On the classification of 2 ( 1 ) n n dimensional non-linear Klein-Gordon equation via Lie and Noether approach

A complete group classification for the Klein-Gordon equation is presented. Symmetry generators, up to equivalence transformations, are calculated for each ) (u f when the principal Lie algebra extends. Further, considered equation is investigated by using Noether approach for the general case 2  n . Conserved quantities are computed for each calculated Noether operator. At the end, a brief conclusion is presented.


INTRODUCTION
The (1+n)-dimensional Klein-Gordon equation In the past, the authors of [3,10] have studied Eq. (1) for different values of n, for exact solutions, compatibility of the conditions for the reduction and reduced equations by consideration of an ansatz which reduces the dimension of the corresponding PDE (see [11]). In [9], the author discussed the symmetry properties and found particular solutions for some cases of Eq. (2). Tajiri [20] proposed some similarity and soliton solutions for the three-dimensional Klein-Gordon equation by means of similarity variables. Fushchych et al. [12] investigated the reductions and solutions by using the broken symmetry for Eq.(1) with n=3. In [8], Fedorchuk considered the reductions of Eq.(1) for n=4 by using decomposable subgroups of the generalized Poincare group P (1,4). Fushchych [10] invoked an ansatz of the form to analyze exact solutions of Eq. (1). Description of such an ansatz for the Eq. (1) can be a difficult problem. That problem can be simplified by using symmetry methods.
Lie symmetry analysis is a systematic way to construct an ansatz which further reduces the dimension of the differential equation. The symmetry method also plays a central role in the algebraic analysis of the differential equation. There are nonlinear equations with arbitrary coefficients which possess nontrivial Lie point symmetries. Such nonlinear differential equations can be classified, with respect to unknown functions, according to the nontrivial Lie point symmetries they admit. This classification is known as group classification. The problem of group classification is one of the central aspects of modern symmetry analysis of differential equations. It was performed in the classical works of Lie.
One of the classical aspects of the Lie theory is the computation of conservation laws. The existence of a large number of conserved quantities of a PDE or system of PDEs is a strong indication of its integrability. An efficient method to compute conservation laws is given by Noether [6,17]. The theorem states that there is a conservation law for the Noether symmetry of the differential equation. Conservation laws for the nonlinear (1+1)-dimensional wave equation viz [15]. Bokhari et al. constructed the conservation laws [5] for the nonlinear (1+n)-dimensional wave equation  via partial Noether approach. Conserved quantities for the (1+1)-dimensional nonlinear Klein-Gordon equation are reported in [14].

Fundamental operators
Consider the 2 nd order PDE of the type The Euler operator is ..., 1,..., are known as the total derivative operators.
The generalized or Lie Backlund operator is defined by: The generalized operator (6) satisfying is known as the Noether operator associated with a Lagrangian L .
In Eq. (8),  (2) is known as the conservation law for Eq. (2) and vector The conserved vectors of the system (2) associated with a Noether operator X can be determined from the formula ( ).
In Eq. (9), 0 , , where W is known as the Lie characteristic function and can be found from The outline of this paper is as follows. In Section 2, the group classification of the (1+n) -dimensional Klein-Gordon equation is given. Section 3 is for the Noether symmetry operators and conserved vectors of Eq. (1). Finally, conclusions are summarized at the end.

Lie point symmetries
In this section, we discuss the group classification for the (1+n)-dimensional Klein-Gordon equation, i.e. Eq. (1) for arbitrary n . We apply the 2 nd prolongation vector i.e ( ) Eq. (11) yields the following determining equations: Eq. (13) forms a set of equations for an infinitesimal conformal transformation on

When f(u) is arbitrary
The minimal algebra for the arbitrary case is: and appeared in all the rest of the considered cases, thus we shall only present the additional algebra(s). The principal Lie algebra for this case is of dimension n(3+n)/2+1.
For 0  c the principal algebra extends and additional generators will be: The Lie algebra is of dimension n(n+3)/2+2.
There is no extension in the principal algebra.
leads to an extension of the principal algebra and additional generators will be: where p is defined in (20) and the Lie algebra is of dimension n(n+3)/2+2.

Noether symmetries
In this section, we will use Noether approach for finding the conserved vectors of Eq. (1) for arbitrary n , taking 2 n  .  [14].

Case 2: n=2
The standard Lagrangian for Eq. (1) will be The Noether determining equation (8) with the help of Eq. (22) after some lengthy manipulation gives the following set of determining equations: Hence doing the routine calculation, Eq. (25) yields: Eq. (26) further divides two cases and discussed in the following sections.

 
For this case, the Noether operators will be: This forms the minimal algebra and thus thus we shall only present the additional algebras in the next section.

 
For this case, the additional Noether operators will be: