Journal of Applied Mathematics

Volume 2012 (2012), Article ID 405978, 10 pages

doi:10.1155/2012/405978

Research Article

Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions

Motlatsi Molati1,2 and Chaudry Masood Khalique¹

¹International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

²Department of Mathematics and Computer Science, National University of Lesotho, Roma 180, Lesotho

Received 18 September 2012; Accepted 13 October 2012

Academic Editor: Fazal M. Mahomed

Copyright © 2012 Motlatsi Molati and Chaudry Masood Khalique. 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 aim of this work is to perform a complete Lie symmetry classification of a generalized Lane-Emden type system in two dimensions which models many physical phenomena in biological and physical sciences. The classical approach of group classification is employed for classification. We show that several cases arise in classifying the arbitrary parameters, the forms of which include amongst others the power law nonlinearity, and exponential and quadratic forms.

1. Introduction

For many years extensive studies using various approaches have been done on the Lane-Emden type equation

which has applications in astrophysics. Recently, the Lane-Emden type systems have attracted a lot of attention in modelling physical phenomena in biological and physical sciences [1, 2]. In many of these investigations the nonlinearity in the system is often assumed. However, the symmetry-based approach provides a systematic way to specify the nonlinearities in the models of physical importance and mathematical interest.

The bidimensional Lane-Emden system [3]:

is studied from the view point of both Lie and Nöether point symmetry classification where are arbitrary constants. In particular, the Lie point symmetry classification is obtained for the cases and where including the special case .

In the current study we generalize the last system by considering the bidimensional Lane-Emden system of the form

where and are nonzero arbitrary functions of their respective arguments. The underlying system (1.3) is a two-dimensional Euler-Lagrange model system in elastostatics [4].

Recently, in [5] Nöether point symmetry classification of system (1.3) is performed and various forms of the arbitrary functions are obtained which include linear, power, exponential, and logarithmic types.

The plan of this work is organized as follows. In Section 2 we generate the classifying relations (determining equations for the arbitrary elements). The computation of the equivalence transformations is presented in Section 3. In Section 4 the Lie group classification of the underlying system is performed. Finally, we summarize our investigations in Section 5.

2. Generator of Symmetry Group and Classifying Relations

According to the Lie algorithm we seek the generator of Lie point symmetries for system (1.3) of the form

The application of the second prolongation of (2.1) on the underlying system yields the determining equations which are solved for , and , see for details [4, 6, 7]. The manual generation and manipulation of determining equations is a tiring task. Fortunately, nowadays the Lie algorithm has been implemented using the computer software packages for symbolic computation such as the YaLie package [8] which is used in this work. Therefore, with the help of the YaLie package written in Mathematica the following determining equations are generated:

where subscripts denote partial differentiation with respect to the indicated variables and “prime” indicates total derivative with respect to the given argument.

The manipulation of (2.2) yields the general generator of symmetry group for system (1.3) in the form

where are arbitrary constants and are the arbitrary smooth functions which satisfy

The determining equations (2.5) and (2.6) are known as the classifying relations/equations.

Since the variables and do not appear explicitly in the underlying system (1.3), the principal symmetry Lie algebra admitted by this system is spanned by at least two operators, namely, and (to be established in Section 3).

3. Equivalence Transformations

Following the infinitesimal approach [9] we consider the generator of equivalence group of the form

where , and for .

The operator (3.1) is the generator of equivalence group for system (1.3) provided it is admitted by the extended system

We require the prolonged operator for the extended system (3.2) having the form

where is the second prolongation of (3.1) given by

The variables and are given by the prolongation formulae

respectively, where

are the total derivative operators and the total derivative operators for the extended system are given by

Upon application of the prolongation (3.3), the invariance conditions of system (3.2) read

The solution of system (3.8) is given by

where are arbitrary constants.

Therefore, system (1.3) has -dimensional equivalence Lie algebra spanned by the operators

The composition of the one-parameter group of transformations for each yields the equivalence group for system (1.3) given by the transformations

where and are arbitrary constants.

Next we use the theorem on projections of equivalence Lie algebras [9] to establish the principal Lie algebra for system (1.3). The projections of the equivalence generator (3.1) are given by

where denotes projection onto the space and onto the space .

An operator spans the principal Lie algebra provided the following condition holds:

In view of (3.1) taking into account (3.9) and (3.13), equation (3.14) is recast as

Thus, we have

Now the generator of equivalence group (3.1) reduces to

and therefore the principal Lie algebra for system (1.3) is three-dimensional and it is spanned by the operators

Remark 3.1. The principal Lie algebra (3.18) can be achieved alternatively by solving the resulting equations obtained from splitting the determining equations (2.5) and (2.6) with respect to the arbitrary elements and their derivatives.

Our goal in Section 4 is to extend the principal Lie algebra, that is, we obtain the functional forms of the arbitrary elements and which provide additional operator(s).

4. Group Classification

Case 1. Following the classical approach of group classification [7], the classifying relations (2.5) and (2.6) become

where , and are arbitrary constants.

It is noted that the analysis of (4.1) and (4.2) is similar.

Upon the use of equivalence transformations (3.11), the classifying equations (4.1) and (4.2) take the form

provided

As an illustration in the analysis of (4.3) we consider the case and . Thus, we obtain

where , and .

Consider also the case and , then we have

where , and .

From (4.5)-(4.6) we obtain the functional forms of the arbitrary parameters and together with their corresponding extra operator(s) given by

where , and are nonzero arbitrary constants.

The cases and yield the classification results given in Table 1.

Table 1: Classification results:    and   .

Condition on const.
Extra operator(s)

,  

,  

,  

Case 2. Suppose that and are nonlinear functions. Differentiation of (2.5) and (2.6) twice with respect to and , respectively, leads to

Thus, the last equations prompt consideration of the following set of cases:

Consider case (4.10) for illustration. We obtain , respectively, where and are arbitrary constants of integration. We make use of the equivalence transformations (3.11) in order to have simplified forms of and .

Firstly consider , then

Therefore, when we drop the bars and set the equivalence relation for is given by , where and . Likewise, for arbitrary constants and . Substitution of the forms of and into (2.4)–(2.6) yields

where

according to  (4.9)

The solution of (4.15) leads to the four-dimensional symmetry Lie algebra spanned by the generators

provided and are quadratic in and , respectively, that is,

However, the last result (4.18) is included in (4.7) for .

Next, in considering case (4.11), we obtain the classification result that if and , then the principal Lie algebra is extended by the operator

The classification results of case (4.11) can be mapped into those of (4.12) by the use of the equivalence transformations and . The last case (4.13) does not yield the forms of and such that the principal Lie algebra is extended.

Note. It should be noted that not included in the preceding classification results are the cases for which the functional forms of the arbitrary elements do not extend the principal Lie algebra, this includes amongst others the case for which both functions are of logarithmic forms. Moreover, the cases which are the same under the equivalence transformations and are also excluded. The constant coefficient case is also excluded.

5. Conclusion

In this work we performed the Lie symmetry classification of a generalized bidimensional Lane-Emden type system. The functional forms of the arbitrary parameters were specified via the classical method of group classification, and these include the combination of power law nonlinearity, exponential, logarithmic, quadratic, linear, and constant forms. Many cases yielded four symmetries apart from the five-dimensional symmetry Lie algebra obtained in the case for which both parameters are of exponential forms. The other cases possess infinite dimensional symmetry Lie algebra.

Acknowledgments

M. Molati and C. M. Khalique would like to thank the Organizing Committee of “Symmetries, Differential Equations and Applications”: Galois Bicentenary (SDEA2012) Conference for their kind hospitality during the conference. M. Molati also acknowledges the financial support from the North-West University, Mafikeng Campus, through the postdoctoral fellowship.

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