Some General New Einstein Walker Manifolds

Abstract

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

1. Introduction

As it is well known, the symmetry group method has an important role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations was developed for the first time by Lie at the end of nineteenth century [1]. By this method, we can construct new solutions from known ones, reduce the order of ODEs, and investigate the invariant solutions (for more information about the other applications of Lie symmetries, see [2–4]). Lie’s method led to an algorithmic approach to find special solutions of differential equation by its symmetry group. These solutions that called group invariant solutions are obtained by solving the reduced system of differential equation having less independent variables than the original system. The fact that for some PDEs, the symmetry reductions are unobtainable by the Lie symmetry method caused the creation of some generalizations of this method. These generalizations are called nonclassical symmetry method and was described in these references [5–8].

In this paper, we apply the Lie symmetry method to find the invariant solutions of a system of PDEs that determines a spacial kind of Walker manifolds, called Einstein Walker manifolds. We continue this section by giving the theoretical notions needed to introduce a system of PDEs that determines Einstein Walker manifolds.

Let ⟨⋅,⋅⟩ be a nondegenerate inner product on a vector space 𝑉. We can choose a basis {𝑒_𝑖} for 𝑉 so that 𝑒_𝑖,𝑒_𝑗=0𝑖≠𝑗±1𝑖=𝑗. (1) We set 𝜀_𝑖∶=⟨𝑒_𝑖,𝑒_𝑖⟩. Let 𝑝 be the number of indices 𝑖 with 𝜀_𝑖=−1 and 𝑞=dim𝑉−𝑝. The inner product is then said to have signature (𝑝,𝑞). We can extend this argument on manifolds. Let ℳ=(𝑀,𝑔) where 𝑀 is an 𝑚-dimensional manifold and 𝑔 is a symmetric nondegenerate smooth bilinear form on 𝑇𝑀 of signature (𝑝,𝑞), such a manifold is called a pseudo-Riemannian manifold of signature (𝑝,𝑞). So we have 𝑝+𝑞=𝑚. If 𝑝=0 then 𝑔 is positive definite and ℳ is a Riemannian manifold. If (𝑥₁,…,𝑥_𝑚) is a system of local coordinates on 𝑀, we may express 𝑔=∑_𝑖,𝑗𝑔_𝑖𝑗𝑑𝑥_𝑖∘𝑑𝑥_𝑗 where 𝑔_𝑖𝑗∶=𝑔(𝜕_𝑥𝑖,𝜕_𝑥𝑗) and “∘’’ is symmetric product. Let 𝑀 be a pseudo-Riemannian manifold of signature (𝑝,𝑞). Suppose that a splitting of the tangent bundle in the form 𝑇𝑀=𝑉₁⊕𝑉₂ is given, where 𝑉₁ and 𝑉₂ are smooth subbundles which are called distribution. This defines two complementary projection 𝜋₁ and 𝜋₂ of 𝑇𝑀 onto 𝑉₁ and 𝑉₂, respectively. 𝑉₁ is called a parallel distribution if ∇𝜋₁=0. Equivalently this means that if 𝑋₁ is any smooth vector field taking values in 𝑉₁, then ∇𝑋₁ again takes values in 𝑉₁. We say that 𝑉₁ is a null parallel distribution if 𝑉₁ is parallel and if the metric restricted to 𝑉₁ vanishes identically. Manifolds which admit null parallel distributions are called Walker manifolds. Let 𝐷 be a connection on manifold 𝑀. The curvature operator ℛ is defined by the formula: ℛ(𝑋,𝑌)𝑍∶=𝐷_𝑋𝐷_𝑌−𝐷_𝑌𝐷_𝑋−𝐷_[𝑋,𝑌]𝑍,𝑋,𝑌,𝑍∈𝐶^∞(𝑇𝑀). (2) Also the associated Ricci tensor 𝜌 is defined by 𝜌(𝑋,𝑌)∶=Tr{𝑍→ℛ(𝑍,𝑋)𝑌}. (3) A Walker manifold is said to be an Einstein Walker manifold if its Ricci tensor is a scaler multiple of the metric at each point; that is, there is a constant 𝑘 so that 𝜌=𝑘𝑔. Walker showed that a canonical form for a 2𝑛-dimensional pseudo-Riemannian Walker manifold which admits an 𝑛-dimensional distribution is given by the metric tensor as 𝑔_𝑖𝑗=0𝐼𝑑_𝑛𝐼𝑑_𝑛𝐵, (4) where 𝐼𝑑_𝑛 is the 𝑛×𝑛 identity matrix and 𝐵 is a symmetric 𝑛×𝑛 matrix whose entries are functions of the (𝑥₁,…,𝑥_2𝑛) (for more details see [9, 10]).

If 𝑛=2 we adopt the following result for 4-dimensional Walker manifolds. Let 𝑀_{𝑎,𝑏,𝑐}∶=(𝒪,𝑔_{𝑎,𝑏,𝑐}), where 𝒪 be an open subset of ℝ⁴ and 𝑎,𝑏,𝑐∈𝐶^∞(𝒪) are smooth functions on 𝒪. Then we can express the general form of metric tensor for 4-dimensional Walker manifolds as follows: 𝑔_{𝑎,𝑏,𝑐𝑖𝑗}=0010000110𝑎𝑐01𝑐𝑏, (5) that is given in the coordinates form as 𝑔_{𝑎,𝑏,𝑐}∶=2𝑑𝑥₁∘𝑑𝑥₃+𝑑𝑥₂∘𝑑𝑥₄+𝑎𝑥₁,𝑥₂,𝑥₃,𝑥₄𝑑𝑥₃∘𝑑𝑥₃+𝑏𝑥₁,𝑥₂,𝑥₃,𝑥₄𝑑𝑥₄∘𝑑𝑥₄+2𝑐𝑥₁,𝑥₂,𝑥₃,𝑥₄𝑑𝑥₃∘𝑑𝑥₄. (6) We can see that 𝑀_{𝑎,𝑏,𝑐} is Einstein if and only if the functions 𝑎, 𝑏, and 𝑐 verify the following system of PDEs [9, page 81]: 𝑎₁₁−𝑏₂₂=0,𝑏₁₂+𝑐₁₁=0,𝑎₁₂+𝑐₂₂𝑎=0,₁𝑐₂+𝑎₂𝑏₂−𝑎₂𝑐₁−𝑐₂²+2𝑐𝑎₁₂+𝑏𝑎₂₂−2𝑎₂₄−𝑎𝑐₁₂+2𝑐₂₃𝑎=0,₂𝑏₁−𝑐₁𝑐₂+𝑐𝑎₁₁−𝑎₁₄−𝑏₂₃−𝑎𝑐₁₁−𝑐𝑐₁₂+𝑐₁₃−𝑏𝑐₂₂+𝑐₂₄𝑎=0,₁𝑏₁−𝑏₁𝑐₂+𝑏₂𝑐₁−𝑐₁²+𝑎𝑏₁₁+2𝑐𝑏₁₂−2𝑏₁₃−𝑏𝑐₁₂+2𝑐₁₄=0. (7) This system is hard to handle, so we consider a spacial case in this paper, where 𝑎, 𝑏, and 𝑐 only depend on 𝑥₁ and 𝑥₂. Therefore the following system must be solved: 𝑎₁₁−𝑏₂₂=0,𝑏₁₂+𝑐₁₁=0,𝑎₁₂+𝑐₂₂𝑎=0,₁𝑐₂+𝑎₂𝑏₂−𝑎₂𝑐₁−𝑐₂²+2𝑐𝑎₁₂+𝑏𝑎₂₂−𝑎𝑐₁₂𝑎=0,₂𝑏₁−𝑐₁𝑐₂+𝑐𝑎₁₁−𝑎𝑐₁₁−𝑐𝑐₁₂−𝑏𝑐₂₂𝑎=0,₁𝑏₁−𝑏₁𝑐₂+𝑏₂𝑐₁−𝑐₁²+𝑎𝑏₁₁+2𝑐𝑏₁₂−𝑏𝑐₁₂=0. (8) This work is organized as follows. In Section 2, some preliminary results about Lie symmetry method are presented. In Section 3, the infinitesimal generators of symmetry algebra of system (8) are determined and some results will be obtained. In Section 4, the optimal system of subalgebras is constructed. In Section 5, we obtain the invariant solutions corresponding to the infinitesimal symmetries of system (8).

2. Lie Symmetries Method

In this section, we recall the procedure for finding symmetries of a system of PDEs (see [2, 11]). To begin, we start with a general case of a system of PDEs of order 𝑛th with 𝑝 independent and 𝑞 dependent variables such as Δ_𝜇𝑥,𝑢^(𝑛)=0,𝜇=1,…,𝑟, (9) involving 𝑥=(𝑥¹,…,𝑥^𝑝) and 𝑢=(𝑢¹,…,𝑢^𝑞) as independent and dependent variables, respectively and all the derivatives of 𝑢 with respect to 𝑥 from order 0 to 𝑛. We consider a one parameter Lie group of transformations which acts on the all variables of (9) 𝑥^𝑖=𝑥^𝑖+𝜀𝜉^𝑖(𝑥,𝑢)+𝑜𝜀²𝑢,𝑖=1,…,𝑝,_𝑗=𝑢^𝑗+𝜀𝜙_𝑗(𝑥,𝑢)+𝑜𝜀²,𝑗=1,…,𝑞, (10) where 𝜉^𝑖 and 𝜙_𝑗 are the infinitesimals of the transformations for the independent and dependent variables, respectively and 𝜀 is the parameter of transformation.

The most general form of infinitesimal generator associated with the above group of transformations is 𝑋=^𝑝_𝑖=1𝜉^𝑖(𝑥,𝑢)𝜕_𝑥𝑖+^𝑞_𝑗=1𝜙_𝑗(𝑥,𝑢)𝜕_𝑢𝑗. (11) Transformations which map solutions of a differential equation to other solutions are called symmetries of this equation. The 𝑛th order prolongation of 𝑋 is defined by 𝑃𝑟^(𝑛)𝑋=𝑋+^𝑞_𝛼=1𝐽𝜙^𝐽_𝛼𝑥,𝑢^(𝑛)𝜕_𝑢^𝛼_𝐽, (12) where 𝐽=(𝑗₁,…,𝑗_𝑘), 1≤𝑗_𝑘≤𝑝, 1≤𝑘≤𝑛 and the sum is over all 𝐽’s of order 0<#𝐽≤𝑛. If #𝐽=𝑘, the coefficient 𝜙^𝐽_𝛼 of 𝜕_𝑢^𝛼_𝐽 will only depend on 𝑘th and lower order derivatives of 𝑢, and 𝜙^𝐽_𝛼𝑥,𝑢^(𝑛)=𝐷_𝐽𝜙_𝛼−^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝑖+^𝑝_𝑖=1𝜉^𝑖𝑢^𝛼_𝐽,𝑖, (13) where 𝑢^𝛼_𝑖∶=𝜕𝑢^𝛼/𝜕𝑥^𝑖 and 𝑢^𝛼_𝐽,𝑖∶=𝜕𝑢^𝛼_𝐽/𝜕𝑥^𝑖. Now, according to Theorem 2.36 of [2], the invariance of the system (9) under the infinitesimal transformations leads to the invariance conditions: 𝑃𝑟^(𝑛)𝑋Δ_𝜇𝑥,𝑢^(𝑛)=0,𝜇=1,…,𝑟,wheneverΔ_𝜇𝑥,𝑢^(𝑛)=0. (14) Also, if the system (9) is a nondegenerate system which is locally solvable and of maximal rank, we can conclude that (14) is a necessary and sufficient condition for 𝐺 to be a (strong) symmetry group of (9) (Theorem 2.71 of [2] and Theorem 1 of [5]).

3. Symmetries of System (8)

In this section, we consider one parameter Lie group of infinitesimal transformations: (𝑥¹=𝑥, 𝑥²=𝑡, 𝑢¹=𝑎, 𝑢²=𝑏, 𝑢³=𝑐, we use 𝑥 and 𝑡 instead of 𝑥₁ and 𝑥₂, resp., in order not to use index) 𝑥=𝑥+𝜀𝜉¹(𝑥,𝑡,𝑎,𝑏,𝑐)+𝑜𝜀²,𝑡̃=𝑡+𝜀𝜉²(𝑥,𝑡,𝑎,𝑏,𝑐)+𝑜𝜀²,𝑎=𝑎+𝜀𝜙₁(𝑥,𝑡,𝑎,𝑏,𝑐)+𝑜𝜀²,𝑏̃=𝑏+𝜀𝜙₂(𝑥,𝑡,𝑎,𝑏,𝑐)+𝑜𝜀²,𝑐̃=𝑐+𝜀𝜙₃(𝑥,𝑡,𝑎,𝑏,𝑐)+𝑜𝜀². (15) The symmetry generator of the above group of transformations is of the form 𝑋=𝜉¹(𝑥,𝑡,𝑎,𝑏,𝑐)𝜕_𝑥+𝜉²(𝑥,𝑡,𝑎,𝑏,𝑐)𝜕_𝑡+𝜙₁(𝑥,𝑡,𝑎,𝑏,𝑐)𝜕_𝑎+𝜙₂(𝑥,𝑡,𝑎,𝑏,𝑐)𝜕_𝑏+𝜙₃(𝑥,𝑡,𝑎,𝑏,𝑐)𝜕_𝑐, (16) and, its second prolongation is the vector field 𝑃𝑟⁽²⁾𝑋=𝑋+𝜙^𝑥₁𝜕_𝑎𝑥+𝜙^𝑡₁𝜕_𝑎𝑡+𝜙^𝑥₂𝜕_𝑏𝑥+𝜙^𝑡₂𝜕_𝑏𝑡+𝜙^𝑥₃𝜕_𝑐𝑥+𝜙^𝑡₃𝜕_𝑐𝑡+𝜙₁^𝑥𝑥𝜕_𝑎𝑥𝑥+𝜙₂^𝑥𝑥𝜕_𝑏𝑥𝑥+𝜙₃^𝑥𝑥𝜕_𝑐𝑥𝑥+𝜙₁^𝑥𝑡𝜕_𝑎𝑥𝑡+𝜙₂^𝑥𝑡𝜕_𝑏𝑥𝑡+𝜙₃^𝑥𝑡𝜕_𝑐𝑥𝑡+𝜙₁^𝑡𝑡𝜕_𝑎𝑡𝑡+𝜙₂^𝑡𝑡𝜕_𝑏𝑡𝑡+𝜙₃^𝑡𝑡𝜕_𝑐𝑡𝑡. (17) Let (for convenience) 𝑄₁=𝜙₁−𝜉¹𝑎_𝑥−𝜉²𝑎_𝑡,𝑄₂=𝜙₂−𝜉¹𝑏_𝑥−𝜉²𝑏_𝑡 and 𝑄₃=𝜙₃−𝜉¹𝑐_𝑥−𝜉²𝑐_𝑡, then by using (13), we can compute the coefficients of (17) as 𝜙^𝑥₁=𝐷_𝑥𝑄₁+𝜉¹𝑎_𝑥𝑥+𝜉²𝑎_𝑥𝑡,𝜙^𝑥₂=𝐷_𝑥𝑄₂+𝜉¹𝑏_𝑥𝑥+𝜉²𝑏_𝑥𝑡,𝜙^𝑥₃=𝐷_𝑥𝑄₃+𝜉¹𝑐_𝑥𝑥+𝜉²𝑐_𝑥𝑡,𝜙^𝑡₁=𝐷_𝑡𝑄₁+𝜉¹𝑎_𝑥𝑡+𝜉²𝑎_𝑡𝑡,𝜙^𝑡₂=𝐷_𝑡𝑄₂+𝜉¹𝑏_𝑥𝑡+𝜉²𝑏_𝑡𝑡,𝜙^𝑡₃=𝐷_𝑡𝑄₃+𝜉¹𝑐_𝑥𝑡+𝜉²𝑐_𝑡𝑡,𝜙₁^𝑥𝑥=𝐷²_𝑥𝑄₁+𝜉¹𝑎_𝑥𝑥𝑥+𝜉²𝑎_𝑥𝑥𝑡,𝜙₂^𝑥𝑥=𝐷²_𝑥𝑄₂+𝜉¹𝑏_𝑥𝑥𝑥+𝜉²𝑏_𝑥𝑥𝑡,𝜙₃^𝑥𝑥=𝐷²_𝑥𝑄₃+𝜉¹𝑐_𝑥𝑥𝑥+𝜉²𝑐_𝑥𝑥𝑡,𝜙₁^𝑡𝑡=𝐷²_𝑡𝑄₁+𝜉¹𝑎_𝑥𝑡𝑡+𝜉²𝑎_𝑡𝑡𝑡,𝜙₂^𝑡𝑡=𝐷²_𝑡𝑄₂+𝜉¹𝑏_𝑥𝑡𝑡+𝜉²𝑏_𝑡𝑡𝑡,𝜙₃^𝑡𝑡=𝐷²_𝑡𝑄₃+𝜉¹𝑐_𝑥𝑡𝑡+𝜉²𝑐_𝑡𝑡𝑡,𝜙₁^𝑥𝑡=𝐷_𝑥𝐷_𝑡𝑄₁+𝜉¹𝑎_𝑥𝑥𝑡+𝜉²𝑎_𝑥𝑡𝑡,𝜙₂^𝑥𝑡=𝐷_𝑥𝐷_𝑡𝑄₂+𝜉¹𝑏_𝑥𝑥𝑡+𝜉²𝑏_𝑥𝑡𝑡,𝜙₃^𝑥𝑡=𝐷_𝑥𝐷_𝑡𝑄₃+𝜉¹𝑐_𝑥𝑥𝑡+𝜉²𝑐_𝑥𝑡𝑡, (18) where 𝐷_𝑥 and 𝐷_𝑡 are the total derivatives with respect to 𝑥 and 𝑡, respectively. By (14) the invariance condition is equivalent with the following equations: 𝑃𝑟⁽²⁾𝑋𝑎_𝑥𝑥−𝑏_𝑡𝑡=0,𝑎_𝑥𝑥−𝑏_𝑡𝑡=0,𝑃𝑟⁽²⁾𝑋𝑏_𝑥𝑦+𝑐_𝑥𝑥=0,𝑏_𝑥𝑦+𝑐_𝑥𝑥⋮=0,𝑃𝑟⁽²⁾𝑋𝑎_𝑥𝑏_𝑥−𝑏_𝑥𝑐_𝑡+𝑏_𝑡𝑐_𝑥−𝑐_𝑥²+𝑎𝑏_𝑥𝑥+2𝑐𝑏_𝑥𝑡−𝑏𝑐_𝑥𝑡𝑎=0,_𝑥𝑏_𝑥−𝑏_𝑥𝑐_𝑡+𝑏_𝑡𝑐_𝑥−𝑐_𝑥²+𝑎𝑏_𝑥𝑥+2𝑐𝑏_𝑥𝑡−𝑏𝑐_𝑥𝑡=0. (19) After substituting 𝑃𝑟⁽²⁾𝑋 in the six equations above, we obtain six polynomial equations involving the various derivatives of 𝑎, 𝑏, and 𝑐, whose coefficients are certain derivatives of 𝜉¹, 𝜉², 𝜙₁, 𝜙₂, and 𝜙₃. Since 𝜉¹, 𝜉², 𝜙₁, 𝜙₂ and 𝜙₃ depend only on 𝑥, 𝑡, 𝑎, 𝑏, 𝑐 we can equate the individual coefficients to zero, leading to the determining the following equations: 𝑎³𝜉¹_𝑎=0,𝑏²𝜉¹_𝑏=0,𝑎𝜉²_𝑏𝑏=0,₂𝜉²_𝑏=0,𝑎𝑐𝜙_2𝑎=0,𝑐𝑏𝜉¹_𝑏=0,…𝑎𝑏𝑐−3𝜉¹_𝑡−𝜙_3𝑏+2𝑐²𝑎−𝜉²_𝑡+𝜉¹_𝑥+𝑏𝑎²𝜙_2𝑏−𝜙_3𝑐−𝜉¹_𝑥+𝜉²_𝑡+𝑐𝑎²−2𝜉²_𝑥+𝜙_2𝑐+𝜙₁𝑎𝑏−2𝑐²−𝑎²𝜙₂+2𝑐𝑎𝜙₃+4𝑐³𝜉¹_𝑡=0. (20) We write some of these equations whereas the number of the whole is 410. By solving this system of PDEs, we have the following.

Theorem 1. The Lie group of point symmetries of system (8) has the Lie algebra generated by (16), which its coefficient functions are 𝜉₁=𝑘₃𝑥+𝑘₁𝑡+𝑘₂,𝜙₂=2𝑘₄−2𝑘₃+𝑘₇𝑏+2𝑘₆𝜙𝑐,₁=2𝑘₁𝑐+𝑘₇𝑎,𝜉₂=𝑘₆𝑥+𝑘₄𝑡+𝑘₅,𝜙₃=𝑘₇+𝑘₄−𝑘₃𝑐+𝑘₆𝑎+𝑘₁𝑏, (21) where 𝑘_𝑖’s, 𝑖=1,…,7 are arbitrary constants.

Corollary 2. Infinitesimal generator of any one-parameter Lie group of point symmetries of (8) is an ℝ-linear combination of 𝑋₁=𝜕_𝑥,𝑋₂=𝜕_𝑡,𝑋₃=𝑥𝜕_𝑥−2𝑏𝜕_𝑏−𝑐𝜕_𝑐,𝑋₄=𝑥𝜕_𝑡+2𝑐𝜕_𝑏+𝑎𝜕_𝑐,𝑋₅=𝑡𝜕_𝑥+2𝑐𝜕_𝑎+𝑏𝜕_𝑐,𝑋₆=𝑡𝜕_𝑡+2𝑏𝜕_𝑏+𝑐𝜕_𝑐,𝑋₇=𝑎𝜕_𝑎+𝑏𝜕_𝑏+𝑐𝜕_𝑐. (22)

It is worthwhile to note that, the Lie group obtained from this method, is a strong symmetry group of system (8). So we can transform solutions of the system to other solutions as well as reduce the system and obtain 𝐺-invariant solutions.

Let 𝔤 be the Lie algebra generated by (22). The commutator table of 𝔤 is given in Table 1, where the entry in the 𝑖th row and 𝑗th column is [𝑋_𝑖,𝑋_𝑗]=𝑋_𝑖𝑋_𝑗−𝑋_𝑗𝑋_𝑖, 𝑖,𝑗=1,…,7.

Table 1 

The commutator table of 𝔤.

The group transformation which is generated by 𝑋_𝑖=𝜉¹_𝑖𝜕_𝑥+𝜉²_𝑖𝜕_𝑡+𝜙_1𝑖𝜕_𝑎+𝜙_2𝑖𝜕_𝑏+𝜙_3𝑖𝜕_𝑐 for 𝑖=1,…,7 is obtained by solving the seven systems of ODEs: 𝑑𝑥(𝑠)𝑑𝑠=𝜉¹_𝑖𝑥(𝑠),𝑡̃(𝑠),𝑎(𝑠),𝑏̃(𝑠),𝑐̃(𝑠),𝑥(0)𝑑𝑡̃(𝑠)=𝑥,𝑑𝑠=𝜉²_𝑖𝑥(𝑠),𝑡̃(𝑠),𝑎(𝑠),𝑏̃(𝑠),𝑐̃(𝑠),𝑡̃(0)𝑑𝑎(𝑠)=𝑡,𝑑𝑠=𝜙_1𝑖𝑥(𝑠),𝑡̃(𝑠),𝑎(𝑠),𝑏̃(𝑠),𝑐̃(𝑠,𝑎(0)𝑑𝑏̃(𝑠))=𝑎,𝑑𝑠=𝜙_2𝑖𝑥(𝑠),𝑡̃(𝑠),𝑎(𝑠),𝑏̃(𝑠),𝑐̃(𝑠),𝑏̃(0)𝑑𝑐̃(𝑠)=𝑏,𝑑𝑠=𝜙_3𝑖𝑥(𝑠),𝑡̃(𝑠),𝑎(𝑠),𝑏̃(𝑠),𝑐̃(𝑠),𝑐̃(0)=𝑐.𝑖=1,…,7. (23) By exponentiating the infinitesimal symmetries (22), we obtain the one-parameter groups 𝑔_𝑘(𝑠) generated by 𝑋_𝑘, 𝑘=1,…,7 as follows: 𝑔₁(𝑠)∶()↦(),𝑔𝑥,𝑡,𝑎,𝑏,𝑐𝑥+𝑠,𝑡,𝑎,𝑏,𝑐₂(𝑠)∶()↦(),𝑔𝑥,𝑡,𝑎,𝑏,𝑐𝑥,𝑡+𝑠,𝑎,𝑏,𝑐₃(𝑠)∶(𝑥,𝑡,𝑎,𝑏,𝑐)↦𝑥𝑒^𝑠,𝑡,𝑎,𝑏𝑒^−2𝑠,𝑐𝑒^−𝑠,𝑔₄(𝑠)∶(𝑥,𝑡,𝑎,𝑏,𝑐)↦𝑥,𝑠𝑥+𝑡,𝑎,𝑎𝑠²,𝑔+2𝑐𝑠+𝑏,𝑎𝑠+𝑐₅(𝑠)∶(𝑥,𝑡,𝑎,𝑏,𝑐)↦𝑡𝑠+𝑥,𝑡,𝑏𝑠²,𝑔+2𝑐𝑠+𝑎,𝑏,𝑏𝑠+𝑐₆(𝑠)∶(𝑥,𝑡,𝑎,𝑏,𝑐)↦𝑥,𝑡𝑒^𝑠,𝑎,𝑏𝑒^2𝑠,𝑐𝑒^𝑠,𝑔₇(𝑠)∶(𝑥,𝑡,𝑎,𝑏,𝑐)↦𝑥,𝑡,𝑎𝑒^𝑠,𝑏𝑒^𝑠,𝑐𝑒^𝑠. (24) Consequently, we can state the following theorem.

Theorem 3. If 𝑎=𝑓=𝑓(𝑥,𝑡), 𝑏=ℎ=ℎ(𝑥,𝑡), and 𝑐=𝑘=𝑘(𝑥,𝑡), is a solution of (8), so are the functions 𝑔₁(𝑠)⋅𝑓=𝑓(𝑥−𝑠,𝑡),𝑔₁(𝑠)(),𝑔⋅ℎ=ℎ𝑥−𝑠,𝑡₁(𝑠)⋅𝑘=𝑘(𝑥−𝑠,𝑡),𝑔₂(𝑠)(),𝑔⋅𝑓=𝑓𝑥,𝑡−𝑠₂(𝑠)⋅ℎ=ℎ(𝑥,𝑡−𝑠),𝑔₂(𝑠)(),𝑔⋅𝑘=𝑘𝑥,𝑡−𝑠₃(𝑠)⋅𝑓=𝑓𝑥𝑒^−𝑠,𝑡,𝑔₃(𝑠)⋅ℎ=ℎ𝑥𝑒^−𝑠,𝑡𝑒^−2𝑠,𝑔₃(𝑠)⋅𝑘=𝑘𝑥𝑒^−𝑠,𝑡𝑒^−𝑠,𝑔₆(𝑠)⋅𝑓=𝑓𝑥,𝑡𝑒^−𝑠,𝑔₆(𝑠)⋅ℎ=ℎ𝑥,𝑡𝑒^−𝑠𝑒^2𝑠,𝑔₆(𝑠)⋅𝑘=𝑘𝑥,𝑡𝑒^−𝑠𝑒^𝑠,𝑔₇(𝑠)⋅𝑓=𝑓(𝑥,𝑡)𝑒^𝑠,𝑔₇(𝑠)⋅ℎ=ℎ(𝑥,𝑡)𝑒^𝑠,𝑔₇(𝑠)⋅𝑘=𝑘(𝑥,𝑡)𝑒^𝑠,𝑔₄(𝑠)(),𝑔⋅𝑓=𝑓𝑥,𝑡−𝑠𝑥₄(𝑠)⋅ℎ=𝑓(𝑥,𝑡−𝑠𝑥)𝑠²()(),𝑔+2𝑘𝑥,𝑡−𝑠𝑥𝑠+ℎ𝑥,𝑡−𝑠𝑥₄(𝑠)()(),𝑔⋅𝑘=𝑓𝑥,𝑡−𝑠𝑥𝑠+𝑘𝑥,𝑡−𝑠𝑥₅(𝑠)⋅𝑓=ℎ(𝑥−𝑠𝑡,𝑡)𝑠²()(),𝑔+2𝑘𝑥−𝑠𝑡,𝑡𝑠+𝑓𝑥−𝑠𝑡,𝑡₅(𝑠)(),𝑔⋅ℎ=ℎ𝑥−𝑠𝑡,𝑡₅(𝑠)⋅𝑘=ℎ(𝑥−𝑠𝑡,𝑡)𝑠+𝑘(𝑥−𝑠𝑡,𝑡). (25)

This theorem is applied to obtain new solutions from known ones.

Example 4. Let 𝑎=𝑓(𝑥,𝑡)=𝑟₁𝑥+𝑟₂,()=𝑟𝑏=ℎ𝑥,𝑡²³𝑟₁𝑟𝑥−²𝑟²₃𝑟²₁ln𝑟₁𝑥+𝑟₂,𝑐=𝑘(𝑥,𝑡)=𝑟₃𝑥+𝑟₄, (26) be a solution of (8), where 𝑟₁,𝑟₂,𝑟₃,𝑟₄∈ℝ are arbitrary constants, we conclude that the functions 𝑔_𝑖(𝑠)⋅𝑓,𝑔_𝑖(𝑠)⋅ℎ, and 𝑔_𝑖(𝑠)⋅𝑘 are also solutions of (8) for 𝑖=1,…,7. For example, 𝑔₅(𝑠)𝑟⋅𝑓=²³𝑟₁()−𝑟𝑥−𝑡𝑠²𝑟²₃𝑟²₁ln𝑟₁(𝑥−𝑡𝑠)+𝑟₂𝑠²+2𝑟₃(𝑥−𝑡𝑠)+𝑟₄𝑠+𝑟₁(𝑥−𝑡𝑠)+𝑟₂,𝑔₅(𝑠)𝑟⋅ℎ=²³𝑟₁()−𝑟𝑥−𝑡𝑠²𝑟²₃𝑟²₁ln𝑟₁(𝑥−𝑠𝑡)+𝑟₂,𝑔₅(𝑠)𝑟⋅𝑘=²³𝑟₁()−𝑟𝑥−𝑡𝑠²𝑟²₃𝑟²₁ln𝑟₁(𝑥−𝑡𝑠)+𝑟₂𝑠+𝑟₃(𝑥−𝑡𝑠)+𝑟₄ (27) is a set of new solutions of (8) and we can obtain many other solutions by arbitrary combination of 𝑔_𝑖(𝑠)’s for 𝑖=1,…,7. So we obtain infinite number of Einstein Walker manifolds just from this example.

4. One-Dimensional Optimal System of (8)

In this section, we obtain the one-parameter optimal system of (8) by using symmetry group. Since every linear combination of infinitesimal symmetries is an infinitesimal symmetry, there is an infinite number of one-dimensional subalgebras for the differential equation. So it is important to determine which subgroups give different types of solutions. Therefore, we must find invariant solutions which are not related by transformation in the full symmetry group. This procedure led to the concept of optimal system for subalgebras. For one-dimensional subalgebras, this classification problem is the same as the problem of classifying the orbits of the adjoint representation [2]. This problem is solved by the simple approach of taking a general element in the Lie algebra and simplify it as much as possible by imposing various adjoint transformation on it ([12, 13]). Optimal set of subalgebras is obtaining from taking only one representative from each class of equivalent subalgebras. Adjoint representation of each 𝑋_𝑖, 𝑖=1,…,7 is defined as follows: Adexp𝑠⋅𝑋_𝑖⋅𝑋_𝑗=𝑋_𝑗−𝑠⋅𝑋_𝑖,𝑋_𝑗+𝑠²2⋅𝑋_𝑖,𝑋_𝑖,𝑋_𝑗−⋯, (28) where 𝑠 is a parameter and [𝑋_𝑖,𝑋_𝑗] is the commutator of 𝔤 defined in Table 1, for 𝑖,𝑗=1,…,7 [2, page 199]. We can write the adjoint action for 𝔤 and show the following.

Theorem 5. A one-dimensional optimal system of (8) is given by (1)𝑋₇,(2)𝑋₁+𝑎𝑋₇,(3)𝑋₂+𝑎𝑋₇,(4)𝜀𝑋₁+𝑋₆+𝑎𝑋₇,(5)𝜀𝑋₂+𝑋₅+𝑎𝑋₆+𝑏𝑋₇,(6)𝜀𝑋₁+𝑋₄+𝑎𝑋₅+𝑏𝑋₆+𝑐𝑋₇,(7)𝜀𝑋₂+𝑋₃+𝜀^′𝑋₄+𝑎𝑋₅+𝑏𝑋₆+𝑐𝑋₇, (29) where 𝜀 and 𝜀^′ are ±1 or zero and 𝑎,𝑏,𝑐∈ℝ are arbitrary numbers.

Proof. Attending to Table 1, we understand that the center of 𝔤 is the subalgebra ⟨𝑋₇⟩, so it is enough to specify the algebras of ⟨𝑋₁,𝑋₂,𝑋₃,𝑋₄,𝑋₅,𝑋₆⟩. Each adjoint transformation is a linear map 𝐹^𝑠_𝑖∶𝔤→𝔤 that defined by 𝑋↦Ad(exp(𝑠𝑋_𝑖)⋅𝑋), for 𝑖=1,…,7. The matrix 𝑀^𝑠_𝑖 of 𝐹^𝑠_𝑖, 𝑖=1,…,7, with respect to basis {𝑋₁,…,𝑋₇} is 𝑀^𝑠₁=,𝑀10000000100000−𝑠0100000−𝑠01000000010000000100000001_𝑠2=,𝑀1000000010000000100000001000−𝑠0001000−𝑠000100000001_𝑠3=𝑒^𝑠00000001000000010000000𝑒^−𝑠0000000𝑒^𝑠,𝑀0000000100000001_𝑠4=1𝑠000000100000001𝑠000000100000−𝑠−𝑠²,𝑀1𝑠−2𝑠000−𝑠0100000001_𝑠5=1000000𝑠1000000010−𝑠0000𝑠1−𝑠²,𝑀−𝑠2𝑠00001000000𝑠100000001_𝑠6=10000000𝑒^𝑠000000010000000𝑒^𝑠0000000𝑒^−𝑠,0000000100000001 (30) respectively and 𝑀^𝑠₇ is the 7×7 identity matrix. Let 𝑋=∑⁷_𝑖=1𝑎_𝑖𝑋_𝑖, then 𝐹^𝑠7₇∘⋯∘𝐹^𝑠1₁↦∶𝑋1+𝑠₅𝑠₄𝑒^𝑠3𝑎₁+𝑒^𝑠6+𝑠3𝑠₄𝑎₂⋅𝑋₁++⋯−𝑠₅𝑠₂𝑎₁−𝑒^𝑠6𝑠₂𝑎₂++⋯1+𝑠₅𝑠₄𝑎₆−2𝑠₅𝑠₄𝑎₇𝑋₆+𝑎₇𝑋₇. (31) Now, we can simplify 𝑋 as follows. If 𝑎₁=𝑎₂=⋯=𝑎₆=0, then 𝑋 is reduced to the case (1), that is center of 𝔤. If 𝑎₃=⋯=𝑎₆=0 and 𝑎₁≠0, then we can make the coefficient of 𝑋₂ vanish by 𝐹^𝑠4₄; by setting 𝑠₄=𝑎₂/𝑎₁. Scaling 𝑋 if necessary, we can assume that 𝑎₁=1. So, 𝑋 is reduced to the case (2). If 𝑎₁=𝑎₃=⋯=𝑎₆=0, then we can make 𝑎₂=±1 by 𝐹^𝑠6₆; by setting 𝑠₆=ln|𝑎₂|. So, 𝑋 is reduced to case (3). If 𝑎₃=⋯=𝑎₅=0 and 𝑎₆≠0, then we can make the coefficient of 𝑋₂ vanish by 𝐹^𝑠2₂; by setting 𝑠₂=−𝑎₂/𝑎₆. Also the coefficient of 𝑋₁ can be vanished or be ±1 by 𝐹^𝑠3₃; by setting 𝑠₃=ln|𝑎₁|. Scaling 𝑋 if necessary, we can assume that 𝑎₆=1. So, 𝑋 is reduced to the case (4). If 𝑎₃=𝑎₄=0 and 𝑎₅≠0, then we can make the coefficient of 𝑋₁ vanish by 𝐹^𝑠2₂; by setting 𝑠₂=−𝑎₁/𝑎₅. Also the coefficient of 𝑋₂ can be vanished or be ±1 by 𝐹^𝑠6₆; by setting 𝑠₆=ln|𝑎₂|. Scaling 𝑋 if necessary, we can assume that 𝑎₅=1. So, 𝑋 is reduced to the case (5). If 𝑎₃=0 and 𝑎₄≠0, then we can make the coefficient of 𝑋₂ vanish by 𝐹^𝑠1₁; by setting 𝑠₁=−𝑎₂/𝑎₄. Also the coefficient of 𝑋₁ can be vanished or be ±1 by 𝐹^𝑠3₃; by setting 𝑠₃=ln|𝑎₁|. Scaling 𝑋 if necessary, we can assume that 𝑎₄=1. So, 𝑋 is reduced to the case (6). If 𝑎₃≠0, then we can make the coefficient of 𝑋₁ vanish by 𝐹^𝑠1₁; by setting 𝑠₁=−𝑎₁/𝑎₃. Also the coefficients of 𝑋₂ and 𝑋₄ can be vanished or be ±1 by 𝐹^𝑠3₃ and 𝐹^𝑠6₆; by setting 𝑠₃=−ln|𝑎₄| and 𝑠₆=ln|𝑎₂|, respectively. Scaling 𝑋 if necessary, we can assume that 𝑎₃=1. So, 𝑋 is reduced to the case (7).

5. Similarity Reduction of System (8)

The system (8) is expressed in the coordinates (𝑥,𝑡,𝑎,𝑏,𝑐), so for reducing this system we must search for its form in suitable coordinates. Those coordinates will be obtained by searching for independent invariants 𝑤, 𝑓, ℎ, 𝑘 corresponding to the infinitesimal generator. Hence by using the chain rule, the expression of the system in the new coordinate leads to the reduced system. Now, we find some of nontrivial solution of system (8). First, consider the operator 𝑋₃=𝑥𝜕_𝑥−2𝑏𝜕_𝑏−𝑐𝜕_𝑐. For determining independent invariants 𝐼, we ought to solve the homogeneous first-order PDE 𝑋₃(𝐼)=0, that is, 𝑥𝜕_𝑥−2𝑏𝜕_𝑏−𝑐𝜕_𝑐𝐼=𝑥𝜕𝐼𝜕𝑥+0𝜕𝐼𝜕𝑡+0𝜕𝐼𝜕𝑎−2𝑏𝜕𝐼𝜕𝑏−𝑐𝜕𝐼.𝜕𝑐 (32) So we must solve the associated characteristic ODE 𝑑𝑥𝑥=𝑑𝑡0=𝑑𝑎0=𝑑𝑏=−2𝑏𝑑𝑐.−𝑐 (33) Hence, four functionally independent invariants 𝑤=𝑡, 𝑓=𝑎, ℎ=𝑏𝑥², and 𝑘=𝑐𝑥 are obtained. If we treat 𝑓, ℎ, and 𝑘 as functions of 𝑤, we can compute formulae for the derivatives of 𝑎, 𝑏, and 𝑐 with respect to 𝑥 and 𝑡 in terms of 𝑤, 𝑓, ℎ, 𝑘, and the derivatives of 𝑓, ℎ, and 𝑘 with respect to 𝑤. We have 𝑎=𝑓(𝑤)=𝑓(𝑡), 𝑏=𝑥⁻²ℎ(𝑤)=𝑥⁻²ℎ(𝑡) and 𝑐=𝑥⁻¹𝑘(𝑤)=𝑥⁻¹𝑘(𝑡). So by using the chain rule, we have 𝑎₁=𝜕𝑎=𝜕𝑥𝜕𝑓𝜕𝑤𝜕𝑤=𝜕𝑥𝜕𝑓𝜕𝑡𝜕𝑡𝜕𝑥=0,𝑎₂=𝑎_𝑡=⋯=𝑓_𝑡,𝑏₁=−2𝑥⁻³ℎ,𝑏₂=𝑥⁻²ℎ_𝑡,𝑐₁=−𝑥⁻²𝑐𝑘,₂=𝑥⁻¹𝑘_𝑡,𝑎₁₁=𝑎_𝑥𝑥=0,𝑎₁₂𝑎=0,₂₂=𝑓_𝑡𝑡,𝑏₁₁=6𝑥⁻⁴ℎ,𝑏₁₂=−2𝑥⁻³ℎ_𝑡,𝑏₂₂=𝑥⁻²ℎ_𝑡𝑡,𝑐₁₁=2𝑥⁻³𝑘,𝑐₁₂=−𝑥⁻²𝑘_𝑡,𝑐₂₂=𝑥⁻¹𝑘_𝑡𝑡. (34) Substituting these in the system (8), the reduced system is obtained as follows: ℎ_𝑡𝑡=0,3ℎ𝑘_𝑡−5ℎ_𝑡𝑘−𝑘²𝑓+6𝑓ℎ=0,_𝑡ℎ_𝑡+𝑓_𝑡𝑘−𝑘²_𝑡+ℎ𝑓_𝑡𝑡+𝑓𝑘_𝑡𝑘=0,_𝑡𝑡=0,ℎ_𝑡−𝑘=0,2𝑓_𝑡ℎ−2𝑘𝑘_𝑡+2𝑓𝑘+ℎ𝑘_𝑡𝑡=0, (35) which is a system of ODEs. Two types of solutions are obtained by solving this system (1)(𝑡)(2)𝑓=𝑓𝑟ℎ=0𝑘=0,𝑓=²²𝑟₂𝑡+𝑟₁ℎ=𝑟₂𝑡+𝑟₁𝑘=𝑟₂, (36) where 𝑟₁ and 𝑟₂ are arbitrary constants and 𝑓(𝑡) is an arbitrary function. We can compute all of invariant solutions for other symmetry generators in a similar way. Some of infinitesimal symmetries and their Lie invariants are listed in Table 2.

Table 2 

Lie invariants and similarity solutions.

In Table 3, we list the reduced form of system (8) corresponding to infinitesimal generators of Table 2 and obtain some of its invariant solutions.

Table 3 

Reduced systems and their group invariant solutions.

Now by using the information of Table 2, we can obtain corresponding solutions for system (8). For example, consider the second invariant solution of case 5. Attending to the Table 2, the following solution is obtained: 𝑟𝑎=−²𝑟₁𝑒^𝑟1𝑥𝑒^𝑡,𝑏=−𝑟₁𝑟₂𝑒^𝑟1𝑥𝑒^𝑡,𝑐=𝑟₂𝑒^𝑟1𝑥𝑒^𝑡, (37) for system (8). Consequently by substituting 𝑎, 𝑏, and 𝑐 in (6), we can determine the general form for metric of Einstein Walker manifolds for case 5. In addition, we can construct many other solutions from this one, by using Theorem 3, as we did in Example 4.

6. Conclusion

In this paper, by applying the method of Lie symmetries, we find the Lie point symmetry group of system (8). This work has been done by applying the criterion of invariance for the system under the prolonged infinitesimal generators. Also, we have obtained the one-parameter optimal system of subalgebras for system (8). Then the Lie invariants and similarity reduced systems and some of the invariant solutions are obtained. In conclusion, we specified a large class of Einstein Walker manifolds.