)𝐷𝑥(𝑌)𝐷2𝑥(𝑋)𝐷𝑥(𝑋)3𝐷=0,𝑥(𝑋)𝐷2𝑥(𝑍)𝐷𝑥(𝑍)𝐷2𝑥(𝑋)𝐷𝑥(𝑋)3=0.(A.1) Substituting 𝐷𝑥(𝑋)=𝑋𝑥+𝑦𝑋𝑦+𝑧𝑋𝑧,𝐷2𝑥(𝑋)=𝑋𝑥𝑥+2𝑦𝑋𝑥𝑦+2𝑧𝑋𝑥𝑧+𝑦2𝑋𝑦𝑦+2𝑦𝑧𝑋𝑦𝑧+𝑧2𝑋𝑧𝑧+𝑦𝑋𝑦+𝑧𝑋𝑧,(A.2) and similar expressions for 𝐷𝑥(𝑌),𝐷𝑥(𝑍),𝐷2𝑥(𝑌), and 𝐷2𝑥(𝑍) in (A.1), yields 𝛼1𝑦+𝛼2𝑧+𝛽1𝑦3+𝛽2𝑦2𝑧+𝛽3𝑦𝑧2+𝛽4𝑧3+𝛾1𝑦2+𝛾2𝑦𝑧+𝛾3𝑧2+𝛿1𝑦+𝛿2𝑧+𝜖1=0,𝛼3𝑦+𝛼4𝑧+𝛽5𝑦3+𝛽6𝑦2𝑧+𝛽7𝑦𝑧2+𝛽8𝑧3+𝛾4𝑦2+𝛾5𝑦𝑧+𝛾6𝑧2+𝛿3𝑦+𝛿4𝑧+𝜖2=0.(A.3)

The coefficients of the above system of ODEs are 𝛼1=𝑋𝑥𝑌𝑦𝑌𝑥𝑋𝑦+𝑧𝑋𝑧𝑌𝑦𝑌𝑧𝑋𝑦,𝛼2=𝑋𝑥𝑌𝑧𝑌𝑥𝑋𝑧+𝑦𝑋𝑦𝑌𝑧𝑌𝑦𝑋𝑧,𝛼3=𝑋𝑥𝑍𝑦𝑍𝑥𝑋𝑦+𝑧𝑋𝑧𝑍𝑦𝑍𝑧𝑋𝑦,𝛼4=𝑋𝑥𝑍𝑧𝑍𝑥𝑋𝑧+𝑦𝑋𝑦𝑍𝑧𝑍𝑦𝑋𝑧,𝛽1=𝑋𝑦𝑌𝑦𝑦𝑌𝑦𝑋𝑦𝑦,𝛽2=𝑋𝑧𝑌𝑦𝑦𝑌𝑧𝑋𝑦𝑦𝑋+2𝑦𝑌𝑦𝑧𝑌𝑦𝑋𝑦𝑧,𝛽3=𝑋𝑦𝑌𝑧𝑧𝑌𝑦𝑋𝑧𝑧𝑋+2𝑧𝑌𝑦𝑧𝑌𝑧𝑋𝑦𝑧,𝛽4=𝑋𝑧𝑌𝑧𝑧𝑌𝑧𝑋𝑧𝑧,𝛽5=𝑋𝑦𝑍𝑦𝑦𝑍𝑦𝑋𝑦𝑦,𝛽6=𝑋𝑧𝑍𝑦𝑦𝑍𝑧𝑋𝑦𝑦𝑋+2𝑦𝑍𝑦𝑧𝑍𝑦𝑋𝑦𝑧,𝛽7=𝑋𝑦𝑍𝑧𝑧𝑍𝑦𝑋𝑧𝑧𝑋+2𝑧𝑍𝑦𝑧𝑍𝑧𝑋𝑦𝑧,𝛽8=𝑋𝑧𝑍𝑧𝑧𝑍𝑧𝑋𝑧𝑧,𝛾1=𝑋𝑥𝑌𝑦𝑦𝑌𝑥𝑋𝑦𝑦𝑋+2𝑦𝑌𝑥𝑦𝑌𝑦𝑋𝑥𝑦,𝛾2𝑋=2𝑥𝑌𝑦𝑧+𝑋𝑦𝑌𝑥𝑧+𝑋𝑧𝑌𝑥𝑦𝑌𝑥𝑋𝑦𝑧+𝑌𝑦𝑋𝑥𝑧+𝑌𝑧𝑋𝑥𝑦,𝛾3=𝑋𝑥𝑌𝑧𝑧𝑌𝑥𝑋𝑧𝑧𝑋+2𝑧𝑌𝑥𝑧𝑌𝑧𝑋𝑥𝑧,𝛾4=𝑋𝑥𝑍𝑦𝑦𝑍𝑥𝑋𝑦𝑦𝑋+2𝑦𝑍𝑥𝑦𝑍𝑦𝑋𝑥𝑦,𝛾5𝑋=2𝑥𝑍𝑦𝑧+𝑋𝑦𝑍𝑥𝑧+𝑋𝑧𝑍𝑥𝑦𝑍𝑥𝑋𝑦𝑧+𝑍𝑦𝑋𝑥𝑧+𝑍𝑧𝑋𝑥𝑦,𝛾6=𝑋𝑥𝑍𝑧𝑧𝑍𝑥𝑋𝑧𝑧𝑋+2𝑧𝑍𝑥𝑧𝑍𝑧𝑋𝑥𝑧,𝛿1=𝑋𝑦𝑌𝑥𝑥𝑌𝑦𝑋𝑥𝑥𝑋+2𝑥𝑌𝑥𝑦𝑌𝑥𝑋𝑥𝑦,𝛿2=𝑋𝑧𝑌𝑥𝑥𝑌𝑧𝑋𝑥𝑥𝑋+2𝑥𝑌𝑥𝑧𝑌𝑥𝑋𝑥𝑧,𝛿3=𝑋𝑦𝑍𝑥𝑥𝑍𝑦𝑋𝑥𝑥𝑋+2𝑥𝑍𝑥𝑦𝑍𝑥𝑋𝑥𝑦,𝛿4=𝑋𝑧𝑍𝑥𝑥𝑍𝑧𝑋𝑥𝑥𝑋+2𝑥𝑍𝑥𝑧𝑍𝑥𝑋𝑥𝑧,𝜖1=𝑋𝑥𝑌𝑥𝑥𝑌𝑥𝑋𝑥𝑥,𝜖2=𝑋𝑥𝑍𝑥𝑥𝑍𝑥𝑋𝑥𝑥.(A.4)

System (A.3) yields a system of the form (3.6) with the following coefficients:𝛼11=𝜏1𝛼4𝛽1𝛼2𝛽5,𝛼12=𝜏1𝛼4𝛽2𝛼2𝛽6,𝛼13=𝜏1𝛼4𝛽3𝛼2𝛽7,𝛼14=𝜏1𝛼4𝛽4𝛼2𝛽8,𝛼21=𝜏2𝛼3𝛽1𝛼1𝛽5,𝛼22=𝜏2𝛼3𝛽2𝛼1𝛽6,𝛼23=𝜏2𝛼3𝛽3𝛼1𝛽7,𝛼24=𝜏2𝛼3𝛽4𝛼1𝛽8,𝛽11=𝜏1𝛼4𝛾1𝛼2𝛾4,𝛽12=𝜏1𝛼4𝛾2𝛼2𝛾5,𝛽13=𝜏1𝛼4𝛾3𝛼2𝛾6,𝛽21=𝜏2𝛼3𝛾1𝛼1𝛾4,𝛽22=𝜏2𝛼3𝛾2𝛼1𝛾5,𝛽23=𝜏2𝛼2𝛾3𝛼1𝛾6,𝛾11=𝜏1𝛼4𝛿1𝛼2𝛿3,𝛾12=𝜏1𝛼4𝛿2𝛼2𝛿4,𝛾21=𝜏2𝛼3𝛿1𝛼1𝛿3,𝛾22=𝜏2𝛼3𝛿2𝛼1𝛿4,𝛿1=𝜏1𝛼4𝜖1𝛼2𝜖2,𝛿2=𝜏2𝛼3𝜖1𝛼1𝜖2,(A.5)

where 𝜏1=𝜏2=𝛼1𝛼4𝛼2𝛼3.

Acknowledgments

The authors are grateful to Fazal Mahomed for useful comments and discussion on this paper. M. Safader is most grateful to NUST for providing financial support.

References

  1. S. Lie, “Klassifikation und integration von gewönlichen differentialgleichungenzwischen x, y, die eine gruppe von transformationen gestaten,” Archiv der Mathematik, vol. VIII-IX, p. 187, 1883. View at Google Scholar
  2. S. Lie, Lectures on Differential Equations with Known Infinitesimal Transformations, Teubner, Leipzig, Germany, 1891, (in German, Lie's Lectures, by G. Sheffers).
  3. S. Lie, Theorie der Transformationsgruppen I, II and III, Teubner, Leipzig, Germany, 1888, (reprinted by Chelsea Publishing, New York, NY, USA, 1970).
  4. S. Lie, Vorlesungen Ü ber Differentialgleichungen mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, Germany, 1891, (reprinted by Chelsea Publishing, New York, NY, USA, 1967).
  5. F. M. Mahomed, “Symmetry group classification of ordinary differential equations: survey of some results,” Mathematical Methods in the Applied Sciences, vol. 30, no. 16, pp. 1995–2012, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. F. M. Mahomed and P. G. L. Leach, “Symmetry Lie algebras of nth order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 80–107, 1990. View at Publisher · View at Google Scholar
  7. V. M. Gorringe and P. G. L. Leach, “Lie point symmetries for systems of second order linear ordinary differential equations,” Quaestiones Mathematicae, vol. 11, no. 1, pp. 95–117, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. C. Wafo Soh and F. M. Mahomed, “Symmetry breaking for a system of two linear second-order ordinary differential equations,” Nonlinear Dynamics, vol. 22, no. 1, pp. 121–133, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. V. Aminova and N. A. M. Aminov, “Projective geometry of systems of differential equations: general conceptions,” Tensor—New Series, vol. 62, no. 1, pp. 65–86, 2000. View at Google Scholar · View at Zentralblatt MATH
  10. F. M. Mahomed and A. Qadir, “Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations,” Nonlinear Dynamics, vol. 48, no. 4, pp. 417–422, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. M. Mahomed and A. Qadir, “Invariant linearization criteria for systems of cubically semi-linear second-order ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 16, pp. 1–16, 2009. View at Google Scholar
  12. Y. Y. Bagderina, “Linearization criteria for a system of two second-order ordinary differential equations,” Journal of Physics, A, vol. 43, no. 46, pp. 465201–465215, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. V. K. Chandrasekar, M. Senthilvelan, and M. Lakshmanan, “On the complete integrability and linearization of nonlinear ordinary differential equations. V. Linearization of coupled second-order equations,” Proceedings of the Royal Society of London, Series A, vol. 465, no. 2108, pp. 2369–2389, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Sookmee and S. V. Meleshko, “Conditions for linearization of a projectable system of two second-order ordinary differential equations,” Journal of Physics, A, vol. 41, no. 40, pp. 402001–402008, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. S. Ali, F. M. Mahomed, and A. Qadir, “Linearizability criteria for systems of two second-order differential equations by complex methods,” Nonlinear Dynamics. In press. View at Publisher · View at Google Scholar
  16. S. Ali, F. M. Mahomed, and A. Qadir, “Complex Lie symmetries for scalar second-order ordinary differential equations,” Nonlinear Analysis—Real World Applications, vol. 10, no. 6, pp. 3335–3344, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Safdar, A. Qadir, and S. Ali, “Inequivalence of classes of linearizable systems of cubically semi-linear ordinary differential equations obtained by real and complex symmetry analysis,” Mathematical and Computational Applications, vol. 16, no. 4, pp. 923–934, 2011. View at Google Scholar
  18. C. W. Soh and F. M. Mahomed, “Linearization criteria for a system of second-order ordinary differential equations,” International Journal of Non-Linear Mechanics, vol. 36, no. 4, pp. 671–677, 2001. View at Publisher · View at Google Scholar
  19. J. Merker, “Characterization of the Newtonian free particle system in m2 dependent variables,” Acta Applicandae Mathematicae, vol. 92, no. 2, pp. 125–207, 2006. View at Publisher · View at Google Scholar
  20. M. Safdar, S. Ali, and A. Qadir, “Symmetry solutions of two-dimensional systems not solvable by symmetry analysis,” arXiv:1104.3837.
  21. H. Stephani, Differential Equations: Their Solutions Using Symmetries, Cambridge University Press, Cambridge, UK, 1996.
  22. F. M. Mahomed and A. Qadir, “Higher dimensional systems of differential equations obtained by iterative use of complex methods,” (preprint of DECMA, CAM, University of Witwatersrand, 2011).