Noether, Partial Noether Operators and First Integrals for the Coupled Lane-emden System

Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as pattern formation, population evolution and chemical reactions. In this paper we construct Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a coupled Lane-Emden system. Then the first integrals with respect to Noether and partial Noether operators are obtained for the Lane-Emden system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. However, the gauge function is different in certain cases.


INTRODUCTION
The generalized Lane-Emden equation where n is a real constant and ) y ( φ is a real-valued continuous function of the variable y, is many problems arising in mathematical physics and astrophysics.For certain fixed values of n and ) y ( φ , Eq. (1) has been used to model several phenomena such as the theory of stellar structure, the thermal behaviour of a spherical cloud of gas, isothermal gaseous sphere and the theory of thermionic currents [1][2][3].Various methods (for example numerical, perturbation, Adomian's decomposition, homotopy analysis, power series, differential transformation, and variational approach) for the solution to the generalized Lane-Emden equation (1) have been widely studied in the literature.See for example [4][5][6][7] and the references therein.
Noether's theorem [8] reveals the general connection between symmetries and conservation laws.In fact, it provides the formula for construction of the conserved quantities (first integrals) for Euler-Lagrange differential equations once their symmetries are known.First integrals are of interest because they tell us something physically about the system [9] and also they reduce the order of the differential equations.
Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as: pattern formation; population evolution; chemical reactions; and so on (see for example [22]) and have attracted much attention in recent years.Several authors have established existence and uniqueness results for the Lane-Emden systems [23,24] and other related systems (see e.g., [25][26][27] and references therein).
The purpose of this study is to investigate the Noether and partial Noether operators and construct first integrals for the coupled Lane-Emden system , 0 dt where n, p and q are real constants, which is a natural extension of the celebrated Lane-Emden equation.
The paper is structured as follows.In Section 2 we briefly recall the preliminaries of the Noether and partial Noether symmetry approach.We obtain all Noether operators and the corresponding first integrals for the system (2) in Section 3. Then in Section 4 we determine all partial Noether operators and the associated first integrals for the same system and comparison is made with the Noether case.Concluding remarks are mentioned in Section 5.

PRELIMINARIES ON NOETHER, PARTIAL NOETHER OPERATORS AND FIRST INTEGRAL
In this section we present some definitions, which we utilize in Section 3.For details the reader is referred to [28][29][30][31][32]. Consider the vector field which has first extension where τ & , ξ & and η & denote total time derivatives of τ, ξ and η respectively.Let us consider the second-order system of differential equations , i.e., Eqs. ( 5) are equivalent to the Euler-Lagrange equations [32] , 0 u

Definition 1
The vector field X of the form (5) Here D is the total differentiation operator defined by [28] .
The generator X as in (3) The following theorems are taken from [8] and [29] respectively.Theorem 1 (Noether [8]) If X as given in ( 3) is a Noether point symmetry generator corresponding to a Lagrangian is a Noether first integral of Eqs.(5) associated with the operator X.

NOETHER POINT SYMMETRIES OF SYSTEM (2)
Consider the Lane-Emden system (2), viz., , 0 dt It can be verified that the natural Lagrangian of this system is The insertion of L from (11) into Eq.( 7) and separation with respect to powers of u & and ν & yields linear overdetermined system of eight PDEs.These are , After some straightforward, albeit tedious and lengthy calculations, the above system gives , t Thus we obtain a single Noether point symmetry for the system (2) with condition (24).Using Theorem 1, due to Noether, we obtain the first integral We now consider the case when 1 p − = and 1 q − = .In this case the Lane-Emden which has a standard Lagrangian The substitution of L from (25) into Eq.( 7) and splitting with respect to the powers of u & and ν & gives the following system of PDEs: The analysis of (34) yields the following two cases: This case provides us with two Noether point symmetries namely, For both cases B=0.Using Theorem 1, we obtain the Noether's first integrals corresponding to 1 X and 2 X as Here we also obtain two Noether operators; .Invoking Theorem 1 due to Noether, we obtain the first integrals associated with 1 X and 2 X as respectively.

PARTIAL NOETHER OPERATORS OF THE LANE-EMDEN SYSTEM (2)
According to definition 2, the operator X in (3) A routine calculation for the above system yields Hence we obtain a single partial Noether operator viz., X for the system (2) with condition (46).Thus we see that in this case the partial Noether operator is the same as in the Noether case.The difference resides in the value of the gauge function B. Using Theorem 2, and the partial Noether operator X we obtain the