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

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 [1–4]. 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 𝑛≥3, 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 [9–11] 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 (𝑚−1)×(𝑚−1) system. In this process the originally homogeneous quadratically semilinear system in 𝑚 dimensions generically becomes a nonhomogeneous, cubically semilinear system in (𝑚−1) 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,ℝ) 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,ℝ). 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 𝐮̈=𝐀𝐮̇+𝐁𝐮+𝐜,(2.1) can be mapped invertibly to one of the following forms: 𝐰𝐯̈̈==𝐂𝐃𝐰𝐯,̇,((22..23)) 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 𝑤𝑤̈̈₁₂=𝑑₁₁(𝑡)𝑤₁+𝑑₁₂(𝑡)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] 𝑦̃=𝑤¹𝜌(𝑡),𝑧̃=𝜌𝑤(𝑡²),𝑥=^𝑡𝜌⁻²(𝑠)𝑑𝑠,(2.5) where 𝜌 satisfies 𝜌^−𝑑¹¹+𝑑₂₂2𝜌=0,(2.6) to the linear system 𝑧𝑦̃̃^_=𝑑₁₁(𝑥)𝑦̃+𝑑₁₂(𝑥)𝑧,̃=𝑑₂₁(𝑥)𝑦̃−𝑑₁₁(𝑥)𝑧,̃(2.7) where 𝑑₁₁=𝜌³𝑑₁₁−𝑑₂₂2,𝑑¹²=𝜌³𝑑₁₂,𝑑₂₁=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 𝑧𝑦^_=Γ₁𝑥,𝑦,𝑧,𝑦^,𝑧^,=Γ₂𝑥,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^=𝐷^𝑥(𝑌)𝐷_𝑥(𝑋)=𝐹¹𝑥,𝑦,𝑧,𝑦^𝑍_,𝑧^,=𝐷^𝑥(𝑍)𝐷_𝑥(𝑋)=𝐹²𝑥,𝑦,𝑧,𝑦^𝑌,𝑧^,(3.4)_=𝐷^𝑥𝐹₁𝐷_𝑥(𝑋),𝑍^=𝐷^𝑥𝐹₂𝐷_𝑥(𝑋),(3.5) where 𝐷_𝑥 is the total derivative operator. The most general linearizable form of (3.2) obtainable from the transformations (3.1) is given by 𝑦^+𝛼₁₁𝑦^3+𝛼₁₂𝑦^2𝑧^+𝛼₁₃𝑦^𝑧^2+𝛼₁₄𝑧^3+𝛽₁₁𝑦^2+𝛽₁₂𝑦^𝑧^+𝛽₁₃𝑧^2𝑧+𝛾₁₁𝑦^+𝛾₁₂𝑧^+𝛿₁=0,_+𝛼₂₁𝑦^3+𝛼₂₂𝑦^2𝑧^+𝛼₂₃𝑦^𝑧^′2+𝛼₂₄𝑧^3+𝛽₂₁𝑦^2+𝛽₂₂𝑦^𝑧^+𝛽₂₃𝑧^2+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^=Γ𝑥,𝑢,𝑢^,(3.7) if and only if Γ₁ and Γ₂ satisfy the CR-equations ΓΓ_{1,𝑦1,𝑦′}=Γ_2,𝑧,Γ_1,𝑧=−Γ_2,𝑦,=Γ_2,𝑧′,Γ_1,𝑧′=−Γ_2,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)𝑢^3+𝐸₂(𝑥,𝑢)𝑢^2+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^+𝛼₁₁𝑦^3−3𝛼₁₂𝑦^2𝑧^−3𝛼₁₁𝑦^𝑧^2+𝛼₁₂𝑧^3+𝛽₁₁𝑦^2−2𝛽₁₂𝑦^𝑧^−𝛽₁₁𝑧^2𝑧+𝛾₁₁𝑦^−𝛾₁₂𝑧^+𝛿₁₁=0,_+𝛼₁₂𝑦^3+3𝛼₁₁𝑦^2𝑧^−3𝛼₁₂𝑦^𝑧^2−𝛼₁₁𝑧^3+𝛽₁₂𝑦^2+2𝛽₁₁𝑦^𝑧^−𝛽₁₂𝑧^2+𝛾₁₂𝑦^′+𝛾₁₁𝑧^′+𝛿₁₂=0,(3.10) here the coefficients 𝛼_1𝑖, 𝛽_1𝑖, 𝛾_1𝑖, and 𝛿_1𝑖 for 𝑖=1,2 are functions of 𝑥, 𝑦, and 𝑧. Clearly, the coefficients 𝛼_1𝑖,𝛽_1𝑖,𝛾_1𝑖, and 𝛿_1𝑖 satisfy the CR-equations, that is, 𝛼_11,𝑦=𝛼_12,𝑧,𝛼_12,𝑦=−𝛼_11,𝑧 and vice versa. It is obvious that (3.9) generates a system by decomposing the complex coefficients 𝐸_𝑗, 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: 𝛼₁₁=−13𝛼¹³=13𝛼²²=−𝛼₂₄=𝛼₁₁,𝛽₂₁=−12𝛽¹²𝛼₁₄=−𝛽₂₃=𝛽₁₂,=−13𝛼¹²=−13𝛼²³=𝛼₂₁=𝛼₁₂,𝛽₁₁=12𝛽²²𝛾₁₁=−𝛽₁₃=𝛽₁₁,=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 𝑢𝑢^=𝜁₄(𝑥)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 𝑧𝑦^_=𝛼₁(𝑥)𝑦^−𝛼₂(𝑥)𝑧^,𝑦=𝛼₂(𝑥)𝑦^+𝛼₁(𝑥)𝑧^,(4.4)_𝑧=𝛼₃(𝑥)𝑦−𝛼₄(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 𝑌=𝑦𝜌(𝑥),𝑍=𝜌(𝑧𝑥),𝑋=^𝑥𝜌⁻²(𝑠)𝑑𝑠,(4.6) where 𝜌 satisfies 𝜌^−𝛼₃𝜌=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)𝑦₁−𝑀₂(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 𝛼𝛼₁₂𝑀₁−𝛼₂𝑀₂=2𝑀^1,𝑀₁+𝛼₁𝑀₂=2𝑀^2,(4.10) and 𝑦^∗, 𝑧^∗are 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 𝛼₃(𝑥)=1𝑀²¹+𝑀²²𝛼𝑀₁𝛼₁𝑀^1−𝛼₂𝑀^2−𝑀^1+𝑀₂𝛼₁𝑀^2+𝛼₂𝑀^1−𝑀^2₄,(𝑥)=1𝑀²¹+𝑀²²𝑀₁𝛼₁𝑀^2+𝛼₂𝑀^1−𝑀^2−𝑀₂𝛼₁𝑀^1−𝛼₂𝑀^2−𝑀^1.(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) 𝜂𝜉_,𝑥𝑥=𝜉_,𝑥𝑦=𝜉_,𝑦𝑦=0=𝜂_1,𝑧𝑧=𝜂_2,𝑦𝑦,(5.1)_1,𝑦𝑦𝜉−2𝜉_𝑥𝑦=𝜂_1,𝑦𝑧−𝜉_,𝑥𝑧=𝜂_2,𝑦𝑧−𝜉_𝑥𝑦=𝜂_2,𝑧𝑧−2𝜉_,𝑥𝑧=0,(5.2)_,𝑥𝑥𝜉−2𝜂_1,𝑥𝑦−3𝑧𝛽(𝑥)𝜉_,𝑦+𝑦𝛽(𝑥)𝜉_,𝑧=𝜂_1,𝑥𝑧+𝑧𝛽(𝑥)𝜉_,𝑧=0,(5.3)_,𝑥𝑥𝜂−2𝜂_2,𝑥𝑧+3𝑦𝛽(𝑥)𝜉_,𝑧−𝑧𝛽(𝑥)𝜉_,𝑦=𝜂_2,𝑥𝑦−𝑦𝛽(𝑥)𝜉_,𝑦=0,(5.4)_1,𝑥𝑥𝜂+𝛽(𝑥)𝑦𝜂_1,𝑧+2𝑧𝜉_,𝑥−𝑧𝜂_1,𝑦+𝜂₂+𝑧𝛽^(𝑥)𝜉=0,(5.5)_2,𝑥𝑥+𝛽(𝑥)𝑦𝜂_2,𝑧−2𝑦𝜉_,𝑥−𝑧𝜂_2,𝑦−𝜂₁−𝑦𝛽^(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 𝜂𝜉=𝛾₁(𝑥)𝑦+𝛾₂(𝑥)𝑧+𝛾₃(𝑥),₁𝜂=𝛾^1(𝑥)𝑦²+𝛾^2(𝑥)𝑦𝑧+𝛾₄(𝑥)𝑦+𝛾₅(𝑥)𝑧+𝛾₆(𝑥),₂=𝛾^1(𝑥)𝑦𝑧+𝛾^2(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 𝜂𝜂_1,𝑥𝑥=𝜂_1,𝑥𝑧=𝜂_1,𝑧𝑧=0,_2,𝑥𝑥2𝜉_,𝑥𝑦=𝜂_2,𝑥𝑦=𝜂_2,𝑦𝑦=0,𝜉_,𝑥𝑧−𝜂_1,𝑦𝑦=0=2𝜉_,𝑥𝑧−𝜂_2,𝑧𝑧,𝜉_,𝑥𝑥−𝜂_1,𝑦𝑧=0=𝜉_,𝑥𝑦−𝜂_2,𝑦𝑧,−2𝜂_1,𝑥𝑦=0=𝜉_,𝑥𝑥−2𝜂_2,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 𝜂𝜉=𝛾₃(𝑥),₁=𝛾^3(𝑥)2+𝑐³𝑦+𝑐¹𝜂𝑧+𝛾₆(𝑥),₂=𝑐₂𝑦+𝛾^3(𝑥)2+𝑐⁴𝑧+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 𝜉𝜂₁=𝐶₁,=𝐶₂𝑦+−𝐶₄𝑒^√𝑥/2−𝐶₃𝑒^√−𝑥/2sin𝑥√2+𝐶⁶𝑒^√𝑥/2cos𝑥+𝐶₅√2𝜂𝑒^√−𝑥/2cos𝑥/√2+𝐶₇𝑧,₂=−𝐶₆𝑒^√𝑥/2+𝐶₅𝑒^√−𝑥/2sin𝑥√2−𝐶⁴𝑒^√𝑥/2cos𝑥√2−𝐶²+𝐶₃𝑧𝑒^√−𝑥/2cos𝑥/√2+𝐶₇𝑦.(5.11) This yields a 7-dimensional symmetry algebra.

Subcase 2.2 (𝛽(𝑥)=𝑥⁻²,𝑥⁻⁴,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)=𝑥⁻¹,𝑥²,𝑥²±𝐶₀ or 𝑒^𝑥). 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)=𝑥⁻¹, 𝑥², 𝑥²±𝐶₀ we get Bessel functions. Similarly for 𝛽(𝑥)=𝑒^𝑥 there are six symmetries, including the generators 𝑦𝜕_𝑦−𝑒^𝑥𝑧𝜕_𝑧, 𝑧𝜕_𝑧+𝑒^𝑥𝑦𝜕_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, 𝑑₁₁(𝑥)=𝑎₀, 𝑑₁₂(𝑥)=𝑏₀, and 𝑑₂₁(𝑥)=𝑐₀, where 𝑎²⁰+𝑏₀𝑐₀≠0.(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 𝑦̃=𝑎(𝑥)𝑦+𝑏(𝑥)𝑧,𝑧̃=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 𝑦𝑧̃̃^_=𝑎₀𝑦^+𝑏₀𝑧^,=𝑐₀𝑦^+𝑑₀𝑧^.(5.15) Using (2.7) and (4.5) in the above equation we find 𝑎₀𝑑₀−𝑏₀𝑐₀𝑦^=+𝑎₀𝑑2₀𝑏₀+𝑏₀𝑐₀𝑑₁₁(𝑥)+𝑐₀𝑑₀𝑑₁₂(𝑥)−𝑎₀𝑏₀𝑑₂₁(𝑥)𝑦𝑎𝑑₀𝑑₁₁(𝑥)+𝑑²⁰𝑑₁₂(𝑥)−𝑏²⁰𝑑₂₁(𝑥)𝑧,₀𝑑₀−𝑏₀𝑐₀𝑧^=+𝑎₀𝑑2₀𝑎₀+𝑏₀𝑐₀𝑑₁₁(𝑥)+𝑐₀𝑑₀𝑑₁₂(𝑥)−𝑎₀𝑏₀𝑑₂₁(𝑥)𝑧𝑐₀𝑑₁₁(𝑥)+𝑐²⁰𝑑₁₂(𝑥)−𝑎²⁰𝑑₂₁(𝑥)𝑦,(5.16) where 𝑎₀𝑑₀−𝑏₀𝑐₀≠0. Using (4.5), (5.16), and the linear independence of the 𝑑's, gives 𝑎𝑎₀𝑏₀=𝑐₀𝑑₀=0,₂₀𝑎−𝑏²⁰=𝑐²⁰−𝑑²⁰=0,₀𝑑₀+𝑏₀𝑐₀=𝑎₀𝑐₀−𝑏₀𝑑₀=0,(5.17) which has a solution 𝑎₀=𝑏₀=𝑐₀=𝑑₀=0, which is inconsistent with (5.16) because the requirement was 𝑎₀𝑑₀−𝑏₀𝑐₀≠0.

Case b. If 𝑎(𝑥), 𝑏(𝑥), 𝑐(𝑥), and 𝑑(𝑥) are arbitrary functions of 𝑥 then 𝑧𝑦̃̃^_=𝑎(𝑥)𝑦^+𝑏(𝑥)𝑧^+𝑎^(𝑥)𝑦+𝑏^(𝑥)𝑧+2𝑎^(𝑥)𝑦^+2𝑏^(𝑥)𝑧^,=𝑐(𝑥)𝑦^+𝑑(𝑥)𝑧^+𝑐^(𝑥)𝑦+𝑑^(𝑥)𝑧+2𝑐^(𝑥)𝑦^+2𝑑^(𝑥)𝑧^.(5.18) Thus we obtain (𝑎𝑑−𝑏𝑐)𝑦^=+(𝑎2𝑑𝑏+𝑑𝑑𝑏𝑐₁₁)𝑑₁₁+𝑐𝑑𝑑₁₂−𝑎𝑏𝑑₂₁−𝑎^𝑑+𝑐^𝑏𝑦(𝑎𝑑−𝑏𝑐)𝑧_+𝑑²𝑑₁₂−𝑏²𝑑₂₁−𝑏^𝑑+𝑑^𝑏𝑧−2𝑑𝑎^𝑦^+𝑏^𝑧^+2𝑏𝑐^𝑦^+𝑑^𝑧^,=+2𝑎(𝑎𝑐𝑑𝑑+₁₁+𝑏𝑐𝑐)²𝑑𝑑₁₂₁₁−𝑎²𝑑₂₁−𝑎^𝑐+𝑐^𝑎𝑦+𝑐𝑑𝑑₁₂−𝑎𝑏𝑑₂₁−𝑏^𝑐+𝑑^𝑎𝑧−2𝑐𝑎^𝑦^+𝑏^𝑧^+2𝑎𝑐^𝑦^+𝑑^𝑧^.(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 𝑦𝑧^+_𝑦^2−𝑧^2=Ω₁𝑥,𝑦,𝑧,𝑦^,𝑧^,+2𝑦^𝑧^=Ω₂𝑥,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^+𝑢^2=Ω𝑥,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)=𝑒^𝑢.(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 𝑍𝑌^_=Ω₁𝜒,𝑌,𝑍,𝑌^,𝑍^−Ω₂𝜒,𝑌,𝑍,𝑌^,𝑍^,=Ω₂𝜒,𝑌,𝑍,𝑌^,𝑍^+Ω₁𝜒,𝑌,𝑍,𝑌^,𝑍^.(6.4) Here Ω₁ and Ω₂ 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 Ω₁=−2𝑥𝑦^Ω₂,=−2𝑥𝑧^,(6.5) it admits a 15-dimensional algebra. The real linearizing transformations 𝜒(𝑥)=1𝑥,𝑌=𝑒^𝑦cos(𝑧),𝑍=𝑒^𝑦sin(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^+𝑢^2+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 ΩΩ₁₂=𝑐₁𝑦^−𝑐₂𝑧^,=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^+𝑢^2−𝑐𝑢^=0.(6.9) Using the transformations (6.3) to generate the real transformations 𝜒(𝑥)=𝑥,𝑌=𝑒^𝑦cos(𝑧),𝑍=𝑒^𝑦sin(𝑧),(6.10) which map the nonlinear system to a linear system of the form (4.4), that is, 𝑍𝑌^_=𝑐₁𝑌^−𝑐₂𝑍^,=𝑐₂𝑌^+𝑐₁𝑍^,(6.11) which also has a 7-dimensional symmetry algebra and corresponds to 𝑈^−𝑐𝑈^=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+𝛽𝑒^{𝑐𝜒(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, ΩΩ₁₂=(1+𝑥)𝑐₁𝑦^−𝑐₂𝑧^,=(1+𝑥)𝑐₂𝑦^+𝑐₁𝑧^,(6.14) in (6.1), then the same transformations (6.10) converts the above system into a linear system 𝑍𝑌^_=(1+𝜒)𝑐₁𝑌^−𝑐₂𝑍^,=(1+𝜒)𝑐₂𝑌^+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 𝑐_𝑖(𝑖=1,2) are non-zero constants, then under the same real transformations (6.10), the nonlinear system (6.1) takes the form 𝑍𝑌^=𝑐₁𝑌−𝑐₂𝑍,_=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 𝐷_𝑥(𝑋)⋅𝐷^2𝑥(𝑌)−𝐷_𝑥(𝑌)⋅𝐷^2𝑥(𝑋)𝐷_𝑥(𝑋)³𝐷=0,_𝑥(𝑋)⋅𝐷^2𝑥(𝑍)−𝐷_𝑥(𝑍)⋅𝐷^2𝑥(𝑋)𝐷_𝑥(𝑋)³=0.(A.1) Substituting 𝐷_2𝑥𝐷_𝑥(𝑋)=𝑋_𝑥+𝑦^𝑋_𝑦+𝑧^𝑋_𝑧,(𝑋)=𝑋_𝑥𝑥+2𝑦^𝑋_𝑥𝑦+2𝑧^𝑋_𝑥𝑧+𝑦^2𝑋_𝑦𝑦+2𝑦^𝑧^𝑋_𝑦𝑧+𝑧^2𝑋_𝑧𝑧+𝑦^𝑋_𝑦+𝑧^𝑋_𝑧,(A.2) and similar expressions for 𝐷_𝑥(𝑌),𝐷_𝑥(𝑍),𝐷^2𝑥(𝑌), and 𝐷^2𝑥(𝑍) in (A.1), yields 𝛼₁𝑦^+𝛼₂𝑧^+𝛽₁𝑦^3+𝛽₂𝑦^2𝑧^+𝛽₃𝑦^𝑧^2+𝛽₄𝑧^3+𝛾₁𝑦^2+𝛾₂𝑦^𝑧^+𝛾₃𝑧^2+𝛿₁𝑦^+𝛿₂𝑧^+𝜖₁=0,𝛼₃𝑦^+𝛼₄𝑧^+𝛽₅𝑦^3+𝛽₆𝑦^2𝑧^+𝛽₇𝑦^𝑧^2+𝛽₈𝑧^3+𝛾₄𝑦^2+𝛾₅𝑦^𝑧^+𝛾₆𝑧^2+𝛿₃𝑦^+𝛿₄𝑧^+𝜖₂=0.(A.3)

The coefficients of the above system of ODEs are 𝛼₁=𝑋_𝑥𝑌_𝑦−𝑌_𝑥𝑋_𝑦+𝑧^𝑋_𝑧𝑌_𝑦−𝑌_𝑧𝑋_𝑦,𝛼₂=𝑋_𝑥𝑌_𝑧−𝑌_𝑥𝑋_𝑧+𝑦^𝑋_𝑦𝑌_𝑧−𝑌_𝑦𝑋_𝑧,𝛼₃=𝑋_𝑥𝑍_𝑦−𝑍_𝑥𝑋_𝑦+𝑧^𝑋_𝑧𝑍_𝑦−𝑍_𝑧𝑋_𝑦,𝛼₄=𝑋_𝑥𝑍_𝑧−𝑍_𝑥𝑋_𝑧+𝑦^𝑋_𝑦𝑍_𝑧−𝑍_𝑦𝑋_𝑧,𝛽₁=𝑋_𝑦𝑌_𝑦𝑦−𝑌_𝑦𝑋_𝑦𝑦,𝛽₂=𝑋_𝑧𝑌_𝑦𝑦−𝑌_𝑧𝑋_𝑦𝑦+2𝑋_𝑦𝑌_𝑦𝑧−𝑌_𝑦𝑋_𝑦𝑧,𝛽₃=𝑋_𝑦𝑌_𝑧𝑧−𝑌_𝑦𝑋_𝑧𝑧+2𝑋_𝑧𝑌_𝑦𝑧−𝑌_𝑧𝑋_𝑦𝑧,𝛽₄=𝑋_𝑧𝑌_𝑧𝑧−𝑌_𝑧𝑋_𝑧𝑧,𝛽₅=𝑋_𝑦𝑍_𝑦𝑦−𝑍_𝑦𝑋_𝑦𝑦,𝛽₆=𝑋_𝑧𝑍_𝑦𝑦−𝑍_𝑧𝑋_𝑦𝑦+2𝑋_𝑦𝑍_𝑦𝑧−𝑍_𝑦𝑋_𝑦𝑧,𝛽₇=𝑋_𝑦𝑍_𝑧𝑧−𝑍_𝑦𝑋_𝑧𝑧+2𝑋_𝑧𝑍_𝑦𝑧−𝑍_𝑧𝑋_𝑦𝑧,𝛽₈=𝑋_𝑧𝑍_𝑧𝑧−𝑍_𝑧𝑋_𝑧𝑧,𝛾₁=𝑋_𝑥𝑌_𝑦𝑦−𝑌_𝑥𝑋_𝑦𝑦+2𝑋_𝑦𝑌_𝑥𝑦−𝑌_𝑦𝑋_𝑥𝑦,𝛾₂=2𝑋_𝑥𝑌_𝑦𝑧+𝑋_𝑦𝑌_𝑥𝑧+𝑋_𝑧𝑌_𝑥𝑦−𝑌_𝑥𝑋_𝑦𝑧+𝑌_𝑦𝑋_𝑥𝑧+𝑌_𝑧𝑋_𝑥𝑦,𝛾₃=𝑋_𝑥𝑌_𝑧𝑧−𝑌_𝑥𝑋_𝑧𝑧+2𝑋_𝑧𝑌_𝑥𝑧−𝑌_𝑧𝑋_𝑥𝑧,𝛾₄=𝑋_𝑥𝑍_𝑦𝑦−𝑍_𝑥𝑋_𝑦𝑦+2𝑋_𝑦𝑍_𝑥𝑦−𝑍_𝑦𝑋_𝑥𝑦,𝛾₅=2𝑋_𝑥𝑍_𝑦𝑧+𝑋_𝑦𝑍_𝑥𝑧+𝑋_𝑧𝑍_𝑥𝑦−𝑍_𝑥𝑋_𝑦𝑧+𝑍_𝑦𝑋_𝑥𝑧+𝑍_𝑧𝑋_𝑥𝑦,𝛾₆=𝑋_𝑥𝑍_𝑧𝑧−𝑍_𝑥𝑋_𝑧𝑧+2𝑋_𝑧𝑍_𝑥𝑧−𝑍_𝑧𝑋_𝑥𝑧,𝛿₁=𝑋_𝑦𝑌_𝑥𝑥−𝑌_𝑦𝑋_𝑥𝑥+2𝑋_𝑥𝑌_𝑥𝑦−𝑌_𝑥𝑋_𝑥𝑦,𝛿₂=𝑋_𝑧𝑌_𝑥𝑥−𝑌_𝑧𝑋_𝑥𝑥+2𝑋_𝑥𝑌_𝑥𝑧−𝑌_𝑥𝑋_𝑥𝑧,𝛿₃=𝑋_𝑦𝑍_𝑥𝑥−𝑍_𝑦𝑋_𝑥𝑥+2𝑋_𝑥𝑍_𝑥𝑦−𝑍_𝑥𝑋_𝑥𝑦,𝛿₄=𝑋_𝑧𝑍_𝑥𝑥−𝑍_𝑧𝑋_𝑥𝑥+2𝑋_𝑥𝑍_𝑥𝑧−𝑍_𝑥𝑋_𝑥𝑧,𝜖₁=𝑋_𝑥𝑌_𝑥𝑥−𝑌_𝑥𝑋_𝑥𝑥,𝜖₂=𝑋_𝑥𝑍_𝑥𝑥−𝑍_𝑥𝑋_𝑥𝑥.(A.4)

System (A.3) yields a system of the form (3.6) with the following coefficients:𝛼𝛼¹¹₁₄=𝜏₁𝛼₄𝛽₁−𝛼₂𝛽₅,𝛼¹²=𝜏₁𝛼₄𝛽₂−𝛼₂𝛽₆,𝛼¹³=𝜏₁𝛼₄𝛽₃−𝛼₂𝛽₇,=𝜏₁𝛼₄𝛽₄𝛼−₂₃𝛼₂𝛽₈,𝛼²¹=𝜏₂𝛼₃𝛽₁−𝛼₁𝛽₅,𝛼²²=𝜏₂𝛼₃𝛽₂−𝛼₁𝛽₆,𝛽₁₁=𝜏₂𝛼₃𝛽₃−𝛼₁𝛽₇,𝛼²⁴=𝜏₂𝛼₃𝛽₄−𝛼₁𝛽₈,𝛽₂₁=𝜏₁𝛼₄𝛾₁−𝛼₂𝛾₄,𝛽₁₂=𝜏₁𝛼₄𝛾₂−𝛼₂𝛾₅,𝛽₁₃=𝜏₁𝛼₄𝛾₃−𝛼₂𝛾₆,𝛾₁₁=𝜏₂𝛼₃𝛾₁−𝛼₁𝛾₄,𝛽₂₂=𝜏₂𝛼₃𝛾₂−𝛼₁𝛾₅,𝛽₂₃=𝜏₂𝛼₂𝛾₃−𝛼₁𝛾₆,𝛾=₂₂𝜏₁𝛼₄𝛿₁−𝛼₂𝛿₃,𝛾¹²=𝜏₁𝛼₄𝛿₂−𝛼₂𝛿₄,𝛾²¹=𝜏₂𝛼₃𝛿₁−𝛼₁𝛿₃,=𝜏₂𝛼₃𝛿₂−𝛼₁𝛿₄,𝛿¹=𝜏₁𝛼₄𝜖₁−𝛼₂𝜖₂,𝛿₂=𝜏₂𝛼₃𝜖₁−𝛼₁𝜖₂,(A.5)

where 𝜏₁=−𝜏₂=𝛼₁𝛼₄−𝛼₂𝛼₃.

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.