On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations ()
1. Introduction
A solution of many problems of the geometric optics, theories of anisotropic media, theory of minimal surfaces, nonlinear electrodynamics, theories of gravity, geometry, unified field theory, string theories, black holes, cosmology, etc. is reduced to the investigation of the Eikonal equations [1] [2] [3] [4] [5] , the Euler-Lagrange equations [6] - [12] , the Born-Infeld equations [13] - [22] , the Monge-Ampère equations [23] - [40] in the spaces of different dimensions and different types (see also the references therein).
Nowadays, there exist a lot of methods for the construction exact solutions of linear and nonlinear partial differential equations (PDEs). More details on this theme can be found in [41] - [46] (see also the references therein).
We consider the following (1 + 3)-dimensional
-invariant PDEs:
• the Eikonal equation,
• the Euler-Lagrange-Born-Infeld equation,
• the homogeneous Monge-Ampère equation,
• the inhomogeneous Monge-Ampère equation.
From the results obtained by Fushchich W.I., Shtelen W.M. and Serov N.I. [40] , it follows, in particular, that the common symmetry group of those equations is the generalized Poincaré group
. Therefore, in the natural way arises the following question: what is the relationship between invariant solutions of the equations under study? In particular, whether those equations have common invariant solutions?
The purpose of this paper is to try to construct and classify the common invariant solutions for the equations under consideration. It is known that the (1 + 3)-dimensional
-invariant Eikonal equation is the simplest one among the equations under study. Therefore, we can use this fact for constructing the common invariant solutions. At the present time, we have constructed invariant solutions for the (1 + 3)-dimensional
-invariant Eikonal equation obtained on the base of low-dimensional (
) nonconjugate subalgebras of the Lie algebra of the Poincaré group
, by using classical Lie-Ovsiannikov approach [41] [42] [43] [44] . This method, in particular, allows us to perform the symmetry reduction of the many-dimensional PDEs with non-trivial symmetry groups to differential equations with a fewer number of independent variables as well as to construct solutions, invariant with respect to nonconjugate subgroups of the symmetry groups, of the equations under study. According to this method, reduced equations (invariant solutions) should be classified with respect to the ranks of the corresponding nonconjugate subalgebras of the Lie algebras of the symmetry groups of the equations under study.
Our contribution in classical Lie-Ovsiannikov method consists in the suggestion to use, for the classification of symmetry reductions (invariant solutions) of PDEs with non-trivial symmetry groups, not only ranks of nonconjugate subalgebras, but also their structural property. Some details on this theme can be found in [47] [48] .
In our paper, we have performed the suggestion for the classification of the common invariant solutions of some P(1, 4)-invariant PDEs by using the structural property of the low-dimensional (
) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4).
The direct checks allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (
) nonconjugate subalgebras of the Lie algebra of the Poincaré group
, satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.
To present the results obtained, we give some information about the Lie algebra of the Poincaré group
and its nonconjugate subalgebras.
2. The Lie Algebra of the Poincaré Group
and Its Nonconjugate Subalgebras
The group
is a group of rotations and translations of the five-dimensional Minkowski space
. It is the smallest group, which contains, as subgroups, the extended Galilei group
[49] (the symmetry group of classical physics) and the Poincaré group
(the symmetry group of relativistic physics).
The Lie algebra of the group
is generated by 15 bases elements
and
, which satisfy the commutation relations
(1)
(2)
where
,
, if
.
In this paper, we consider the following representation [40] of the Lie algebra of the group
:
(3)
(4)
In the following, we will use the next bases elements:
(5)
(6)
(7)
The Lie algebra of the extended Galilei group
is generated by the following bases elements:
(8)
The classification of all nonconjugate subalgebras of the Lie algebra of the group
of dimensions ≤ 3 was performed in [50] .
3. On the Construction and Classification of the Common Invariant Solutions for Some (1 + 3)-Dimensional
-Invariant PDEs
In this Section, We Consider the Following PDEs
• the Eikonal equation
• the Euler-Lagrange-Born-Infeld equation
• the homogeneous Monge-Ampère equation
• the inhomogeneous Monge-Ampère equation
where
,
,
,
,
,
,
,
is the d’Alembert operator.
Here, and in what follows,
is a four-dimensional Minkowski space,
is a real number axis of the depended variable u.
From the results obtained by Fushchich W.I., Shtelen W.M. and Serov N.I. [40] it follows, in particule, that the common symmetry group of those equations is the generalised Poincaré group
.
In this section we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions. To obtain those results, we used the nonconjugate subalgebras of the Lie algebra of the group
, structural properties of its low-dimensional (
) nonconjugate subalgebras as well as the results of the classification of symmetry reductions of the eikonal equation. More details on this theme can be found in [47] [48] .
Bellow we present the results obtained.
3.1. Classification of the Common Invariant Solutions for the Equations under Study Using One-Dimensional Nonconjugate Subalgebras of the Lie Algebra of the Group
1)
:
The common invariant solution for the equations under study:
where
and
are arbitrary real constants.
2)
:
The common invariant solution for the equations under study:
where
and
are arbitrary real constants.
3)
:
The common invariant solution for the equations under study:
where
and
are arbitrary real constants.
4)
:
The common invariant solution for the equations under study:
5)
:
The common invariant solution for the equations under study:
6)
:
The common invariant solution for the equations under study:
7)
:
The common invariant solution for the equations under study:
where
and
are arbitrary real constants.
8)
:
The common invariant solution for the equations under study:
,
where
and
are arbitrary real constants.
9)
:
The common invariant solution for the equations under study:
where:
are arbitrary real constants, f is an arbitrary smooth function.
3.2. Classification of the Common Invariant Solutions for the Equations under Study Using Two-Dimensional Nonconjugate Subalgebras of the Lie Algebra of the Group
3.2.1. Lie Algebras of the Type 2A1
1)
:
The common invariant solution for the equations under study:
where
are arbitrary real constants.
2)
:
The common invariant solution for the equations under study:
where
are arbitrary real constants.
3)
:
The common invariant solution for the equations under study:
where
are arbitrary real constants.
4)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
5)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
6)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
7)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
8)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
9)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
10)
:
The common invariant solution for the equations under study:
11)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
12)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
13)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
14)
:
The common invariant solution for the equations under study:
15)
:
The common invariant solution for the equations under study:
16)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
17)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
18)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
19)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
20)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
21)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
22)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
23)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
24)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
25)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
26)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
3.2.2. Lie Algebras of the Type A2
1)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
2)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
3)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
4)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
5)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
3.3. Classification of the Common Invariant Solutions for the Equations under Study Using Three-Dimensional Nonconjugate Subalgebras of the Lie Algebra of the Group
3.3.1. Lie Algebras of the Type 3A1
1)
:
The common invariant solution for the equations under study:
2)
:
The common invariant solution for the equations under study:
3)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
4)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
5)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
6)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
7)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
8)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
9)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
10)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
11)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
12)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
13)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
14)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
15)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
16)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3.3.2. Lie Algebras of the Type
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
2)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
4)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
5)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
6)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3.3.3. Lie Algebras of the Type A3,1
1)
:
The common invariant solution for the equations under study:
2)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
4)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
5)
:
The common invariant solution for the equations under study:
where f is an arbitrary smooth function.
3.3.4. Lie Algebras of the Type A3,2
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3.3.5. Lie Algebras of the Type A3,3
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
2)
:
The common invariant solution for the equations under study:
3)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3.3.6. Lie Algebras of the Type A3,6
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
2)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
4)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
5)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
6)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
7)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
8)
:
The common invariant solution for the equations under study:
3.3.7. Lie Algebras of the Type
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
2)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
3.3.8. Lie Algebras of the Type A3,8
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
3.3.9. Lie Algebras of the Type A3,9
1)
:
The common invariant solution for the equations under study:
where c is an arbitrary constant.
2)
:
The common invariant solution for the equations under study:
where
are arbitrary constants.
4. Conclusions
In this paper, we have presented obtained common invariant solutions of the following (1 + 3)-dimensional equations: the Eikonal equations, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation and the inhomogeneous Monge-Ampère equation. We have used the structural properties of the low-dimensional (
) nonconjugate subalgebras of the same ranks of the Lie algebra of the Poincaré group
for classification of the obtained common invariant solutions.
Since the group
contains, as subgroups, the extended Galilei group
[49] (the symmetry group of classical physics) and the Poincaré group
(the symmetry group of relativistic physics), the results obtained can be useful in construction and investigation of corresponding physical models.