Journal of Applied Mathematics

Volume 2012 (2012), Article ID 484805, 11 pages

doi:10.1155/2012/484805

Research Article

Some Results on Equivalence Groups

J. C. Ndogmo

School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

Received 28 August 2012; Accepted 1 October 2012

Academic Editor: Fazal M. Mahomed

Copyright © 2012 J. C. Ndogmo. 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.

The comparison of two common types of equivalence groups of differential equations is discussed, and it is shown that one type can be identified with a subgroup of the other type, and a case where the two groups are isomorphic is exhibited. A result on the determination of the finite transformations of the infinitesimal generator of the larger group, which is useful for the determination of the invariant functions of the differential equation, is also given. In addition, the Levidecomposition of the Lie algebra associated with the larger group is found; the Levi factor of which is shown to be equal, up to a constant factor, to the Lie algebra associated with the smaller group.

1. Introduction

An invertible point transformation that maps every element in a family of differential equations of a specified form into the same family is commonly referred to as an equivalence transformation of the equation [1–3]. Elements of the family are generally labeled by a set of arbitrary functions, and the set of all equivalence transformations forms, in general, an infinite dimensional Lie group called the equivalence group of . One type of equivalence transformations usually considered [1, 4, 5] is that in which the arbitrary functions are also transformed. More specifically, if we denote by the arbitrary functions specifying the family element in , then for given independent variables and dependent variable , this type of equivalence transformations takes the form

where is the new set of independent variables, is the new dependent variable, and represents the new set of arbitrary functions. The original arbitrary functions may be functions of , , and the derivatives of up to a certain order, although quite often they arise naturally as functions of alone, and for the equivalence transformations of the type (1.1a), (1.1b), and (1.1c), the corresponding equivalence group that we denote by is simply the symmetry pseudogroup of the equation, in which the arbitrary functions are also considered as additional dependent variables.

The other type of equivalence transformations commonly considered [2, 6–8] involves only the ordinary variables of the equation, that is, the independent and the dependent variables, and thus with the notation already introduced, it consists of point transformations of the form

If we let denote the resulting equivalence group, then it follows from a result of Lie [9] that induces another group of transformations acting on the arbitrary functions of the equation. The invariants of the group are what are referred to as the invariants of the family of differential equations, and they play a crucial role in the classification and integrability of differential equations [1, 6, 10–14].

In the recent scientific literature, there has been a great deal of interest for finding infinitesimal methods for the determination of invariant functions of differential equations [2, 7, 15–17]. Some of these methods consist in finding the infinitesimal (generic) generator of , and then using it in one way or another [7, 18] to obtain the infinitesimal generator of , which gives the determining equations for the invariant functions. Most of these methods remain computationally demanding and in some cases quite inefficient, perhaps just because the connection between the three groups , , and does not seem to have been fully investigated.

We therefore present in this paper a comparison of the groups and and show in particular that can be identified with a subgroup of , and we exhibit a case where the two groups are isomorphic. We also show that the generator of admits a simple linear decomposition of the form , where is an operator uniquely associated with , and we also give a very simple and systematic method for extracting from . This decomposition also turns out to be intimately associated with the Lie algebraic structure of the equation, as we show that and each generate a Lie algebra, the two of which are closely related to the components of the Levi decomposition of the Lie algebra of .

2. The Relationship between and

We will call type I the equivalence transformations of the form (1.2a) and (1.2b) and type II those of the form (1.1a), (1.1b), and (1.1c), whose equivalence groups we have denoted by and , respectively. When the coordinates system in which a vector field is expressed is clearly understood, it will be represented only by its components, so that a vector field

will be represented simply by . On the other hand, for a vector representing a subset of coordinates, the notation will mean

Hence, with the notation introduced in the previous section, we may represent the generator of as

Let be the projection of this generator into the -space, and let be the infinitesimal generator of . Elements of may be thought of as differential equations of the form

where denotes and all its derivatives up to the order . We have the following result.

Theorem 2.1. (a) The group can be identified with a subgroup of .

(b) The component functions and are particular values of the functions and , respectively.

Proof. Suppose that the action of induced by that of on the arbitrary functions of the equation is given by the transformations

Then, since (1.2a) and (1.2b) leave the equation invariant except for the arbitrary functions, by also viewing the functions as dependent variables, (1.2a) and (1.2b) together with (2.5) constitute a symmetry transformation of the equation. This is more easily seen if we consider the inverse transformations of (1.2a) and (1.2b) which may be put in the form

If we now denote by

the resulting arbitrary functions in the transformed equation, it follows that in terms of the new set of variables , , and , any element of is locally invariant under (2.6a), (2.6b), and (2.7), and this proves the first part of the theorem. The second part of the theorem is an immediate consequence of the first part, for we can associate with any element of a triplet in , where is the action in (2.5) induced by on the arbitrary functions of the equation. The result thus follows by first recalling that has generic generator and by considering the infinitesimal counterpart of the finite transformations , which must be of the form for a certain function .

On the basis of Theorem 2.1, it is clear that one can obtain the generator of by imposing on the projection of the set of minimum conditions that reduces it to the infinitesimal generator of the equivalence group of , so that . It was also observed (see [7]) that in case is the function of alone, if we let denote the resulting value of when these minimum conditions are also imposed on , then the generator of can be obtained by setting . However, the problem that arises is that of finding the simplest and most systematic way of extracting from .

To begin with, we note that the coefficient is an -component vector that depends in general on variables, and finding its corresponding finite transformations by integrating the vector field can be a very complicated task. Fortunately, once the finite transformations of the generator of which are easier to find are known, we can easily obtain those associated with using the following result.

Lemma 2.2. The finite transformations associated with the component of are precisely given by the action (2.7) of induced by that of (2.6a) and (2.6b).

Proof. Since , where is the set of minimum conditions to be imposed on to reduce it into an infinitesimal generator of , it first follows that once the finite transformations (2.6a) and (2.6b) corresponding to are applied to the equation, the resulting equation is invariant, except for the expressions of the arbitrary functions which are now given by (2.7). Thus if are the new variables generated by the symmetry operator , where , then the only way to have an invariant equation is to set

where is the same function appearing in (2.7), and this readily proves the lemma.

3. Case of the General Third Order Linear ODE

We will look more closely at the connection between the two operators and by considering the case of the family of third-order linear ordinary differential equations (ODEs) of the form

which is said to be in its normal reduced form. Here, the arbitrary functions of the previous section are simply the coefficients of the equation. This form of the equation is in no way restricted, for any general linear third order ODE can be transformed into (3.1) by a simple change of the dependent variable [8, 16]. If we consider the arbitrary functions as additional dependent variables, then by applying known procedures for finding Lie point symmetries [19–21], the infinitesimal generator of the symmetry group in the coordinates system is found to be of the form

where

and where and are arbitrary functions of . The projection of in the -space is therefore

and a simple observation of this expression shows that due to the homogeneity of (3.1), (3.3) may represent an infinitesimal generator of the equivalence group only if . A search for the one-parameter subgroup , satisfying and generated by the resulting reduced vector field , readily gives

where

Integrating these last two equations while taking into account the initial conditions gives

where

Differentiating both sides of (3.6a) with respect to shows that . Thus, if we assume that is explicitly given by

for some function , then this leads to

and we thus recover the well-known equivalence transformation [6, 8, 14] of (3.1). Therefore, the condition is the necessary and sufficient condition for the vector in (3.3) to represent the infinitesimal generator of . In other words, the set of necessary and sufficient conditions to be imposed on to obtain is reduced in this case to setting . More explicitly, we have

where .

We would now like to derive some results on the algebraic structure of , the Lie algebra of the group related to (3.1), and its connection with that for the corresponding group . Thus, for any generator of , set , where is given by (3.9c), while takes the form

Since depends on and while depends on , we set

for any arbitrary functions and and arbitrary constant . Let , , and be the vector spaces generated by , , and , respectively, where , , and are viewed as parameters. Let

be the subspace of the Lie algebra of . We note that is obtained from simply by setting in the generator of , which according to (3.8b) amounts to ignoring the constant factor in the transformation of the dependent variable under . Moreover, we have , while itself is infinite dimensional, in general.

Theorem 3.1. (a) The vector spaces , , and are all Lie subalgebras of .

(b) and are the components of the Levi decomposition of the Lie algebra , that is,

and is a solvable ideal while is semisimple.

Proof. A computation of the commutation relations of the vector fields shows that

where the and are arbitrary functions, while the are arbitrary constants. Consequently, it readily follows from (3.14a) and (3.14b) that and are Lie subalgebras of , while setting in (3.14a) shows that is also a Lie subalgebra, and this proves the first part of the theorem. Moreover, it follows from the commutation relations (3.14a), (3.14b), and (3.14c) that , and hence that is an ideal of , while (3.14b) and (3.14c) show that is an abelian ideal in , and in particular in . Thus, we are only left with showing that is a semisimple subalgebra of . Clearly, , and if had a proper ideal , then for a given nonzero operator in , all operators would be in for all possible functions . However, since for every function of the equation

admits a solution in , it follows that would be equal to . This contradiction shows that has no proper ideal and is therefore a simple subalgebra of .

Note that part (b) of Theorem 3.1 can also be interpreted as stating that up to a constant factor, and generate the components of the Levi decomposition of . The theorem therefore shows that the decomposition is not fortuitous, as it is intimately associated with the the Levi decomposition of , and this decomposition is unique up to isomorphism for any given Lie algebra.

Although we have stated the results of this theorem only for the general linear third order equation (3.1) in its normal reduced form, these results can certainly be extended to the general linear ODE

of an arbitrary order . We first note that if we write the infinitesimal generator of the symmetry group of this equation in the form

where is the set of all arbitrary functions, then on account of the linearity of the equation, we must have

for some arbitrary functions and . Now, let again and be given by

and set . We have shown in another recent paper [16] that thus obtained using as the minimum set of conditions is the infinitesimal generator of the group for . This should certainly also hold for the linear equation (3.16) of a general order, and we thus propose the following.

Conjecture 3.2. For the general linear ODE (3.16), is the infinitesimal generator of , where is the generator of .

As already noted, it has been proved [7] that for any family of (linear or nonlinear) differential equations of any order in which the arbitrary functions depend on the independent variables alone, if is obtained by setting for some set of minimum conditions that reduce into a generator of , then is the generator of . However, the difficulty lies in finding the set of minimum conditions, and we have proved that for (3.1), is given by and extended this as a conjecture for a general linear homogeneous ODE.

Moreover, calculations done for equations of low order up to five suggest that all subalgebras appearing in Theorem 3.1 can also be defined in a similar way for the general linear equation (3.16) and that all the results of the theorem also hold for this general equation.

We now wish to pay some attention to the converse of part (a) of Theorem 2.1 which states that for any given family of differential equations, the group can be identified with a subgroup of . From the proof of that theorem, it appears that the symmetry group is much larger in general, because there are symmetry transformations that do not arise from type I equivalence transformations. A simple example of such a symmetry is given by the term appearing in (3.10) of the symmetry generator of (3.1). Indeed, by construction, its projection in the -space does not match any particular form of the generic infinitesimal generator of , where is an arbitrary function and an arbitrary constant.

Nevertheless, although (3.1) gives an example in which the inclusion is strict, there are equations for which the two groups are isomorphic. Such an equation is given by the nonhomogeneous version of (3.1) which may be put in the form

where is also an arbitrary function, in addition to and . The linearity of this equation forces its equivalence transformations to be of the form

and the latter change of variables transforms (3.20) into an equation of the form

where the , for are functions of and

The required vanishing of shows that the necessary and sufficient condition for (3.21) to represent an equivalence transformation of (3.20) is to have for some arbitrary constant . The equivalence transformations of (3.20) are therefore given by

On the other hand, the generator of the symmetry group of the nonhomogeneous equation (3.20) in the coordinates system is found to be of the form

where

and where and are arbitrary functions of and is an arbitrary constant, while is an arbitrary function of , and . Thus, has projection on -space and this is exactly the infinitesimal transformation of (3.24). Consequently, the minimum set of conditions to be imposed on to reduce it into the infinitesimal generator of is void in this case, and hence

It thus follows from Lemma 2.2 that the finite transformations associated with are given precisely by (3.24), together with the corresponding induced transformations of the arbitrary functions , and . Consequently, to each symmetry transformation in , there corresponds a unique equivalence transformation in and vice versa. We have thus proved the following results.

Proposition 3.3. For the nonhomogeneous equation (3.20), the groups and are isomorphic.

This proposition should certainly also hold for the nonhomogeneous version of the general linear equation (3.16) of an arbitrary order . In such cases, invariants of the differential equation are determined simply by searching the symmetry generator of , which must satisfy (3.26) and then solving the resulting system of linear first-order partial differential equations (PDEs) resulting from the determining equation of the form

where is the generator of prolonged to the desired order of the unknown invariants .

4. Concluding Remarks

Because type I equivalence group can be identified with a subgroup of type II equivalence group , every function invariant under must be invariant under , and hence has much more invariant functions than , and functions invariant under are naturally much easier to find than those invariant under . If we consider for instance the third order linear equation (3.1), it is well known [6] that its first nontrivial invariant function is given by the third order differential invariant

where , while at order four [16] it has two differential invariants,

It can be verified on the other hand that has no nontrivial differential invariants up to the order four.

Acknowledgment

This publication was made possible in part by a grant from the NRF CSUR program of South Africa.

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