Lie Group Classification of a Generalized Lane-Emden Type System in Two Dimensions
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
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
Since the variables x and y 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, X1 = ∂x and X2 = ∂y (to be established in Section 3).
3. Equivalence Transformations
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 f(v) and g(u) 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
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
Consider also the case and , then we have
From (4.5)-(4.6) we obtain the functional forms of the arbitrary parameters f(v) and g(u) together with their corresponding extra operator(s) given by
The cases k ≠ 0 (p = 0) and k = 0 (p ≠ 0) yield the classification results given in Table 1.
| f | g | Condition on const. | Extra operator(s) |
|---|---|---|---|
| g0u−γ | λ ≠ 0 | X4 = λγx∂x + λγy∂y + 2λu∂u + 2(γ − 1)∂v | |
| g0 | λ ≠ 0 | X4 = λu∂u − ∂v, Xc = c(x, y)∂u | |
| f0v−1 | δln u | δ = ±1 | X4 = x∂x + y∂y + 2v∂v |
| g0u−γ | μ = ±1 | X4 = (1 + γ)x∂x + (1 + γ)y∂y + 4u∂u + 2(1 − γ)v∂v | |
| μv | ν ≠ 0 | X4 = νx∂x + νy∂y + 4u∂u − 2νv∂v | |
| ρu | ρ = ±1 | X4 = u∂u + v∂v, Xd = d(x, y)∂v | |
| βln v | g0 | β = ±1 | X4 = x∂x + y∂y + 2u∂u + 2v∂v, Xc = c(x, y)∂u |
| f(v) | g0u−1 | X4 = x∂x + y∂y + 2u∂u | |
| g0 | Xc = c(x, y)∂u |
Case 2. Suppose that f(v) and g(u) are nonlinear functions. Differentiation of (2.5) and (2.6) twice with respect to v and u, respectively, leads to
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 f and g.
Firstly consider f, then
The solution of (4.15) leads to the four-dimensional symmetry Lie algebra spanned by the generators
Next, in considering case (4.11), we obtain the classification result that if f = v2 and g = eu, then the principal Lie algebra is extended by the operator
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 u ↦ v and f ↦ g 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.
