Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2013, Article ID 591852, 8 pages
http://dx.doi.org/10.1155/2013/591852
Research Article
Some General New Einstein Walker Manifolds
Mehdi Nadjafikhah1 and Mehdi Jafari2
1
2
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 1684613114, Iran
Department of Complementary Education, Payame Noor University, Tehran 19395-3697, Iran
Correspondence should be addressed to Mehdi Nadjafikhah; m nadjafikhah@iust.ac.ir
Received 1 October 2012; Accepted 7 November 2012
Academic Editor: Anatol Odzijewicz
Copyright © 2013 M. Nadjafikhah and M. Jafari. 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.
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 onedimensional 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 {𝑒𝑖 } for 𝑉 so that
⟨𝑒𝑖 , 𝑒𝑗 ⟩ = {
0
𝑖 ≠ 𝑗
±1 𝑖 = 𝑗.
(1)
We set 𝜀𝑖 := ⟨𝑒𝑖 , 𝑒𝑖 ⟩. Let 𝑝 be the number of indices 𝑖 with 𝜀𝑖 =
−1 and 𝑞 = dim 𝑉 − 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 pseudoRiemannian 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 , . . . , 𝑥𝑚 ) is a system of local
coordinates on 𝑀, we may express 𝑔 = ∑𝑖,𝑗 𝑔𝑖𝑗 𝑑𝑥𝑖 ∘ 𝑑𝑥𝑗
where 𝑔𝑖𝑗 := 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
2
Advances in Mathematical Physics
R is defined by the formula:
R (𝑋, 𝑌) 𝑍 := (𝐷𝑋 𝐷𝑌 − 𝐷𝑌 𝐷𝑋 − 𝐷[𝑋,𝑌] ) 𝑍,
𝑋, 𝑌, 𝑍 ∈ 𝐶∞ (𝑇𝑀) .
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:
(2)
(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 𝑛dimensional distribution is given by the metric tensor as
0 𝐼𝑑𝑛
),
(𝑔𝑖𝑗 ) = (
𝐼𝑑𝑛 𝐵
(4)
where 𝐼𝑑𝑛 is the 𝑛 × 𝑛 identity matrix and 𝐵 is a symmetric
𝑛 × 𝑛 matrix whose entries are functions of the (𝑥1 , . . . , 𝑥2𝑛 )
(for more details see [9, 10]).
If 𝑛 = 2 we adopt the following result for 4-dimensional
Walker manifolds. Let 𝑀𝑎,𝑏,𝑐 := (O, 𝑔𝑎,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:
0
0
(𝑔𝑎,𝑏,𝑐 )𝑖𝑗 = (
1
0
0
0
0
1
1
0
𝑎
𝑐
0
1
),
𝑐
𝑏
(5)
that is given in the coordinates form as
𝑔𝑎,𝑏,𝑐 := 2 (𝑑𝑥1 ∘ 𝑑𝑥3 + 𝑑𝑥2 ∘ 𝑑𝑥4 )
+ 𝑎 (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) 𝑑𝑥3 ∘ 𝑑𝑥3
+ 𝑏 (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) 𝑑𝑥4 ∘ 𝑑𝑥4
(6)
+ 2𝑐 (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) 𝑑𝑥3 ∘ 𝑑𝑥4 .
We can see that 𝑀𝑎,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,
𝑎1 𝑐2 + 𝑎2 𝑏2 − 𝑎2 𝑐1 − 𝑐2 2 + 2𝑐𝑎12
𝑎1 𝑏1 − 𝑏1 𝑐2 + 𝑏2 𝑐1 − 𝑐1 2 + 𝑎𝑏11
+ 2𝑐𝑏12 − 2𝑏13 − 𝑏𝑐12 + 2𝑐14 = 0.
𝑎2 𝑏1 − 𝑐1 𝑐2 + 𝑐𝑎11 − 𝑎𝑐11 − 𝑐𝑐12 − 𝑏𝑐22 = 0,
(8)
𝑎1 𝑏1 − 𝑏1 𝑐2 + 𝑏2 𝑐1 − 𝑐1 2 + 𝑎𝑏11 + 2𝑐𝑏12 − 𝑏𝑐12 = 0.
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 𝜇 (𝑥, 𝑢(𝑛) ) = 0,
𝜇 = 1, . . . , 𝑟,
(9)
involving 𝑥 = (𝑥1 , . . . , 𝑥𝑝 ) and 𝑢 = (𝑢1 , . . . , 𝑢𝑞 ) 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)
𝑥̃𝑖 = 𝑥𝑖 + 𝜀𝜉𝑖 (𝑥, 𝑢) + 𝑜 (𝜀2 ) ,
𝑖 = 1, . . . , 𝑝,
𝑢̃𝑗 = 𝑢𝑗 + 𝜀𝜙𝑗 (𝑥, 𝑢) + 𝑜 (𝜀2 ) ,
𝑗 = 1, . . . , 𝑞,
(10)
where 𝜉𝑖 and 𝜙𝑗 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
𝑝
𝑞
𝑖=1
𝑗=1
(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
+ 𝑏𝑎22 − 2𝑎24 − 𝑎𝑐12 + 2𝑐23 = 0,
− 𝑐𝑐12 + 𝑐13 − 𝑏𝑐22 + 𝑐24 = 0,
𝑎12 + 𝑐22 = 0,
𝑋 = ∑𝜉𝑖 (𝑥, 𝑢) 𝜕𝑥𝑖 + ∑ 𝜙𝑗 (𝑥, 𝑢) 𝜕𝑢𝑗 .
𝑎12 + 𝑐22 = 0,
𝑎2 𝑏1 − 𝑐1 𝑐2 + 𝑐𝑎11 − 𝑎14 − 𝑏23 − 𝑎𝑐11
𝑏12 + 𝑐11 = 0,
𝑎1 𝑐2 + 𝑎2 𝑏2 − 𝑎2 𝑐1 − 𝑐2 2 + 2𝑐𝑎12 + 𝑏𝑎22 − 𝑎𝑐12 = 0,
Also the associated Ricci tensor 𝜌 is defined by
𝜌 (𝑋, 𝑌) := Tr {𝑍 󳨀→ R (𝑍, 𝑋) 𝑌} .
𝑎11 − 𝑏22 = 0,
(7)
𝑞
𝑃𝑟(𝑛) 𝑋 = 𝑋 + ∑ ∑𝜙𝛼𝐽 (𝑥, 𝑢(𝑛) ) 𝜕𝑢𝐽𝛼 ,
(12)
𝛼=1 𝐽
where 𝐽 = (𝑗1 , . . . , 𝑗𝑘 ), 1 ≤ 𝑗𝑘 ≤ 𝑝, 1 ≤ 𝑘 ≤ 𝑛 and the sum is
over all 𝐽’s of order 0 < #𝐽 ≤ 𝑛. If #𝐽 = 𝑘, the coefficient 𝜙𝛼𝐽 of
Advances in Mathematical Physics
3
𝜕𝑢𝐽𝛼 will only depend on 𝑘th and lower order derivatives of 𝑢,
and
𝜙𝛼𝐽
𝑝
(𝑛)
(𝑥, 𝑢 ) = 𝐷𝐽 (𝜙𝛼 −
∑𝜉𝑖 𝑢𝑖𝛼 )
𝑖=1
𝑝
+
𝛼
,
∑𝜉𝑖 𝑢𝐽,𝑖
𝑖=1
(13)
𝛼
:= 𝜕𝑢𝐽𝛼 /𝜕𝑥𝑖 . Now, according to
where 𝑢𝑖𝛼 := 𝜕𝑢𝛼 /𝜕𝑥𝑖 and 𝑢𝐽,𝑖
Theorem 2.36 of [2], the invariance of the system (9) under
the infinitesimal transformations leads to the invariance
conditions:
𝑃𝑟(𝑛) 𝑋 [Δ 𝜇 (𝑥, 𝑢(𝑛) )] = 0,
𝜇 = 1, . . . , 𝑟,
(14)
(𝑛)
whenever Δ 𝜇 (𝑥, 𝑢 ) = 0.
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]).
compute the coefficients of (17) as
𝜙1𝑥 = 𝐷𝑥 𝑄1 + 𝜉1 𝑎𝑥𝑥 + 𝜉2 𝑎𝑥𝑡 ,
𝜙2𝑥 = 𝐷𝑥 𝑄2 + 𝜉1 𝑏𝑥𝑥 + 𝜉2 𝑏𝑥𝑡 ,
𝜙3𝑥 = 𝐷𝑥 𝑄3 + 𝜉1 𝑐𝑥𝑥 + 𝜉2 𝑐𝑥𝑡 ,
𝜙1𝑡 = 𝐷𝑡 𝑄1 + 𝜉1 𝑎𝑥𝑡 + 𝜉2 𝑎𝑡𝑡 ,
𝜙2𝑡 = 𝐷𝑡 𝑄2 + 𝜉1 𝑏𝑥𝑡 + 𝜉2 𝑏𝑡𝑡 ,
𝜙3𝑡 = 𝐷𝑡 𝑄3 + 𝜉1 𝑐𝑥𝑡 + 𝜉2 𝑐𝑡𝑡 ,
𝜙1𝑥𝑥 = 𝐷𝑥2 𝑄1 + 𝜉1 𝑎𝑥𝑥𝑥 + 𝜉2 𝑎𝑥𝑥𝑡 ,
𝜙2𝑥𝑥 = 𝐷𝑥2 𝑄2 + 𝜉1 𝑏𝑥𝑥𝑥 + 𝜉2 𝑏𝑥𝑥𝑡 ,
𝜙3𝑥𝑥 = 𝐷𝑥2 𝑄3 + 𝜉1 𝑐𝑥𝑥𝑥 + 𝜉2 𝑐𝑥𝑥𝑡 ,
𝜙1𝑡𝑡 = 𝐷𝑡2 𝑄1 + 𝜉1 𝑎𝑥𝑡𝑡 + 𝜉2 𝑎𝑡𝑡𝑡 ,
3. Symmetries of System (8)
𝜙2𝑡𝑡 = 𝐷𝑡2 𝑄2 + 𝜉1 𝑏𝑥𝑡𝑡 + 𝜉2 𝑏𝑡𝑡𝑡 ,
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)
𝜙3𝑡𝑡 = 𝐷𝑡2 𝑄3 + 𝜉1 𝑐𝑥𝑡𝑡 + 𝜉2 𝑐𝑡𝑡𝑡 ,
𝑥̃ = 𝑥 + 𝜀𝜉1 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) + 𝑜 (𝜀2 ) ,
2
(18)
𝜙1𝑥𝑡 = 𝐷𝑥 𝐷𝑡 𝑄1 + 𝜉1 𝑎𝑥𝑥𝑡 + 𝜉2 𝑎𝑥𝑡𝑡 ,
𝜙2𝑥𝑡 = 𝐷𝑥 𝐷𝑡 𝑄2 + 𝜉1 𝑏𝑥𝑥𝑡 + 𝜉2 𝑏𝑥𝑡𝑡 ,
𝜙3𝑥𝑡 = 𝐷𝑥 𝐷𝑡 𝑄3 + 𝜉1 𝑐𝑥𝑥𝑡 + 𝜉2 𝑐𝑥𝑡𝑡 ,
2
̃𝑡 = 𝑡 + 𝜀𝜉 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) + 𝑜 (𝜀 ) ,
𝑎̃ = 𝑎 + 𝜀𝜙1 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) + 𝑜 (𝜀2 ) ,
(15)
̃𝑏 = 𝑏 + 𝜀𝜙 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) + 𝑜 (𝜀2 ) ,
2
where 𝐷𝑥 and 𝐷𝑡 are the total derivatives with respect to 𝑥 and
𝑡, respectively. By (14) the invariance condition is equivalent
with the following equations:
𝑐̃ = 𝑐 + 𝜀𝜙3 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) + 𝑜 (𝜀2 ) .
𝑃𝑟(2) 𝑋 [𝑎𝑥𝑥 − 𝑏𝑡𝑡 ] = 0,
𝑎𝑥𝑥 − 𝑏𝑡𝑡 = 0,
The symmetry generator of the above group of transformations is of the form
𝑃𝑟(2) 𝑋 [𝑏𝑥𝑦 + 𝑐𝑥𝑥 ] = 0,
𝑏𝑥𝑦 + 𝑐𝑥𝑥 = 0,
..
.
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
(16)
+ 𝜙3 (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 𝜕𝑐 ,
𝑃𝑟(2) 𝑋 [𝑎𝑥 𝑏𝑥 − 𝑏𝑥 𝑐𝑡 + 𝑏𝑡 𝑐𝑥 − 𝑐𝑥 2 + 𝑎𝑏𝑥𝑥 + 2 𝑐𝑏𝑥𝑡 − 𝑏𝑐𝑥𝑡 ] = 0,
𝑎𝑥 𝑏𝑥 − 𝑏𝑥 𝑐𝑡 + 𝑏𝑡 𝑐𝑥 − 𝑐𝑥 2 + 𝑎𝑏𝑥𝑥 + 2 𝑐𝑏𝑥𝑡 − 𝑏𝑐𝑥𝑡 = 0.
(19)
and, its second prolongation is the vector field
𝑃𝑟(2) 𝑋 = 𝑋 + 𝜙1𝑥 𝜕𝑎𝑥 + 𝜙1𝑡 𝜕𝑎𝑡 + 𝜙2𝑥 𝜕𝑏𝑥
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:
+ 𝜙2𝑡 𝜕𝑏𝑡 + 𝜙3𝑥 𝜕𝑐𝑥 + 𝜙3𝑡 𝜕𝑐𝑡 + 𝜙1𝑥𝑥 𝜕𝑎𝑥𝑥 + 𝜙2𝑥𝑥 𝜕𝑏𝑥𝑥
+ 𝜙3𝑥𝑥 𝜕𝑐𝑥𝑥 + 𝜙1𝑥𝑡 𝜕𝑎𝑥𝑡 + 𝜙2𝑥𝑡 𝜕𝑏𝑥𝑡
+ 𝜙3𝑥𝑡 𝜕𝑐𝑥𝑡 + 𝜙1𝑡𝑡 𝜕𝑎𝑡𝑡 + 𝜙2𝑡𝑡 𝜕𝑏𝑡𝑡 + 𝜙3𝑡𝑡 𝜕𝑐𝑡𝑡 .
(17)
1
2
1
Let (for convenience) 𝑄1 = 𝜙1 − 𝜉 𝑎𝑥 − 𝜉 𝑎𝑡 , 𝑄2 = 𝜙2 − 𝜉 𝑏𝑥 −
𝜉2 𝑏𝑡 and 𝑄3 = 𝜙3 − 𝜉1 𝑐𝑥 − 𝜉2 𝑐𝑡 , then by using (13), we can
𝑎3 𝜉𝑎1 = 0,
𝑏2 𝜉𝑏2 = 0,
𝑏2 𝜉𝑏1 = 0,
𝑎𝑐𝜙2𝑎 = 0,
𝑎𝜉𝑏2 = 0,
𝑐𝑏𝜉𝑏1 = 0, . . .
4
Advances in Mathematical Physics
Table 1: The commutator table of g.
[ ,]
𝑋1
𝑋2
𝑋3
𝑋4
𝑋5
𝑋6
𝑋7
𝑋1
0
0
−𝑋1
−𝑋2
0
0
0
𝑋2
0
0
0
0
−𝑋1
−𝑋2
0
𝑋3
𝑋1
0
0
−𝑋4
𝑋5
0
0
𝑋4
𝑋2
0
𝑋4
0
−𝑋3 + 𝑋6 − 2𝑋7
−𝑋4
0
𝑋5
0
𝑋1
−𝑋5
𝑋3 − 𝑋6 + 2𝑋7
0
𝑋5
0
𝑋6
0
𝑋2
0
𝑋4
−𝑋5
0
0
𝑋7
0
0
0
0
0
0
0
Table 2: Lie invariants and similarity solutions.
𝑖
1
2
3
4
5
6
𝑉𝑖
𝑋1
𝑋2
𝑋6
𝑋1 + 𝑋7
𝑋2 + 𝑋7
𝑋6 + 𝑋7
𝑤𝑖
𝑡
𝑥
𝑥
𝑡
𝑥
𝑥
𝑓𝑖 (𝑤)
𝑎
𝑎
𝑎
𝑎𝑒−𝑥
𝑎𝑒−𝑡
𝑎𝑡−1
ℎ𝑖 (𝑤)
𝑏
𝑏
𝑏𝑡−2
𝑏𝑒−𝑥
𝑏𝑒−𝑡
𝑏𝑡−3
𝑎𝑏𝑐 (−3𝜉𝑡1 − 𝜙3𝑏 ) + 2𝑐2 𝑎 (−𝜉𝑡2 + 𝜉𝑥1 )
+ 𝑐𝑎2 (−2𝜉𝑥2 + 𝜙2𝑐 ) + 𝜙1 (𝑎𝑏 − 2𝑐2 )
− 𝑎2 𝜙2 + 2𝑐𝑎𝜙3 + 4𝑐3 𝜉𝑡1 = 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
𝜙1 = 2𝑘1 𝑐 + 𝑘7 𝑎,
𝑎𝑖
𝑓(𝑤)
𝑓(𝑤)
𝑓(𝑤)
𝑒𝑥 𝑓(𝑤)
𝑒𝑡 𝑓(𝑤)
𝑡𝑓(𝑤)
𝑏𝑖
ℎ(𝑤)
ℎ(𝑤)
2
𝑡 ℎ(𝑤)
𝑒𝑥 ℎ(𝑤)
𝑒𝑡 ℎ(𝑤)
𝑡3 ℎ(𝑤)
𝜙2 = (2𝑘4 − 2𝑘3 + 𝑘7 ) 𝑏 + 2𝑘6 𝑐,
𝜉2 = 𝑘6 𝑥 + 𝑘4 𝑡 + 𝑘5 ,
𝜙3 = (𝑘7 + 𝑘4 − 𝑘3 ) 𝑐 + 𝑘6 𝑎 + 𝑘1 𝑏,
𝑑𝑥̃ (𝑠)
= 𝜉𝑖1 (𝑥̃ (𝑠) , ̃𝑡 (𝑠) , 𝑎̃ (𝑠) , ̃𝑏 (𝑠) , 𝑐̃ (𝑠)) ,
𝑑𝑠
𝑥̃ (0) = 𝑥,
𝑑̃𝑡 (𝑠)
= 𝜉𝑖2 (𝑥̃ (𝑠) , ̃𝑡 (𝑠) , 𝑎̃ (𝑠) , ̃𝑏 (𝑠) , 𝑐̃ (𝑠)) ,
𝑑𝑠
̃𝑡 (0) = 𝑡,
𝑑̃
𝑎 (𝑠)
= 𝜙1𝑖 (𝑥̃ (𝑠) , ̃𝑡 (𝑠) , 𝑎̃ (𝑠) , ̃𝑏 (𝑠) , 𝑐̃ (𝑠)) ,
𝑑𝑠
𝑎̃ (0) = 𝑎,
𝑑̃𝑏 (𝑠)
= 𝜙2𝑖 (𝑥̃ (𝑠) , ̃𝑡 (𝑠) , 𝑎̃ (𝑠) , ̃𝑏 (𝑠) , 𝑐̃ (𝑠)) ,
𝑑𝑠
̃𝑏 (0) = 𝑏,
𝑑̃𝑐 (𝑠)
= 𝜙3𝑖 (𝑥̃ (𝑠) , ̃𝑡 (𝑠) , 𝑎̃ (𝑠) , ̃𝑏 (𝑠) , 𝑐̃ (𝑠)) ,
𝑑𝑠
𝑐̃ (0) = 𝑐.
𝑖 = 1, . . . , 7.
(23)
(21)
where 𝑘𝑖 ’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 .
𝑐𝑖
𝑘(𝑤)
𝑘(𝑤)
𝑡𝑘(𝑤)
𝑒𝑥 𝑘(𝑤)
𝑒𝑡 𝑘(𝑤)
𝑡2 𝑘(𝑤)
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 [𝑋𝑖 , 𝑋𝑗 ] = 𝑋𝑖 𝑋𝑗 − 𝑋𝑗 𝑋𝑖 , 𝑖, 𝑗 = 1, . . . , 7.
The group transformation which is generated by 𝑋𝑖 =
𝜉𝑖1 𝜕𝑥 + 𝜉𝑖2 𝜕𝑡 + 𝜙1𝑖 𝜕𝑎 + 𝜙2𝑖 𝜕𝑏 + 𝜙3𝑖 𝜕𝑐 for 𝑖 = 1, . . . , 7 is obtained
by solving the seven systems of ODEs:
+ 𝑏𝑎2 (𝜙2𝑏 − 𝜙3𝑐 − 𝜉𝑥1 + 𝜉𝑡2 )
𝜉1 = 𝑘3 𝑥 + 𝑘1 𝑡 + 𝑘2 ,
𝑘𝑖 (𝑤)
𝑐
𝑐
𝑐𝑡−1
𝑐𝑒−𝑥
𝑐𝑒−𝑡
𝑐𝑡−2
(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.
By exponentiating the infinitesimal symmetries (22), we
obtain the one-parameter groups 𝑔𝑘 (𝑠) generated by 𝑋𝑘 , 𝑘 =
1, . . . , 7 as follows:
𝑔1 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥 + 𝑠, 𝑡, 𝑎, 𝑏, 𝑐) ,
𝑔2 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥, 𝑡 + 𝑠, 𝑎, 𝑏, 𝑐) ,
𝑔3 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥𝑒𝑠 , 𝑡, 𝑎, 𝑏𝑒−2𝑠 , 𝑐𝑒−𝑠 ) ,
𝑔4 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥, 𝑠𝑥 + 𝑡, 𝑎, 𝑎𝑠2 + 2𝑐𝑠 + 𝑏, 𝑎𝑠 + 𝑐) ,
Advances in Mathematical Physics
5
𝑔5 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑡𝑠 + 𝑥, 𝑡, 𝑏𝑠2 + 2𝑐𝑠 + 𝑎, 𝑏, 𝑏𝑠 + 𝑐) ,
𝑔5 (𝑠) ⋅ ℎ =
𝑔6 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥, 𝑡𝑒𝑠 , 𝑎, 𝑏𝑒2𝑠 , 𝑐𝑒𝑠 ) ,
𝑠
𝑠
𝑟32
𝑟2 𝑟2
(𝑥 − 𝑡𝑠) − 23 ln (𝑟1 (𝑥 − 𝑠𝑡) + 𝑟2 ) ,
𝑟1
𝑟1
𝑔5 (𝑠) ⋅ 𝑘 = (
𝑠
𝑔7 (𝑠) : (𝑥, 𝑡, 𝑎, 𝑏, 𝑐) 󳨃󳨀→ (𝑥, 𝑡, 𝑎𝑒 , 𝑏𝑒 , 𝑐𝑒 ) .
(24)
𝑟32
𝑟2 𝑟2
(𝑥 − 𝑡𝑠) − 23 ln (𝑟1 (𝑥 − 𝑡𝑠) + 𝑟2 )) 𝑠
𝑟1
𝑟1
+ 𝑟3 (𝑥 − 𝑡𝑠) + 𝑟4
(27)
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
is a set of new solutions of (8) and we can obtain many other
solutions by arbitrary combination of 𝑔𝑖 (𝑠)’s for 𝑖 = 1, . . . , 7.
So we obtain infinite number of Einstein Walker manifolds
just from this example.
𝑔1 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥 − 𝑠, 𝑡) ,
𝑔1 (𝑠) ⋅ ℎ = ℎ (𝑥 − 𝑠, 𝑡) ,
𝑔1 (𝑠) ⋅ 𝑘 = 𝑘 (𝑥 − 𝑠, 𝑡) ,
𝑔2 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥, 𝑡 − 𝑠) ,
4. One-Dimensional Optimal System of (8)
𝑔2 (𝑠) ⋅ ℎ = ℎ (𝑥, 𝑡 − 𝑠) ,
𝑔2 (𝑠) ⋅ 𝑘 = 𝑘 (𝑥, 𝑡 − 𝑠) ,
𝑔3 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥𝑒−𝑠 , 𝑡) ,
𝑔3 (𝑠) ⋅ ℎ = ℎ (𝑥𝑒−𝑠 , 𝑡) 𝑒−2𝑠 ,
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 onedimensional 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 𝑋𝑖 ,
𝑖 = 1, . . . , 7 is defined as follows:
𝑔3 (𝑠) ⋅ 𝑘 = 𝑘 (𝑥𝑒−𝑠 , 𝑡) 𝑒−𝑠 ,
𝑔6 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥, 𝑡𝑒−𝑠 ) ,
𝑔6 (𝑠) ⋅ ℎ = ℎ (𝑥, 𝑡𝑒−𝑠 ) 𝑒2𝑠 ,
𝑔6 (𝑠) ⋅ 𝑘 = 𝑘 (𝑥, 𝑡𝑒−𝑠 ) 𝑒𝑠 ,
𝑔7 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥, 𝑡) 𝑒𝑠 ,
𝑔7 (𝑠) ⋅ ℎ = ℎ (𝑥, 𝑡) 𝑒𝑠 ,
𝑔7 (𝑠) ⋅ 𝑘 = 𝑘 (𝑥, 𝑡) 𝑒𝑠 ,
𝑔4 (𝑠) ⋅ 𝑓 = 𝑓 (𝑥, 𝑡 − 𝑠𝑥) ,
𝑔4 (𝑠) ⋅ ℎ = 𝑓 (𝑥, 𝑡 − 𝑠𝑥) 𝑠2 + 2𝑘 (𝑥, 𝑡 − 𝑠𝑥) 𝑠 + ℎ (𝑥, 𝑡 − 𝑠𝑥) ,
𝑔4 (𝑠) ⋅ 𝑘 = 𝑓 (𝑥, 𝑡 − 𝑠𝑥) 𝑠 + 𝑘 (𝑥, 𝑡 − 𝑠𝑥) ,
𝑔5 (𝑠) ⋅ 𝑓 = ℎ (𝑥 − 𝑠𝑡, 𝑡) 𝑠2 + 2𝑘 (𝑥 − 𝑠𝑡, 𝑡) 𝑠 + 𝑓 (𝑥 − 𝑠𝑡, 𝑡) ,
𝑔5 (𝑠) ⋅ ℎ = ℎ (𝑥 − 𝑠𝑡, 𝑡) ,
𝑔5 (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) = 𝑟1 𝑥 + 𝑟2 ,
𝑏 = ℎ (𝑥, 𝑡) =
𝑟 𝑟2
𝑟32
𝑥 − 2 23 ln (𝑟1 𝑥 + 𝑟2 ) ,
𝑟1
𝑟1
= 𝑋𝑗 − 𝑠 ⋅ [𝑋𝑖 , 𝑋𝑗 ] +
𝑠2
⋅ [𝑋𝑖 , [𝑋𝑖 , 𝑋𝑗 ]] − ⋅ ⋅ ⋅ ,
2
(28)
where 𝑠 is a parameter and [𝑋𝑖 , 𝑋𝑗 ] 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
(26)
𝑐 = 𝑘 (𝑥, 𝑡) = 𝑟3 𝑥 + 𝑟4 ,
be a solution of (8), where 𝑟1 , 𝑟2 , 𝑟3 , 𝑟4 ∈ R are arbitrary
constants, we conclude that the functions 𝑔𝑖 (𝑠)⋅𝑓, 𝑔𝑖 (𝑠)⋅ℎ, and
𝑔𝑖 (𝑠) ⋅ 𝑘 are also solutions of (8) for 𝑖 = 1, . . . , 7. For example,
𝑟2
𝑟 𝑟2
𝑔5 (𝑠) ⋅ 𝑓 = ( 3 (𝑥 − 𝑡𝑠) − 2 23 ln (𝑟1 (𝑥 − 𝑡𝑠) + 𝑟2 )) 𝑠2
𝑟1
𝑟1
+ 2 (𝑟3 (𝑥 − 𝑡𝑠) + 𝑟4 ) 𝑠 + 𝑟1 (𝑥 − 𝑡𝑠) + 𝑟2 ,
Ad (exp (𝑠 ⋅ 𝑋𝑖 ) ⋅ 𝑋𝑗 )
(1) 𝑋7 ,
(2) 𝑋1 + 𝑎𝑋7 ,
(4) 𝜀𝑋1 + 𝑋6 + 𝑎𝑋7 ,
(3) 𝑋2 + 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
6
Advances in Mathematical Physics
of ⟨𝑋1 , 𝑋2 , 𝑋3 , 𝑋4 , 𝑋5 , 𝑋6 ⟩. Each adjoint transformation is a
linear map 𝐹𝑖𝑠 : g → g that defined by 𝑋 󳨃→ Ad(exp(𝑠𝑋𝑖 )⋅𝑋),
for 𝑖 = 1, . . . , 7. The matrix 𝑀𝑖𝑠 of 𝐹𝑖𝑠 , 𝑖 = 1, . . . , 7, with respect
to basis {𝑋1 , . . . , 𝑋7 } is
1
[0
[
[−𝑠
[
𝑀1𝑠 = [
[0
[0
[
[0
[0
0
1
0
−𝑠
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0]
]
0]
]
0]
],
0]
]
0]
1]
1
[0
[
[0
[
𝑠
𝑀2 = [
[0
[−𝑠
[
[0
[0
0
1
0
0
0
−𝑠
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0]
]
0]
]
0]
],
0]
]
0]
1]
𝑒𝑠
[0
[
[0
[
𝑠
𝑀3 = [
[0
[0
[
[0
[0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
𝑒−𝑠
0
0
0
0
0
0
0
𝑒𝑠
0
0
0
0
0
0
0
1
0
0
0]
]
0]
]
0]
],
0]
]
0]
1]
1
[0
[
[0
[
𝑀4𝑠 = [
[0
[0
[
[0
[0
𝑠
1
0
0
0
0
0
0 0 0 0 0
0 0 0 0 0 ]
]
1 𝑠 0 0 0 ]
]
0 1 0 0 0 ]
],
−𝑠 −𝑠2 1 𝑠 −2𝑠]
]
0 −𝑠 0 1 0 ]
0 0 0 0 1 ]
1
[𝑠
[
[0
[
𝑠
𝑀5 = [
[0
[0
[
[0
[0
0
1
0
0
0
0
0
0
0
1
𝑠
0
0
0
0 0 0 0
0 0 0 0]
]
0 −𝑠 0 0 ]
]
1 −𝑠2 −𝑠 2𝑠]
],
0 1 0 0]
]
0 𝑠 1 0]
0 0 0 1]
1
[0
[
[0
[
𝑠
𝑀6 = [
[0
[0
[
[0
[0
0
𝑒𝑠
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
𝑒𝑠
0
0
0
0
0
0
0
𝑒−𝑠
0
0
0
0
0
0
0
1
0
𝑠
󳨃󳨀→ ((1 + 𝑠5 𝑠4 ) 𝑒𝑠3 𝑎1 + 𝑒𝑠6 +𝑠3 𝑠4 𝑎2 ) ⋅ 𝑋1 + ⋅ ⋅ ⋅
(31)
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
𝑠
coefficient of 𝑋2 vanish by 𝐹44 ; 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
𝑠
by 𝐹66 ; by setting 𝑠6 = ln |𝑎2 |. So, 𝑋 is reduced to case
(3).
If 𝑎3 = ⋅ ⋅ ⋅ = 𝑎5 = 0 and 𝑎6 ≠ 0, then we can make the
𝑠
coefficient of 𝑋2 vanish by 𝐹22 ; by setting 𝑠2 = −𝑎2 /𝑎6 .
Also the coefficient of 𝑋1 can be vanished or be ±1 by
𝑠
𝐹33 ; by setting 𝑠3 = ln |𝑎1 |. Scaling 𝑋 if necessary, we
can assume that 𝑎6 = 1. So, 𝑋 is reduced to the case
(4).
(30)
respectively and 𝑀7𝑠 is the 7 × 7 identity matrix. Let 𝑋 =
∑7𝑖=1 𝑎𝑖 𝑋𝑖 , then
𝑠
+ (1 + 𝑠5 𝑠4 ) 𝑎6 − 2𝑠5 𝑠4 𝑎7 ) 𝑋6 + 𝑎7 𝑋7 .
Now, we can simplify 𝑋 as follows.
0
0]
]
0]
]
0]
],
0]
]
0]
1]
𝐹77 ∘ ⋅ ⋅ ⋅ ∘ 𝐹11 : 𝑋
+ (−𝑠5 𝑠2 𝑎1 − 𝑒𝑠6 𝑠2 𝑎2 + ⋅ ⋅ ⋅
If 𝑎3 = 𝑎4 = 0 and 𝑎5 ≠ 0, then we can make the
𝑠
coefficient of 𝑋1 vanish by 𝐹22 ; by setting 𝑠2 = −𝑎1 /𝑎5 .
Also the coefficient of 𝑋2 can be vanished or be ±1 by
𝑠
𝐹66 ; 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
𝑠
of 𝑋2 vanish by 𝐹11 ; by setting 𝑠1 = −𝑎2 /𝑎4 . Also the
𝑠
coefficient of 𝑋1 can be vanished or be ±1 by 𝐹33 ; 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
𝑠
by 𝐹11 ; by setting 𝑠1 = −𝑎1 /𝑎3 . Also the coefficients of
𝑠
𝑠
𝑋2 and 𝑋4 can be vanished or be ±1 by 𝐹33 and 𝐹66 ;
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𝜕𝑐 ) 𝐼 = 𝑥
𝜕𝐼
𝜕𝐼
𝜕𝐼
𝜕𝐼
𝜕𝐼
+ 0 + 0 − 2𝑏 − 𝑐 .
𝜕𝑥
𝜕𝑡
𝜕𝑎
𝜕𝑏
𝜕𝑐
(32)
󸀠
(5)
where 𝑓 ,. . . are derivatives with respect to w and 𝑟𝑖 ’s are arbitrary constants.
(6)
𝑓𝑥𝑥 − ℎ = 0, ℎ𝑥 + 𝑘𝑥𝑥 = 0, 𝑓𝑥 + 𝑘 = 0
2ℎ𝑓 − 2𝑓𝑘𝑥 + 3𝑓𝑥 𝑘 − 𝑘2 = 0
𝑓ℎ𝑥 − 2𝑘𝑥 𝑘 + 𝑘𝑓𝑥𝑥 − 𝑓𝑘𝑥𝑥 − ℎ𝑘 = 0
ℎ𝑥 𝑓𝑥 + ℎ𝑥 𝑘 − 𝑘𝑥2 + 𝑓ℎ𝑥𝑥 = 0
(4)
𝑓𝑥𝑥 − 6ℎ = 0, 𝑓𝑥 + 2𝑘 = 0
3𝑓𝑘𝑥 − 4𝑓𝑥 𝑘 − 3𝑓ℎ + 4𝑘2 = 0
3ℎ𝑥 + 𝑘𝑥𝑥 = 0
4𝑘𝑥 𝑘 − 𝑓ℎ𝑥 − 𝑘𝑓𝑥𝑥 + 𝑓𝑘𝑥𝑥 + 2ℎ𝑘 = 0
𝑓𝑥 ℎ𝑥 + 4ℎ𝑥 𝑘 + ℎ𝑘𝑥 − 𝑘𝑥2 + 𝑓ℎ𝑥𝑥 = 0
ℎ𝑡𝑡 − 𝑓 = 0, ℎ𝑡 + 𝑘 = 0, 𝑓𝑡 + 𝑘𝑡𝑡 = 0
𝑓𝑡 ℎ𝑡 + 𝑓𝑡 𝑘 − 𝑘𝑡2 + ℎ𝑓𝑡𝑡 = 0
ℎ𝑘𝑡𝑡 − 𝑓𝑡 ℎ + 2𝑘𝑘𝑡 = 0
2𝑓ℎ − 2ℎ𝑘𝑡 + 3ℎ𝑡 𝑘 − 𝑘2 = 0
(3)
ℎ𝑡𝑡 = 0
𝑘𝑡𝑡 = 0
𝑓𝑡 ℎ𝑡 − 𝑘𝑡2 + ℎ𝑓𝑡𝑡 = 0
ℎ𝑘𝑡𝑡 = 0
𝑓𝑥𝑥 = 0
𝑘𝑥𝑥 = 0
𝑓𝑥 ℎ𝑥 − 𝑘𝑥2 + 𝑓ℎ𝑥𝑥 = 0
𝑘𝑓𝑥𝑥 − 𝑓𝑘𝑥𝑥 = 0
Reduced system
𝑓𝑥𝑥 − 2ℎ = 0
2𝑘𝑥 𝑘 − 𝑘𝑓𝑥𝑥 + 𝑓𝑘𝑥𝑥 = 0
2ℎ𝑥 + 𝑘𝑥𝑥 = 0
𝑓𝑥 𝑘 − 𝑘2 − 𝑓𝑘𝑥 = 0
𝑘𝑥2 − 𝑓𝑥 ℎ𝑥 − 3ℎ𝑥 𝑘 − ℎ𝑘𝑥 − 𝑓ℎ𝑥𝑥 = 0
(2)
(1)
𝑖
𝑓 = 𝑟1
ℎ=0
𝑘=0
ℎ=0
𝑓 = 𝑟1
𝑘=0
𝑓=0
ℎ = 𝑟1
𝑘=0
ℎ = 𝑟2
𝑓 = 𝑘(𝑥 + 𝑟1 )
𝑘 = 𝑟2 𝑥 + 𝑟3
𝑘 = 𝑟1
ℎ = ℎ(𝑥)
ℎ=0
𝑘 = 𝑟1
𝑓=0
𝑓 = 𝑓(𝑡)
Table 3: Reduced systems and their group invariant solutions.
27
(𝑟1 𝑥 + 𝑟2 )
1
ℎ = − 𝑘𝑥
3 2
3𝑘
𝑓=−
𝑘𝑥
𝑘=−
𝑘 = 𝑟2 𝑒𝑟1 𝑥
𝑟
𝑓 = − 2 𝑒𝑟1 𝑥
𝑟1
ℎ = −𝑟1 𝑟2 𝑒𝑟1 𝑥
𝑘 = 𝑟2 𝑒𝑟1 𝑡
𝑟
ℎ = − 2 𝑒𝑟1 𝑡
𝑟1
𝑓 = −𝑟1 𝑟2 𝑒𝑟1 𝑡
𝑓=(
3
𝑟1 𝑟3 − 𝑟4 𝑟52
𝑟52 𝑡
)
ln(𝑟
𝑡
+
𝑟
)
+
+ 𝑟2
3
4
𝑟32
𝑟3
ℎ = 𝑟3 𝑡 + 𝑟4
𝑘 = 𝑟5 𝑡 + 𝑟6
𝑓 = 𝑟3 𝑥 + 𝑟4
𝑟2 𝑥
𝑟 𝑟 − 𝑟 𝑟2
ℎ = ( 1 3 2 4 5 ) ln(𝑟3 𝑥 + 𝑟4 ) + 5 + 𝑟2
𝑟3
𝑟3
𝑘 = 𝑟5 𝑥 + 𝑟6
128 − 𝑟1
𝑘=
+ 𝑟3 𝑥 + 𝑟4
2
3(𝑥 + 𝑟2 )
4𝑘𝑘𝑥𝑥
𝑓=−
𝑘𝑥𝑥𝑥
2
𝑘𝑥 𝑘𝑥𝑥𝑥 − 2𝑘𝑥𝑥
ℎ=
𝑘𝑥𝑥𝑥
Group invariant solutions
Advances in Mathematical Physics
7
8
Advances in Mathematical Physics
So we must solve the associated characteristic ODE
𝑑𝑥 𝑑𝑡 𝑑𝑎
𝑑𝑏
𝑑𝑐
=
=
=
=
.
𝑥
0
0
−2𝑏 −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−2 ℎ(𝑤) = 𝑥−2 ℎ(𝑡) and 𝑐 = 𝑥−1 𝑘(𝑤) =
𝑥−1 𝑘(𝑡). So by using the chain rule, we have
𝑎1 =
𝜕𝑓 𝜕𝑤 𝜕𝑓 𝜕𝑡
𝜕𝑎
=
=
= 0,
𝜕𝑥 𝜕𝑤 𝜕𝑥
𝜕𝑡 𝜕𝑥
𝑎2 = 𝑎𝑡 = ⋅ ⋅ ⋅ = 𝑓𝑡 ,
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.
𝑏1 = −2𝑥−3 ℎ,
𝑏2 = 𝑥−2 ℎ𝑡 ,
𝑐1 = −𝑥−2 𝑘,
𝑐2 = 𝑥−1 𝑘𝑡 ,
𝑎11 = 𝑎𝑥𝑥 = 0,
𝑎12 = 0,
𝑎22 = 𝑓𝑡𝑡 ,
𝑏11 = 6𝑥−4 ℎ,
𝑏12 = −2𝑥−3 ℎ𝑡 ,
References
𝑏22 = 𝑥−2 ℎ𝑡𝑡 ,
𝑐11 = 2𝑥−3 𝑘,
𝑐12 = −𝑥−2 𝑘𝑡 ,
[1] S. Lie, “On integration of a class of linear partial differential
equations by means of definite integrals,” Archiv der Mathematik, vol. 6, no. 3, pp. 328–368, 1881, translation by N. H.
Ibragimov.
[2] P. J. Olver, Applications of Lie Groups to Differential Equations,
vol. 107 of Graduate Texts in Mathematics, Springer, New York,
NY, USA, 1986.
[3] G. W. Bluman and J. D. Cole, Similarity Methods for Differential
Equations, vol. 13 of Applied Mathematical Sciences, Springer,
New York, NY, USA, 1974.
[4] G. W. Bluman and S. Kumei, Symmetries and Differential
Equations, vol. 81 of Applied Mathematical Sciences, Springer,
New York, NY, USA, 1989.
[5] P. J. Olver and P. Rosenau, “Group-invariant solutions of
differential equations,” SIAM Journal on Applied Mathematics,
vol. 47, no. 2, pp. 263–278, 1987.
[6] R. Z. Zhdanov, I. M. Tsyfra, and R. O. Popovych, “A precise
definition of reduction of partial differential equations,” Journal
of Mathematical Analysis and Applications, vol. 238, no. 1, pp.
101–123, 1999.
[7] G. Cicogna, “A discussion on the different notions of symmetry
of differential equations,” Proceedings of Institute of Mathematics
of NAS of Ukraine, vol. 50, pp. 77–84, 2004.
[8] G. W. Bluman and J. D. Cole, “The general similarity solution of
the heat equation,” Journal of Mathematics and Mechanics, vol.
18, pp. 1025–1042, 1969.
[9] M. Brozos-Vázquez, E. Garcı́a-Rı́o, P. Gilkey, S. Nikčević, and R.
Vázquez-Lorenzo, The Geometry of Walker Manifolds, vol. 5 of
Synthesis Lectures on Mathematics and Statistics Series, Morgan
& Claypool, 2009.
[10] A. G. Walker, “Canonical form for a Riemannian space with a
parallel field of null planes,” The Quarterly Journal of Mathematics, vol. 1, no. 2, pp. 69–79, 1950.
[11] M. Nadjafikhah and V. Shirvani-Sh, “Lie symmetry analysis of
Kudryashov-Sinelshchikov equation,” Mathematical Problems
in Engineering, vol. 2011, Article ID 457697, 9 pages, 2011.
[12] L. V. Ovsiannikov, Group Analysis of Differential Equations,
Academic Press, New York, NY, USA, 1982.
[13] M. Nadjafikhah, “Lie symmetries of inviscid Burgers’ equation,”
Advances in Applied Clifford Algebras, vol. 19, no. 1, pp. 101–112,
2009.
𝑐22 = 𝑥−1 𝑘𝑡𝑡 .
(34)
Substituting these in the system (8), the reduced system is
obtained as follows:
ℎ𝑡𝑡 = 0,
3ℎ𝑘𝑡 − 5ℎ𝑡 𝑘 − 𝑘2 + 6𝑓ℎ = 0,
𝑓𝑡 ℎ𝑡 + 𝑓𝑡 𝑘 − 𝑘𝑡2 + ℎ𝑓𝑡𝑡 + 𝑓𝑘𝑡 = 0,
𝑘𝑡𝑡 = 0,
(35)
ℎ𝑡 − 𝑘 = 0,
2𝑓𝑡 ℎ − 2𝑘𝑘𝑡 + 2𝑓𝑘 + ℎ𝑘𝑡𝑡 = 0,
which is a system of ODEs. Two types of solutions are
obtained by solving this system
𝑓 = 𝑓 (𝑡)
{
{
(1) {ℎ = 0
{
{𝑘 = 0,
𝑟22
{
{
𝑓
=
{
{
𝑟2 𝑡 + 𝑟1
(2) {
ℎ
=
𝑟
{
2 𝑡 + 𝑟1
{
{
𝑘
=
𝑟
2,
{
(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.
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.
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:
𝑎=−
𝑟2 𝑟1 𝑥 𝑡
𝑒 𝑒,
𝑟1
𝑏 = −𝑟1 𝑟2 𝑒𝑟1 𝑥 𝑒𝑡 ,
𝑐 = 𝑟2 𝑒𝑟1 𝑥 𝑒𝑡 , (37)
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