We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional ( dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), 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.
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 [
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 [
We consider the following (1 + 3)-dimensional P ( 1,4 ) -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. [
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 P ( 1,4 ) -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 P ( 1,4 ) -invariant Eikonal equation obtained on the base of low-dimensional ( d i m L ≤ 3 ) nonconjugate subalgebras of the Lie algebra of the Poincaré group P ( 1,4 ) , by using classical Lie-Ovsiannikov approach [
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 [
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 ( d i m L ≤ 3 ) 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 ( d i m L ≤ 3 ) nonconjugate subalgebras of the Lie algebra of the Poincaré group P ( 1,4 ) , 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 P ( 1,4 ) and its nonconjugate subalgebras.
The group P ( 1,4 ) is a group of rotations and translations of the five-dimensional Minkowski space M ( 1,4 ) . It is the smallest group, which contains, as subgroups, the extended Galilei group G ˜ ( 1,3 ) [
The Lie algebra of the group P ( 1,4 ) is generated by 15 bases elements M μ ν = − M ν μ ( μ , ν = 0 , 1 , 2 , 3 , 4 ) and P μ ( μ = 0 , 1 , 2 , 3 , 4 ) , which satisfy the commutation relations
[ P μ , P ν ] = 0 , [ M μ ν , P σ ] = g ν σ P μ − g μ σ P ν , (1)
[ M μ ν , M ρ σ ] = g μ σ M ν ρ + g ν ρ M μ σ − g μ ρ M ν σ − g ν σ M μ ρ , (2)
where g 00 = − g 11 = − g 22 = − g 33 = − g 44 = 1 , g μ ν = 0 , if μ ≠ ν .
In this paper, we consider the following representation [
P 0 = ∂ ∂ x 0 , P 1 = − ∂ ∂ x 1 , P 2 = − ∂ ∂ x 2 , P 3 = − ∂ ∂ x 3 , (3)
P 4 = − ∂ ∂ u , M μ ν = x μ P ν − x ν P μ , x 4 ≡ u . (4)
In the following, we will use the next bases elements:
G = M 04 , L 1 = M 23 , L 2 = − M 13 , L 3 = M 12 , (5)
P a = M a 4 − M 0 a , C a = M a 4 + M 0 a , ( a = 1 , 2 , 3 ) , (6)
X 0 = 1 2 ( P 0 − P 4 ) , X k = P k ( k = 1 , 2 , 3 ) , X 4 = 1 2 ( P 0 + P 4 ) . (7)
The Lie algebra of the extended Galilei group G ˜ ( 1,3 ) is generated by the following bases elements:
L 1 , L 2 , L 3 , P 1 , P 2 , P 3 , X 0 , X 1 , X 2 , X 3 , X 4 . (8)
The classification of all nonconjugate subalgebras of the Lie algebra of the group P ( 1,4 ) of dimensions ≤ 3 was performed in [
In this Section, We Consider the Following PDEs
• the Eikonal equation
u 0 2 − u 1 2 − u 2 2 − u 3 2 = 1 ;
• the Euler-Lagrange-Born-Infeld equation
□ u ( 1 − u ν u ν ) + u μ u ν u μ ν = 0 ;
• the homogeneous Monge-Ampère equation
det ( u μ ν ) = 0 ;
• the inhomogeneous Monge-Ampère equation
det ( u μ ν ) = λ ( 1 − u ν u ν ) 3 , λ ≠ 0,
where u = u ( x ) , x = ( x 0 , x 1 , x 2 , x 3 ) ∈ M ( 1 , 3 ) , u μ ≡ ∂ u ∂ x μ , u μ ν ≡ ∂ 2 u ∂ x μ ∂ x ν , u μ = g μ ν u ν , g μ ν = ( 1, − 1, − 1, − 1 ) δ μ ν , μ , ν = 0,1,2,3 , □ is the d’Alembert operator.
Here, and in what follows, M ( 1,3 ) is a four-dimensional Minkowski space, R ( u ) 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. [
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 P ( 1,4 ) , structural properties of its low-dimensional ( d i m L ≤ 3 ) 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 [
Bellow we present the results obtained.
1) 〈 G 〉 :
The common invariant solution for the equations under study:
( x 0 2 − u 2 ) 1 / 2 = − ( 1 − c 2 2 − c 3 2 ) 1 / 2 x 1 + c 2 x 2 + c 3 x 3 + c 1 ,
where c 1 , c 2 and c 3 are arbitrary real constants.
2) 〈 G + α X 1 , α > 0 〉 :
The common invariant solution for the equations under study:
α ln ( 2 α ( ( c 1 2 + c 2 2 + 1 ) ( x 0 2 − u 2 ) + α 2 + α ) x 0 − u ) − ( c 1 2 + c 2 2 + 1 ) ( x 0 2 − u 2 ) + α 2 − x 1 + c 1 x 2 + c 2 x 3 + c 3 ,
where c 1 , c 2 and c 3 are arbitrary real constants.
3) 〈 L 3 〉 :
The common invariant solution for the equations under study:
u = ( c 2 2 + c 3 2 + 1 ) 1 / 2 x 0 + c 2 x 3 + c 3 ( x 1 2 + x 2 2 ) 1 / 2 + c 1 ,
where c 1 , c 2 and c 3 are arbitrary real constants.
4) 〈 L 3 + α ( X 0 + X 4 ) , α > 0 〉 :
The common invariant solution for the equations under study:
u = i α c 2 arctanh c 2 α ( ( c 1 2 − c 2 2 + 1 ) ( x 1 2 + x 2 2 ) + c 2 2 α 2 ) 1 / 2 − i ( ( c 1 2 − c 2 2 + 1 ) ( x 1 2 + x 2 2 ) + c 2 2 α 2 ) 1 / 2 + c 2 ( x 0 − α arctan x 1 x 2 ) + c 1 x 3 + c 3 .
5) 〈 L 3 + α X 3 , α > 0 〉 :
The common invariant solution for the equations under study:
u = ( c 1 2 − c 2 2 − 1 ) ( x 1 2 + x 2 2 ) − α 2 c 2 2 + c 2 α arctan x 1 x 2 − c 2 α arctan ( ( c 1 2 − c 2 2 − 1 ) ( x 1 2 + x 2 2 ) − α 2 c 2 2 c 2 α ) + c 2 x 3 + c 1 x 0 + c 3 .
6) 〈 L 3 + 2 X 4 〉 :
The common invariant solution for the equations under study:
x 0 − u + 2 arctan x 2 x 1 = i ( c 2 2 + 4 c 1 ) ( x 1 2 + x 2 2 ) + 4 − 2 i arctanh ( 2 ( c 2 2 + 4 c 1 ) ( x 1 2 + x 2 2 ) + 4 ) + c 1 ( x 0 + u ) + c 2 x 3 + c 3 .
7) 〈 P 3 − 2 X 0 〉 :
The common invariant solution for the equations under study:
1 6 ( x 0 + u ) 3 + x 3 ( x 0 + u ) + x 0 − u = − 1 6 ( ( x 0 + u ) 2 + 4 x 3 − c 1 2 − c 2 2 ) 3 / 2 + c 1 x 1 + c 2 x 2 + c 3 ,
where c 1 , c 2 and c 3 are arbitrary real constants.
8) 〈 X 0 + X 4 〉 :
The common invariant solution for the equations under study:
u = i ( c 2 2 + c 3 2 + 1 ) 1 / 2 x 1 + c 2 x 2 + c 3 x 3 + c 1 ,
where c 1 , c 2 and c 3 are arbitrary real constants.
9) 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 3 = − i ( c 2 2 + 1 ) 1 / 2 x 1 + c 2 x 2 + c 1 + f ( x 0 + u ) ,
where: c 1 , c 2 are arbitrary real constants, f is an arbitrary smooth function.
1) 〈 G 〉 ⊕ 〈 L 3 〉 :
The common invariant solution for the equations under study:
( x 0 2 − u 2 ) 1 / 2 = ( 1 − c 2 2 ) 1 / 2 x 3 + c 2 ( x 1 2 + x 2 2 ) 1 / 2 + c 1 ,
where c 1 , c 2 are arbitrary real constants.
2) 〈 G + α X 3 , α > 0 〉 ⊕ 〈 L 3 〉 :
The common invariant solution for the equations under study:
x 3 − α ln ( x 0 + u ) = − ( c 1 2 + 1 ) ( x 0 2 − u 2 ) + α 2 + α ln ( 2 α ( ( c 1 2 + 1 ) ( x 0 2 − u 2 ) + α 2 + α ) x 0 2 − u 2 ) + c 1 x 1 2 + x 2 2 + c 2 ,
where c 1 , c 2 are arbitrary real constants.
3) 〈 G 〉 ⊕ 〈 L 3 + α X 3 , α > 0 〉 :
The common invariant solution for the equations under study:
x 3 + α arctan x 1 x 2 = α arctan α ( c 2 2 − 1 ) ( x 1 2 + x 2 2 ) − α 2 + c 2 ( x 0 2 − u 2 ) 1 / 2 + ( c 2 2 − 1 ) ( x 1 2 + x 2 2 ) − α 2 + c 1 ,
where c 1 , c 2 are arbitrary real constants.
4) 〈 G 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
( x 0 2 − u 2 ) 1 / 2 = ε ( 1 − c 2 2 ) 1 / 2 x 2 + c 2 x 3 + c 1 , ε = ± 1 ,
where c 1 , c 2 are arbitrary constants.
5) 〈 G + α X 2 , α > 0 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
x 3 + ( c 2 2 + 1 ) ( x 0 2 − u 2 ) + α 2 c 2 2 = α c 2 arctanh α c 2 ( c 2 2 + 1 ) ( x 0 2 − u 2 ) + α 2 c 2 2 + α c 2 2 ln x 0 − u x 0 + u + c 2 x 2 + c 1 ,
where c 1 , c 2 are arbitrary constants.
6) 〈 L 3 〉 ⊕ 〈 P 3 + C 3 〉 :
The common invariant solution for the equations under study:
( u 2 + x 3 2 ) 1 / 2 = ( c 2 2 + 1 ) 1 / 2 x 0 + c 2 ( x 1 2 + x 2 2 ) 1 / 2 + c 1 ,
where c 1 , c 2 are arbitrary constants.
7) 〈 L 3 + α ( X 0 + X 4 ) , α > 0 〉 ⊕ 〈 P 3 + C 3 〉 :
The common invariant solution for the equations under study:
x 0 − α arctan x 1 x 2 = α arctan α ( 1 − c 2 2 ) ( x 1 2 + x 2 2 ) − α 2 + c 2 u 2 + x 3 2 + ( 1 − c 2 2 ) ( x 1 2 + x 2 2 ) − α 2 + c 1 ,
where c 1 , c 2 are arbitrary constants.
8) 〈 L 3 〉 ⊕ 〈 X 0 + X 4 〉 :
The common invariant solution for the equations under study:
u = i ε ( c 2 2 + 1 ) 1 / 2 ( x 1 2 + x 2 2 ) 1 / 2 + c 2 x 3 + c 1 , ε = ± 1 ,
where c 1 , c 2 are arbitrary constants.
9) 〈 L 3 + α ( X 0 + X 4 ) , α > 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
u = α arctan x 1 x 2 + i c 1 2 ( x 1 2 + x 2 2 ) + α 2 − i α arctanh α c 1 2 ( x 1 2 + x 2 2 ) + α 2 − x 0 + c 1 x 3 + c 2 ,
where c 1 , c 2 are arbitrary constants.
10) 〈 L 3 + α X 3 , α > 0 〉 ⊕ 〈 X 0 + X 4 〉 :
The common invariant solution for the equations under study:
u = α c 1 arctan ( x 1 ( c 1 2 + 1 ) ( x 1 2 + x 2 2 ) + α 2 − i α x 2 x 2 ( c 1 2 + 1 ) ( x 1 2 + x 2 2 ) + α 2 + i α x 1 ) + i c 1 ( c 1 2 + 1 ) ( x 1 2 + x 2 2 ) + α 2 + x 3 c 1 + c 2 , c 1 ≠ 0.
11) 〈 L 3 + 2 X 4 〉 ⊕ 〈 X 3 〉 :
The common invariant solution for the equations under study:
x 0 − u + 2 arctan x 2 x 1
= 2 i arctanh 1 c 1 ( x 1 2 + x 2 2 ) + 1 − 2 i c 1 ( x 1 2 + x 2 2 ) + 1 + c 1 ( x 0 + u ) + c 2 ,
where c 1 , c 2 are arbitrary constants.
12) 〈 L 3 − P 3 + 2 α X 0 , α ≠ 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 0 + u − 2 α arctan x 1 x 2 = 2 i α ε 4 c 2 2 ( x 1 2 + x 2 2 ) + 1 − 2 i α ε arctanh 1 4 c 2 2 ( x 1 2 + x 2 2 ) + 1 + c 2 ( ( x 0 + u ) 2 + 4 α x 3 ) + c 1 , ε = ± 1,
where c 1 , c 2 are arbitrary constants.
13) 〈 L 3 〉 ⊕ 〈 P 3 − 2 X 0 〉 :
The common invariant solution for the equations under study:
1 6 ( x 0 + u ) 3 + x 3 ( x 0 + u ) + x 0 − u = c 1 x 1 2 + x 2 2 − 1 6 ( ( x 0 + u ) 2 + 4 x 3 − c 1 2 ) 3 / 2 + c 2 ,
where c 1 , c 2 are arbitrary constants.
14) 〈 P 1 〉 ⊕ 〈 P 2 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − u 2 = 0.
15) 〈 P 1 − X 3 〉 ⊕ 〈 P 2 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − u 2 = 0.
16) 〈 P 1 − X 3 〉 ⊕ 〈 P 2 − γ X 2 − β X 3 , β > 0 , γ > 0 〉 :
The common invariant solution for the equations under study:
x 1 ( x 1 − 2 c 2 ) x 0 + u + ( x 2 − β c 2 ) 2 + c 2 2 x 0 + u + γ + γ c 2 2 ( x 0 + u ) ( x 0 + u + γ ) − ( c 2 2 + 1 ) ( x 0 + u ) + 2 c 2 x 3 + 2 u + c 1 = 0,
where c 1 , c 2 are arbitrary constants.
17) 〈 P 1 − X 3 〉 ⊕ 〈 P 2 − γ X 2 , γ > 0 〉 :
The common invariant solution for the equations under study:
( x 1 − c 2 ) 2 x 0 + u + x 2 2 x 0 + u + γ + 2 u = ( c 2 2 + 1 ) ( x 0 + u ) − 2 c 2 x 3 + c 1 ,
where c 1 , c 2 are arbitrary constants.
18) 〈 P 1 〉 ⊕ 〈 P 2 − X 2 − β X 3 , β > 0 〉 :
The common invariant solution for the equations under study:
x 1 2 x 0 + u + 2 u = ( c 2 2 4 + 1 ) ( x 0 + u ) − ( β c 2 + 2 x 2 ) 2 4 ( x 0 + u + 1 ) + c 2 x 3 + c 1 ,
where c 1 , c 2 are arbitrary constants.
19) 〈 P 1 〉 ⊕ 〈 P 2 − X 2 〉 :
The common invariant solution for the equations under study:
x 1 2 x 0 + u + x 2 2 x 0 + u + 1 + 2 u = ( c 2 2 4 + 1 ) ( x 0 + u ) + c 2 x 3 + c 1 ,
where c 1 , c 2 are arbitrary constants.
20) 〈 P 3 − 2 X 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
u = ± c 2 x 2 − 4 x 3 − i c 2 2 + 16 x 1 + c 1 − x 0 ,
where c 1 , c 2 are arbitrary constants.
21) 〈 P 3 − 2 X 0 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
1 6 ( x 0 + u ) 3 + x 3 ( x 0 + u ) + x 0 − u = ε c 1 x 2 − ε 6 ( ( x 0 + u ) 2 + 4 x 3 − c 1 2 ) 3 / 2 + c 2 ,
where c 1 , c 2 are arbitrary constants.
22) 〈 L 3 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
( x 1 2 + x 2 2 ) 1 / 2 = i ε x 3 + f ( x 0 + u ) , ε = ± 1 ,
where f is an arbitrary smooth function.
23) 〈 L 3 + α X 3 , α > 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 3 + α arctan x 1 x 2 = i ε x 1 2 + x 2 2 + α 2 − i ε α arctanh α x 1 2 + x 2 2 + α 2 + f ( x 0 + u ) , ε = ± 1,
where f is an arbitrary smooth function.
24) 〈 P 3 − X 1 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 1 − x 3 x 0 + u = i ε x 2 1 ( x 0 + u ) 2 + 1 + f ( x 0 + u ) , ε = ± 1 ,
where f is an arbitrary smooth function.
25) 〈 P 3 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 1 = i ε x 2 + f ( x 0 + u ) , ε = ± 1 ,
where f is an arbitrary smooth function.
26) 〈 X 1 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 3 = i ε x 2 + f ( x 0 + u ) , ε = ± 1 ,
where f is an arbitrary smooth function.
1) 〈 − G , P 3 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 3 2 − u 2 ) 1 / 2 = ε 1 − c 2 2 x 1 + c 2 x 2 + c 1 , ε = ± 1 ,
where c 1 , c 2 are arbitrary constants.
2) 〈 − G − 1 λ L 3 , X 4 , λ > 0 〉 :
The common invariant solution for the equations under study:
ln ( x 0 + u ) = i λ arctanh λ c 1 2 ( x 1 2 + x 2 2 ) + λ 2 − i c 1 2 ( x 1 2 + x 2 2 ) + λ 2 − λ arctan x 1 x 2 + c 1 x 3 + c 2 ,
where c 1 , c 2 are arbitrary constants.
3) 〈 − G − α X 1 , X 4 , α > 0 〉 :
The common invariant solution for the equations under study:
x 1 − α ln ( x 0 + u ) = i ε ( c 2 2 + 1 ) 1 / 2 x 2 + c 2 x 3 + c 1 , ε = ± 1,
where c 1 , c 2 are arbitrary constants.
4) 〈 − 1 λ ( L 3 + λ G + α X 3 ) , X 4 , α > 0 , λ > 0 〉 :
The common invariant solution for the equations under study:
ln ( x 0 + u ) = i ε c 2 2 ( x 1 2 + x 2 2 ) + ( α c 2 − λ ) 2 − i ε ( α c 2 − λ ) arctanh α c 2 − λ c 2 2 ( x 1 2 + x 2 2 ) + ( α c 2 − λ ) 2 + ( α c 2 − λ ) arctan x 1 x 2 + c 2 x 3 + c 1 , ε = ± 1,
where c 1 , c 2 are arbitrary constants.
5) 〈 − G − α X 1 , P 3 , α > 0 〉 :
The common invariant solution for the equations under study:
x 1 − α ln ( x 0 + u ) = α ln ( 2 α ( c 1 2 + 1 ) ( x 0 2 − x 3 2 − u 2 ) + α 2 + α x 0 2 − x 3 2 − u 2 ) − ( c 1 2 + 1 ) ( x 0 2 − x 3 2 − u 2 ) + α 2 + c 1 x 2 + c 2 ,
where c 1 , c 2 are arbitrary constants.
1) 〈 P 1 − γ X 3 , γ > 0 〉 ⊕ 〈 P 2 − X 2 − δ X 3 , δ ≠ 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 4 + 2 ( x 0 + u ) 3 + ( γ 2 + δ 2 + 1 ) ( x 0 + u ) 2 + 2 γ 2 ( x 0 + u ) + γ 2 = 0.
2) 〈 P 1 〉 ⊕ 〈 P 2 − X 2 − δ X 3 , δ > 0 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 2 + 2 ( x 0 + u ) + δ 2 + 1 = 0.
3) 〈 P 1 〉 ⊕ 〈 P 2 〉 ⊕ 〈 X 3 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − u 2 = c ( x 0 + u ) ,
where c is an arbitrary constant.
4) 〈 P 3 〉 ⊕ 〈 X 1 〉 ⊕ 〈 X 2 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 3 2 − u 2 = c ( x 0 + u ) ,
where c is an arbitrary constant.
5) 〈 P 1 〉 ⊕ 〈 P 2 〉 ⊕ 〈 P 3 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − x 3 2 − u 2 = c ( x 0 + u ) ,
where c is an arbitrary constant.
6) 〈 P 1 〉 ⊕ 〈 P 2 − X 2 〉 ⊕ 〈 X 3 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − u 2 x 0 + u − x 2 2 x 0 + u + 1 = c ,
where c is an arbitrary constant.
7) 〈 P 1 〉 ⊕ 〈 P 2 − α X 2 , α > 0 〉 ⊕ 〈 P 3 − γ X 3 , γ ≠ 0 〉 :
The common invariant solution for the equations under study:
2 u + x 1 2 x 0 + u + x 2 2 x 0 + u + α + x 3 2 x 0 + u + γ = x 0 + u + c ,
where c is an arbitrary constant.
8) 〈 P 1 〉 ⊕ 〈 P 2 − α X 2 , α > 0 〉 ⊕ 〈 P 3 〉 :
The common invariant solution for the equations under study:
2 u + x 1 2 + x 3 2 x 0 + u + x 2 2 x 0 + u + α = x 0 + u + c ,
where c is an arbitrary constant.
9) 〈 G 〉 ⊕ 〈 X 2 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
( x 0 2 − u 2 ) 1 / 2 = ε x 3 + c , ε = ± 1,
where c is an arbitrary constant.
10) 〈 G 〉 ⊕ 〈 L 3 〉 ⊕ 〈 X 3 〉 :
The common invariant solution for the equations under study:
( x 0 2 − u 2 ) 1 / 2 = ε ( x 1 2 + x 2 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
11) 〈 P 3 − 2 X 0 〉 ⊕ 〈 X 1 〉 ⊕ 〈 X 2 〉 :
The common invariant solution for the equations under study:
1 6 ( x 0 + u ) 3 + x 3 ( x 0 + u ) + x 0 − u = ε 6 ( ( x 0 + u ) 2 + 4 x 3 ) 3 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
12) 〈 G + α X 3 , α > 0 〉 ⊕ 〈 X 1 〉 ⊕ 〈 X 2 〉 :
The common invariant solution for the equations under study:
x 3 − α ln ( x 0 + u ) = ε ( α 2 + x 0 2 − u 2 ) 1 / 2 − α 2 ln ( x 0 2 − u 2 ) − ε α arctanh ( α 2 + x 0 2 − u 2 ) 1 / 2 α + c , ε = ± 1,
where c is an arbitrary constant.
13) 〈 L 3 〉 ⊕ 〈 P 3 + C 3 〉 ⊕ 〈 X 0 + X 4 〉 :
The common invariant solution for the equations under study:
( x 3 2 + u 2 ) 1 / 2 = i ε ( x 1 2 + x 2 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
14) 〈 L 3 + α ( X 0 + X 4 ) , α > 0 〉 ⊕ 〈 X 3 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
x 0 + u + α arctan x 2 x 1 = i ε α 2 ln ( x 1 2 + x 2 2 ) + c , ε = ± 1,
where c is an arbitrary constant.
15) 〈 P 3 − 2 X 0 〉 ⊕ 〈 X 1 〉 ⊕ 〈 X 4 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 2 + 4 x 3 = 4 i ε x 2 + c , ε = ± 1,
where c is an arbitrary constant.
16) 〈 L 3 〉 ⊕ 〈 − P 3 + 2 X 0 〉 ⊕ 〈 2 X 4 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 2 + 4 x 3 = 4 i ε ( x 1 2 + x 2 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
1) 〈 − G , P 3 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 3 2 − u 2 ) 1 / 2 = ε x 2 + c , ε = ± 1,
where c is an arbitrary constant.
2) 〈 − G , P 3 〉 ⊕ 〈 L 3 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 3 2 − u 2 ) 1 / 2 = ε ( x 1 2 + x 2 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
3) 〈 − ( G + α X 2 ) , P 3 , α > 0 〉 ⊕ 〈 X 1 〉 :
The common invariant solution for the equations under study:
x 2 − α ln ( x 0 + u ) = ε ( x 0 2 − x 3 2 − u 2 + α 2 ) 1 / 2 − α 2 ln ( x 0 2 − x 3 2 − u 2 ) − ε α arctanh α ( x 0 2 − x 3 2 − u 2 + α 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
4) 〈 − 1 λ L 3 − G , 2 X 4 , λ > 0 〉 ⊕ 〈 X 3 〉 :
The common invariant solution for the equations under study:
ln ( x 0 + u ) + λ arctan x 1 x 2 = i ε λ 2 ln ( x 1 2 + x 2 2 ) + c , ε = ± 1,
where c is an arbitrary constant.
5) 〈 − ( G + α X 3 ) , X 4 , α > 0 〉 ⊕ 〈 L 3 + β X 3 , β > 0 〉 :
The common invariant solution for the equations under study:
x 3 − α ln ( x 0 + u ) + β arctan x 1 x 2 = − i ε β arctanh β ( x 1 2 + x 2 2 + β 2 ) 1 / 2 + i ε ( x 1 2 + x 2 2 + β 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
6) 〈 − ( G + α X 3 ) , X 4 , α > 0 〉 ⊕ 〈 L 3 〉 :
The common invariant solution for the equations under study:
x 3 − α ln ( x 0 + u ) = i ε ( x 1 2 + x 2 2 ) 1 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
1) 〈 4 X 4 , P 1 − X 2 − γ X 3 , P 2 + X 1 − μ X 2 − δ X 3 , γ > 0 , δ ≠ 0 , μ > 0 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 4 + 2 μ ( x 0 + u ) 3 + ( γ 2 + μ 2 + δ 2 + 2 ) ( x 0 + u ) 2 + 2 μ ( γ 2 + 1 ) ( x 0 + u ) + ( γ μ − δ ) 2 + γ 2 + 1 = 0.
2) 〈 2 μ X 4 , P 3 − 2 X 0 , X 1 + μ X 3 , μ > 0 〉 :
The common invariant solution for the equations under study:
u = 2 ( i ε x 2 μ 2 + 1 + μ x 1 − x 3 + c ) 1 / 2 − x 0 , ε = ± 1,
where c is an arbitrary constant.
3) 〈 2 X 4 , P 3 − L 3 − 2 α X 0 , X 3 , α > 0 〉 :
The common invariant solution for the equations under study:
u = 2 α arctan x 1 x 2 + i ε α ln ( x 1 2 + x 2 2 ) − x 0 + c , ε = ± 1,
where c is an arbitrary constant.
4) 〈 − 2 β X 4 , L 3 + β X 3 , P 3 − 2 X 0 , β > 0 〉 :
The common invariant solution for the equations under study:
β arctan x 1 x 2 + 1 4 ( x 0 + u ) 2 = i ε x 1 2 + x 2 2 + β 2 − i ε β arctanh β x 1 2 + x 2 2 + β 2 − x 3 + c , ε = ± 1,
where c is an arbitrary constant.
5) 〈 2 X 4 , P 3 , X 3 〉 :
The common invariant solution for the equations under study:
x 2 = i ε x 1 + f ( x 0 + u ) , ε = ± 1,
where f is an arbitrary smooth function.
1) 〈 2 α X 4 , λ P 3 , 1 λ L 3 + G + α λ X 3 , α > 0 , λ > 0 〉 :
The common invariant solution for the equations under study:
ln ( x 0 + u ) + λ arctan x 1 x 2 = i ε λ 2 ln ( x 1 2 + x 2 2 ) + c , ε = ± 1,
where c is an arbitrary constant.
1) 〈 P 1 , P 2 , G 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 1 2 − x 2 2 − u 2 ) 1 / 2 = ε x 3 + c , ε = ± 1,
where c is an arbitrary constant.
2) 〈 P 1 , P 2 , G + α X 3 , α > 0 〉 :
The common invariant solution for the equations under study:
x 3 − α ln ( x 0 + u ) = ε ( x 0 2 − x 1 2 − x 2 2 − u 2 + α 2 ) 1 / 2 − i ε α arctan ( x 0 2 − x 1 2 − x 2 2 − u 2 + α 2 ) 1 / 2 i α − α 2 ln ( x 0 2 − x 1 2 − x 2 2 − u 2 ) + c , ε = ± 1.
3) 〈 P 3 , X 4 , 1 λ L 3 + G , λ > 0 〉 :
The common invariant solution for the equations under study:
ln ( x 0 + u ) + λ arctan x 1 x 2 = i ε λ 2 ln ( x 1 2 + x 2 2 ) + c , ε = ± 1,
where c is an arbitrary constant.
1) 〈 P 1 − X 1 , P 2 − X 2 , − P 3 + L 3 〉 :
The common invariant solution for the equations under study:
x 1 2 + x 2 2 x 0 + u + 1 + x 3 2 x 0 + u + 2 u = x 0 + u + c ,
where c is an arbitrary constant.
2) 〈 P 1 , − P 2 , − ( L 3 + α X 3 ) , α > 0 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − u 2 = c ( x 0 + u ) ,
where c is an arbitrary constant.
3) 〈 P 1 , P 2 , − P 3 + L 3 〉 :
The common invariant solution for the equations under study:
x 0 2 − x 1 2 − x 2 2 − x 3 2 − u 2 = c ( x 0 + u ) ,
where c is an arbitrary constant.
4) 〈 X 1 , − X 2 , P 3 − L 3 − 2 α X 0 , α > 0 〉 :
The common invariant solution for the equations under study:
( x 0 + u ) 3 + 6 α x 3 ( x 0 + u ) + 6 α 2 ( x 0 − u ) = ε ( ( x 0 + u ) 2 + 4 α x 3 ) 3 / 2 + c , ε = ± 1,
where c is an arbitrary constant.
5) 〈 X 1 , − X 2 , − L 3 − 1 2 ( P 3 + C 3 ) − α ( X 0 + X 4 ) , α > 0 〉 :
The common invariant solution for the equations under study:
α arctan x 3 u − x 0 = ε x 3 2 + u 2 − α 2 + ε α arctan α x 3 2 + u 2 − α 2 + c , ε = ± 1,
where c is an arbitrary constant.
6) 〈 X 1 , X 2 , L 3 + λ 2 ( P 3 + C 3 ) + α ( X 0 + X 4 ) , α > 0 , 0 < λ < 1 〉 :
The common invariant solution for the equations under study:
α arctan x 3 u − λ x 0 = ε λ 2 ( x 3 2 + u 2 ) − α 2 + ε α arctan α λ 2 ( x 3 2 + u 2 ) − α 2 + c , ε = ± 1,
where c is an arbitrary constant.
7) 〈 X 1 , X 2 , L 3 + λ G + α X 3 , α > 0 , λ > 0 〉 :
The common invariant solution for the equations under study:
λ x 3 − α ln ( x 0 + u ) = ε λ 2 ( x 0 2 − u 2 ) + α 2 − ε α arctanh α λ 2 ( x 0 2 − u 2 ) + α 2 − α 2 ln ( x 0 2 − u 2 ) + c , ε = ± 1.
where c is an arbitrary constant.
8) 〈 P 1 , P 2 , L 3 〉 :
The common invariant solution for the equations under study:
x 3 = c 1 ln ( x 0 + u ) − ε ( x 0 2 − x 1 2 − x 2 2 − u 2 + c 1 2 ) 1 / 2 + ε c 1 arctanh x 0 2 − x 1 2 − x 2 2 − u 2 + c 1 2 c 1 − c 1 2 ln ( x 0 2 − x 1 2 − x 2 2 − u 2 ) + c 2 , ε = ± 1, c 1 ≠ 0.
1) 〈 P 1 , P 2 , L 3 + λ G , λ > 0 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 1 2 − x 2 2 − u 2 ) 1 / 2 = ε x 3 + c , ε = ± 1,
where c is an arbitrary constant.
2) 〈 P 1 , P 2 , L 3 + λ G + α X 3 , α > 0 , λ > 0 〉 :
The common invariant solution for the equations under study:
λ x 3 − α ln ( x 0 + u ) = ε ( λ 2 ( x 0 2 − x 1 2 − x 2 2 − u 2 ) + α 2 ) 1 / 2 − ε α arctanh λ 2 ( x 0 2 − x 1 2 − x 2 2 − u 2 ) + α 2 α − α 2 ln ( x 0 2 − x 1 2 − x 2 2 − u 2 ) + c , ε = ± 1,
where c is an arbitrary constant.
〈 P 3 , G , − C 3 〉 :
The common invariant solution for the equations under study:
( x 0 2 − x 3 2 − u 2 ) 1 / 2 = ε ( 1 − c 2 2 ) 1 / 2 x 1 + c 2 x 2 + c 1 , ε = ± 1,
where c 1 , c 2 are arbitrary constants.
1) 〈 − 1 2 ( L 3 + 1 2 ( P 3 + C 3 ) ) , 1 2 ( L 2 + 1 2 ( P 2 + C 2 ) ) , 1 2 ( L 1 + 1 2 ( P 1 + C 1 ) ) 〉 :
The common invariant solution for the equations under study:
( x 1 2 + x 2 2 + x 3 2 + u 2 ) 1 / 2 = ε x 0 + c , ε = ± 1,
where c is an arbitrary constant.
2) 〈 − L 3 , − L 2 , − L 1 〉 :
The common invariant solution for the equations under study:
u = ε ( c 2 2 + 1 ) 1 / 2 x 0 + c 2 ( x 1 2 + x 2 2 + x 3 2 ) 1 / 2 + c 1 , ε = ± 1,
where c 1 , c 2 are arbitrary constants.
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 ( d i m L ≤ 3 ) nonconjugate subalgebras of the same ranks of the Lie algebra of the Poincaré group P ( 1,4 ) for classification of the obtained common invariant solutions.
Since the group P ( 1,4 ) contains, as subgroups, the extended Galilei group G ˜ ( 1,3 ) [
The authors declare no conflicts of interest regarding the publication of this paper.
Fedorchuk, V.M. and Fedorchuk, V.I. (2023) On the Construction and Classification of the Common Invariant Solutions for Some-Invariant Partial Differential Equations. Applied Mathematics, 14, 728-747. https://doi.org/10.4236/am.2023.1411044