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

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.


Introduction
Lie used algebraic symmetry properties of differential equations to extract their solutions [8,9,10,11]. 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 [12], i.e. 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 [13].
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 [6]. 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 [19]. 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 [5,14,15] 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 semi-linear system in m dimensions generically becomes a non-homogeneous, cubically semi-linear 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, IR ) and hence the number of generators is n 2 + 4n + 3. Thus for a system of two equations to be linearizable by this method it must have 15 generators.
A scalar complex ODE involves two real functions of two real variables, yielding a system of two partial differential equations (PDEs) [1,2]. 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 with the above mentioned systems obtained earlier (by geometric means) [15], has been proved [17]. Thus their symmetry structures are not the same. We prove that a general two-dimensional system of secondorder 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 for the systems of ODEs 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 [19] 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 we provide here, are characterized by complex transformations of the form 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.

Preliminaries
The simplest form of a second-order equation has the maximal-dimensional algebra, sl(3, IR ). 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 2n 2 + n arbitrary coefficients and some canonical forms that have fewer arbitrary coefficients [20]. Any system of n second-order non-homogeneous linear ODEs can be mapped invertibly to one of the following forms v = Cv, where A, B, C, D are n × n matrix functions, u, v, w, c are vector functions and dot represents differentiation relative to the independent variable t. For a system of two secondorder ODEs (n = 2) there are a total of 10 coefficients for the system represented by equation (1). It is reducible to the first and second canonical forms, (2) and (3) respectively. Thus a system with 4 arbitrary coefficients of the form can be obtained by using the equivalence of (1) and the counterpart of the Laguerre-Forsyth second canonical form (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 [19] where ρ satisfies to the linear systemỹ This procedure of reduction of arbitrary coefficients for linearizable systems simplifies the classification problem enormously. System (7) is called the optimal canonical form for linear systems of two second-order ODEs, as it has the fewest arbitrary coefficients, namely three.

Systems of ODEs obtainable by CSA
Following the classical Lie procedure, one uses point transformations to map the general linearizable system of two second-order ODEs [16], which is (at most) cubically semi-linear in both the dependent variables, where prime denotes differentiation relative to x, to the simplest form where the prime now denotes differentiation with respect to X and the mappings (9) are invertible. The derivatives transform as and where D x is the total derivative operator. This yields the coefficients being functions of the independent and dependent variables. System (14) is the most general candidate for two second-order ODEs that may be linearizable. While another candidate of linearizability of two dimensional systems obtainable from the most general form of a complex linearizable equation where u is a complex function of the real independent variable x, is also cubically semi-linear i.e. a system of the form here the coefficientsᾱ 1i ,β 1i ,γ 1i andδ 1i for i = 1, 2 are functions of x, y and z. Clearly, the system (16) corresponds to (15) if the coefficientsᾱ 1i ,β 1i ,γ 1i andδ 1i satisfy the CRequations i.e. α 11,y = α 12,z , α 12,y = −α 11,z and vice versa. It is obvious as (15) generates a system by breaking the complex coefficients E j , for j = 0, 1, 2, 3 into real and imaginary parts where all the functions are analytic. Hence we can state the following theorem.
Theorem 1. A general two dimensional system of second-order ODEs (10) corresponds to a complex equation if and only if ω 1 and ω 2 satisfy the CR-equations For the correspondence of both the cubic forms (14) and (16) of two dimensional systems we state the following theorem.
Proof. It can be trivially proved if we rewrite the above equations asᾱ 1i ,β 1i andγ 1i , respectively. These coefficients correspond to complex coefficients of (15) if and only if they satisfy the CR-equations.
Thus Theorem (1) and (2) identify those two dimensional systems which are obtainable from complex equations.

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 where prime denotes differentiation relative to x and u(x) = y(x)+iz(x) is a complex function of the real independent variable x. As all the linear scalar second-order ODEs are equivalent, so equation (21) is equivalent to the following scalar second-order complex ODEs where all the three forms (21), (22) and (23) 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, i.e. 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 (22) and (23) we put ζ 4 (x) = α 1 (x) + iα 2 (x) and ζ 5 (x) = α 3 (x) + iα 4 (x) to obtain two linear forms of system of two linear second-order ODEs and thus we state the following theorem.
Theorem 3. 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 (24) or (25).
Notice that here we have only two arbitrary coefficients in both the linear forms, while the minimum number obtained before was three i.e. a system of the form (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 (25) can be reduced further by the change of variables where ρ satisfies to where β = ρ 3 α 4 . We state this result in the form of a theorem.
Theorem 4. Any linear system of two second-order ODEs of the form (25) with two arbitrary coefficients is transformable to a simplest system of two linear ODEs (28) with one arbitrary coefficient via real point transformations (26) and (27).
Equation (28) is the reduced optimal canonical form for systems associated with complex ODEs, with just one coefficient which is an arbitrary function of x. The equivalence of systems (24) and (25) can be established via invertible point transformations, so we state the following theorem.
of the dependent variables only, where M 1 (x), M 2 (x) are two linearly independent solutions of and y * , z * are the particular solutions of (24).
Proof. Differentiating the set of equations (30) and using the result in the linear form (24), Routine calculations show that (24) can be mapped to (25) where Thus the linear form (24) is reducible to (28).
Remark 1. Any nonlinear system of two second-order ODEs that is linearizable by complex methods can be mapped invertibly to a system of the form (28) with one coefficient which is an arbitrary function of the independent variable.

Symmetry structure of linear systems obtained by CSA
To use the reduced canonical form [18] 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. Proof. The symmetry conditions provide the following set of PDEs for the system (28) ξ xx = ξ xy = ξ yy = 0 = η 1,zz = η 2,yy , Equations (34)-(37) involve an arbitrary function of the independent variable and its first derivatives. Using equations (32) and (33) we have the following solution set Using equations (34) and (35), we get Now assuming β(x) to be zero, non-zero constant and arbitrary function of x will generate the following cases.
Here two subcases arise. and This yields a 7-dimensional symmetry algebra.
Equations (34)-(37) and (41) 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.
Using equations (34)- (37) and (41), we arrive at a 6-dimensional Lie point symmetry algebra. The explicit expressions involve special functions, e.g for β(x) = x −1 , x 2 , x 2 ± C 0 we get Bessel functions. Similarly for β(x) = e x there are six symmetries, including the generators y∂ y − e x z∂ z , z∂ z + e x y∂ y . The remaining four generators come from the solution of an ODE of order four.
Thus there is only a 6, 7 or 15-dimensional algebra for linearizable systems of two secondorder ODEs transformable to (28) via invertible complex point transformations. We are not investigating the remaining two linear forms (24) and (25), because these are transformable to system (28) i.e. all these forms have the same symmetry structures. The linear forms providing 6 or 7-dimensional algebras here are obtainable by linear forms extractable from (7), with a 6 or 7 dimensional algebra respectively. Consider (7) with all the coefficients to be non-zero constants i.e.d 11 (x) = a 0 ,d 12 (x) = b 0 andd 21 (x) = c 0 , where This system provides seven symmetry generators. The linear form (28) also provides a 7dimensional algebra with constant coefficients satisfying (44), while the 8-dimensional symmetry algebra was extracted [19] by assuming Such linear forms cannot be obtained from (28). These two examples explain why a 7dimensional algebra can be obtained from (28), 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 Case a. If a(x) = a 0 , b(x) = b 0 , c(x) = c 0 and d(x) = d 0 are constants then (46) implies Using (7) and (25) in the above equation we find where a 0 d 0 − b 0 c 0 = 0. Using (25), (48) and the linear independence of thed's, gives Thus we obtain Comparing the coefficients as before and using the linear independence ofd's we obtain which implies that it reduces to a system of the form (48), which leaves us again with the same result. Thus we have the theorem.
Theorem 7. The linear forms for systems of two second-order ODEs obtainable by CSA are in general inequivalent to those linear forms obtained by real symmetry analysis.
Before presenting some illustrative applications of the theory developed we refine Theorem 6 by using Theorem 7 to make the following remark.

Remark 2.
There are only 6, 7 or 15-dimensional algebras for linearizable systems obtainable by scalar complex linearizable ODEs, i.e. there are no 5 or 8-dimensional Lie point symmetry algebras for such systems.

Applications
Consider a system of non-homogeneous geodesic-type differential equations where Ω 1 and Ω 1 are linear functions of the dependent variables and their derivatives. This system corresponds to a complex scalar equation which is either transformable to the free particle equation or one of the linear forms (21)-(23), by means of the complex transformations Which are further transformable to the free particle equation by utilizing another set of invertible complex point transformations. Generally, the system (54) is transformable to a system of the free particle equations or a linear system of the form Here Ω 1 and Ω 2 are linear functions of the dependent variables and their derivatives, via an invertible change of variables obtainable from (56). The linear form (57) can be mapped to a maximally symmetric system if and only if there exist some invertible complex transformations of the form (56), otherwise these forms can not 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 (56). Then we consider four applications to nonlinear systems of quadratically semi-linear ODEs transformable to (57) via (56) that are not further reducible to the free particle system.

1.
Consider (54) with it admits a 15-dimensional algebra. The real linearizing transformations obtainable from the complex transformations (56) with U(χ) = Y (χ) + iZ(χ), map the above nonlinear system to Y ′′ = 0, Z ′′ = 0. Moreover, the solution of (58) corresponds to the solution of the corresponding complex equation 2. Now consider Ω 1 and Ω 2 to be linear functions of the first derivatives y ′ , z ′ , i.e., system (54) with which admits a 7-dimensional algebra, provided both c 1 and c 2 , are not simultaneously zero. It is associated with the complex equation Using the transformations (56) to generate the real transformations which map the nonlinear system to a linear system of the form (24), i.e., which also has a 7-dimensional symmetry algebra and corresponds to 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 where α, β and c are complex. But these complex transformations can not generate real transformations to reduce the corresponding system (64) to a maximally symmetric system.

3.
A system with a 6−dimensional Lie algebra is obtainable from (54) by introducing a linear function of x in the above coefficients i.e., , in (54), then the same transformations (63) converts the above system into a linear system where both systems (67) and (68) 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 (24).

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 [19]. 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 non-constant function. The non existence 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 [19] with 5 and 8 generators.
Work is in progress [4] 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).
Notice that these transformations are different from (56). 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.