1. Introduction
The homogenous wave equation on the sphere is given by the equation
H. Azad and M. Mustafa [
1], using the package MathLie [
2], determined the Lie algebra of the Lie point symmetries of (
1), and classified its subalgebras up to conjugancy. Subsequently, they performed similarity reduction for each subalgebra, and in some cases of the two-dimensional subalgebra, they provided the invariant solution.
In [
3], Freire observed that Equation (
1) is a particular case of the Poisson equation
for the metric
The Lie point and the Noether point symmetries of the nonlinear Poisson equation of the form
have been studied in [
4] for certain forms of the function
and
, where
n is the dimension of the space. Using the results of [
4], Freire studied the Lie point symmetries of (
1), which is a particular case of (
3) for
and metric (
2).
However, it should be pointed out that the situation here is different than the one considered in [
4]. Indeed, having the Lie symmetry conditions for a generic metric and a general function
, one has two possibilities: either to leave the metric unspecified and consider various types of functions
, or to specify the metric and determine the functions
for which the symmetry conditions are satisfied.
In [
4], the authors considered the first scenario, that is, they assumed a generic metric and considered certain forms of
In the present work, the situation is different because the metric is specified. Therefore, one has to use the second approach, that is, to fix the metric (
2) and use the Lie symmetry conditions to find for which functions
Lie point symmetries are admitted. This is what it is carried out in the present work.
In the second approach, one needs the conformal Killing vectors (CKVs) of the metric (
2). To our knowledge, there is no a package available to perform that, at least for a complex metric and/or a significant number of independent variables, and one has to use available results from Riemannian geometry. This case, although more demanding, has the advantage that it is applicable to higher dimensions and more complex metrics due to the existence of the plethora of general relevant geometric results. In the
Appendix A, we use a systematic method, which determines the CKVs in a
decomposable Riemannian space to compute the CKVs of (
2).
The structure of the paper is as follows. In
Section 3,
Section 4,
Section 5 and
Section 6, we present the geometric results, which shall be used in the “solution” of the system of Lie conditions. In
Section 6, we use these results to compute the CKVs of the metric defined by (
2). In
Section 7, we write the Lie point symmetry conditions and find that the only homogeneous wave equations on the sphere that admit Lie point symmetries are the ones for which
where
k is a constant. Furthermore, the Lie symmetry generators are the Killing vectors (KVs) of the metric (
2) and the vector
, where
is a solution of the wave equation. In
Section 8, it is shown that the Lie symmetries are also Noether symmetries, and it is demonstrated how the conserved Noether currents are computed. Finally, in
Section 10, we draw our conclusions.
3. Maximally Symmetric Metrics
Maximally symmetric metrics are the units of “symmetry”, and have a direct relation with the standard Euclidean concepts of symmetry, e.g., spherical symmetry.
Definition 1. A maximally symmetric metric is a metric whose curvature tensor satisfies the relationwhere is a constant, is the curvature scalar, and the tensor A flat metric is a maximally symmetric metric for which . The covariant definition of a flat metric is
A maximally symmetric metric is characterized by the number of admitted KVs as follows (see pp. 238–239 in [
5]):
Theorem 1. A non-degenerate metric is a metric of constant curvature iff it is a maximally symmetric metric iff it admits KVs iff the equations of geodesics admit linearly independent linear first integrals ( where ).
All maximally symmetric metrics are conformally flat and have the global property that they can be written in the form
where
and
where
R is the scalar curvature of the space and
for a negative, positive, and zero curvature, respectively.
4. The Conformal Algebra of Maximally Symmetric Metrics
Definition 2. Two metrics, , are conformally related if they satisfy the conditionwhere is the conformal factor. Two conformally related metrics share the same CKVs. From (
9), it follows that a maximally symmetric metric is conformally related to the flat metric with the conformal factor
Therefore, the conformal algebra of the metric is the same with conformal algebra of the flat metric.
It is well known [
5,
6,
7,
8] that a flat metric of dimension
n admits
KVs, 1 gradient HV and
Special CKVs (a CKV is called special if the conformal factor
satisfies the condition
).
Specifically, the conformal algebra of the flat space
where
and
consists of the following vectors.
gradient KVs: .
non-gradient KVs: .
One HV: .
Special CKVs: where with conformal factor and
From these, by taking into consideration the conformal factor
, we have that the conformal algebra of a maximally symmetric metric
consists of the vectors of
Table 1.
The metric
being maximally symmetric must admit
KVs. In
Table 1, we have only the
KVs
. Therefore, there must exist another
n KVs. Furthermore, we note that the HV is a gradient CKV. Because gradient vector fields are always convenient, we are looking for combinations of these CKVs, which produce KVs and gradient CKVs. We find that the vectors
are the required ones. Indeed,
is a KV and
is a proper gradient CKV with the conformal functor
We collect these results in
Table 2. The non-tensorial indices
count vector fields.
is the antisymmetric part
of the non-gradient CKV, called the bivector of
X.
We note that the new basis consists of the KVs , no HV and proper CKVs These vectors are in total as it is necessary for a metric of constant curvature. Furthermore all KVs of this basis are not gradient and all proper CKVs are gradient. This observation is very useful in applications.
We note that the vectors
are not SCKVs except in the case
Indeed from the last column we compute
where we have set
A similar calculation applies to
Relations (
17) and (
18) show two more facts:
The gradient vectors and are gradient CKVs of the metric (not the flat metric!);
For non-flat metrics ( the gradient vector is non-null.
6. The Geometry of the Metric (2)
The metric
defined by the homogenous wave Equation (
3) is 1 + 2 decomposable along the gradient KV
Furthermore, the 2-metric
is the metric of a maximally symmetric space with positive Gaussian curvature
therefore admits
CKVs. Three of these vectors are Killing vectors (KVs) and the remaining three are proper CKVs. The homothetic vector (HV) is not admitted. It is easily found (using any algebraic computing program) that the KVs
(say) are
The 1 + 3 metric (
2) is conformally flat (this can be shown easily by showing that the Cotton tensor vanishes), and therefore admits
CKVs. These are the four KVs
and six proper CKVs.
The computation of the six CKVs are computed using Theorem 2. It is found that there are two sets of proper CKVs, the non-gradient and the gradient ones. The non-gradient CKVs are
with conformal factors
and the gradient CKVs are
with conformal factors
It is important to note that the conformal factors of the all proper CKVs satisfy the relation
that is, the vectors
are gradient CKVs.
In a Riemannian space, which admits CKVs the following results are well known [
7,
8].
Lemma 1. Assume that the vector is a CKV of the metric with conformal factor i.e., Then, the following relations hold:where is the dimension of the space. Proof. The proof of (25) is straightforward by using the definition of and contracting with . To prove (26), one uses the identityand replaces to find
□
Contracting with the result follows.
We note that for
the
, that is, the connection coefficients are disassociated from the conformal factor. This is a singular case, and it is the reason that the case
was not considered in [
4].
7. The Lie Point Symmetries
As it is done in [
3], we consider the Equation (
3) for an arbitrary function
and the metric (
2), that is, we consider the equation
, or equivalently,
where
and
The Lie point symmetry vector is
and the Lie symmetry condition is
where
is a function to be determined.
is the second prolongation of
X given by the expression
where
The introduction of the function
in the defining equation allows one to consider the variables
to be independent [
10].
When the symmetry condition
is applied to (
28), the following system of equations results [
10,
11,
12,
13,
14,
15,
16,
17,
18] in
These relations coincide with those given in [
10] (p. 115) if we consider the following correspondence:
This Notation | Ibragimov Notation |
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The strategy to solve these conditions is the following.
Because we know the metric, it is possible (in principle!) to solve condition (
34) and determine the CKVs
Using these vectors in (
33), one determines the value of
Having these results and using (
32), one determines the possible functions
, and consequently, the equations of the form (
28), which admit Lie point symmetries. Having the admitted Lie symmetries, one may use similarity reduction and possibly determine invariant solutions of (
28). This latter part has been carried out for the case
in [
1]. Finally, using the fact that Equation (
28) follows from the Lagrangian [
4]
one determines the Noether point symmetries and finds the corresponding Noether currents, which reduce this equation possibly to one that can be solved with quadratures. This has been considered in [
3] for
, where
k is a constant.
In the present case, it is possible to compute the CKVs of the decomposable metric; therefore, we follow the above algorithm.
We start with (
34). This states that
is a CKV of (
2). These vectors have been determined in
Section 6.
Next, we consider condition (
33).
Using property (
25) of Lemma 1, condition (
33) becomes
Again, using (
26) of Lemma 1 to replace
, we find that eventually, condition (
33) gives
where
is the dimension of the space. In our case,
; therefore,
or
where
C is a constant, which counts for the KVs and the HV. We conclude that condition (
33) expresses the conformal factor in terms of
a, and furthermore provides the value of
:
Due to (
38), condition (
34) becomes
Still, we have to consider the remaining Lie symmetry condition (
32):
from which follows
The obvious implication of (
41) is (because
)
; therefore, the function
, where
k is a constant.
In the last section, it has been shown that the conformal factors of the CKVs of the metric (
2) satisfy the relation
from which follows
From (
34) we have
, and replacing
from from (
39), we find
Combining (
44) and (
45), we find
Then, condition (
41) becomes
or
We consider two cases.
Case a.
Then, a is a constant, which means that (homothetic vector is not admitted), and subsequently, In this case, only the KVs survive. Condition (42) gives .
We conclude that for metric (
2), the wave equations that admit Lie point symmetries are of the form
where
The Lie symmetry vectors are
where
are the KVs (
23) and
is a solution of Equation (
48). These vectors coincide with the ones found in [
1,
3], except the Lie symmetry
This is reasonable, because
Note: In [
3], the author refers to [
4]’s case
, where
. In our case,
, which also gives
, and therefore agrees with our result.
Case b:
In this case, we have
, and (
47) is trivially satisfied; therefore
a stays unspecified. Now, we use (
42), from which follows
We conclude that the Lie symmetries are the same as in the case
Therefore the result of applies to all values of k.
10. Conclusions
We addressed the problem of finding the Lie point symmetries of the partial differential equation
where
and
are arbitrary functions of their argument. In (
68), there are two sets of unknown quantities, that is, the tensor
and the functions
This means that in order for it to be possible for the Lie point symmetries to be determined, one of these sets must be specified. If the functions of the second set are assumed, one determines the CKVs of the metrics for which Lie point symmetries are admitted. Because the CKVs do not specify completely the metric, one finds essentially families of metrics. This approach has been taken in [
4]. Here, we are given the metric, which we read from the Equation (
3), which is a special case of (
68), and we assume that
Therefore, we compute the Lie point symmetries, and consequently, we determine the functions
for which Lie point symmetries are admitted. This completes the work of [
3].
From the Lie symmetry condition (
34), it follows that the Lie point symmetries are the CKVs of the metric
. This is a
decomposable metric whose reduced 2D metric is a metric of maximal symmetry with curvature scalar
Using Theorem 2, we computed the CKVs of (
34), and consequently, the Lie point symmetries. Using the rest of the Lie symmetry conditions, we found that
, where
k is a constant. Finally, it has been shown that the Lie symmetries are also Noether symmetries, and it has been demonstrated how one computes the conserved Noether currents.
The geometric method we have considered in computing the Lie point symmetries and the functions
is general and can be applied to other known differential equations of the form (
68), especially to the ones in which the metric defined by
is more complex and of a higher dimension where the algebraic computing programmes might not answer.