On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs
Connections between structure properties of three-dimensional subalgebras of the Poincare algebra ${\mathfrak p}(1,4)$ and Lie reductions of the Euler-Lagrange-Born-Infeld equation are studied. We concentrate our attention on Lie reductions with respect to three-dimensional subalgebras...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552874588766208 |
|---|---|
| author | Fedorchuk, V. M. Fedorchuk, V. I. Федорчук, В. М. Федорчук, В. І. |
| author_facet | Fedorchuk, V. M. Fedorchuk, V. I. Федорчук, В. М. Федорчук, В. І. |
| author_institution_txt_mv | [
{
"author": "V. M. Fedorchuk",
"institution": "Pedagogical University, Cracow; Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine"
},
{
"author": "V. I. Fedorchuk",
"institution": "Ya.S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine"
}
] |
| author_sort | Fedorchuk, V. M. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | Connections between structure properties of three-dimensional subalgebras of the Poincare algebra ${\mathfrak p}(1,4)$ and Lie reductions of the Euler-Lagrange-Born-Infeld equation are studied. We concentrate our attention on Lie reductions with respect to three-dimensional subalgebras that reduce the Euler-Lagrange-Born-Infeld equation to linear ordinary differential equations. |
| first_indexed | 2026-08-04T01:06:22Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 193�202
ÓÄÊ 512.813:517.957.6
On symmetry reduction of the
Euler�Lagrange�Born�Infeld equation
to linear ODEs
V.M. Fedorchuk †‡, V.I. Fedorchuk ‡
† Pedagogical University, Cracow, Poland
E-mail: vasyl.fedorchuk@up.krakow.pl
‡ Pidstryhach Institute for Applied Problems of Mechanics
and Mathematics, Lviv, Ukraine
E-mail: volfed@gmail.com
Âèâ÷à¹òüñÿ çâ'ÿçîê ìiæ ñòðóêòóðíèìè âëàñòèâîñòÿìè òðèâèìiðíèõ
ïiäàëãåáð àëãåáðè Ïóàíêàðå p(1, 4) i ñèìåòðiéíîþ ðåäóêöi¹þ ðiâíÿí-
íÿ Åéëåðà�Ëàãðàíæà�Áîðíà�Iíôåëüäà. Îñíîâíó óâàãó çîñåðåäæåíî íà
ðåäóêöÿõ çà òðèâèìiðíèìè ïiäàëãåáðàìè, ùî çâîäÿòü ðiâíÿííÿ Åéëå-
ðà�Ëàãðàíæà�Áîðíà�Iíôåëüäà äî ëiíiéíèõ äèôåðåíöiàëüíèõ ðiâíÿíü.
Connections between structure properties of three-dimensional subalge-
bras of the Poincar�e algebra p(1, 4) and Lie reductions of the Euler�
Lagrange�Born�Infeld equation are studied. We concentrate our attenti-
on on Lie reductions with respect to three-dimensional subalgebras that
reduce the Euler�Lagrange�Born�Infeld equation to linear ordinary di-
�erential equations.
1. Introduction. Symmetry reduction is the most universal tool
for finding exact solutions of partial differential equations (PDEs). We
focus our attention on some applications of the classical Lie method
to investigation of PDEs with non-trivial symmetry groups. In 1895,
Lie [19] considered solutions of PDEs that are invariant with respect to
symmetry groups admitted by these PDEs. It turned out that the prob-
lem of symmetry reduction and construction of independent invariant
solutions for a PDE with a non-trivial symmetry group is reduced to the
algebraic problem of classification of inequivalent subalgebras of the Lie
invariance algebra of this equation [23, 24].
194 V.M. Fedorchuk, V.I. Fedorchuk
In 1975, Patera, Winternitz, and Zassenhaus [25] proposed a gene-
ral method for describing inequivalent subalgebras of Lie algebras with
nontrivial ideals. It turned out that reduced equations obtained from
inequivalent subalgebras of the same dimension were of different types.
Grundland, Harnad and Winternitz [17] were the first who pointed out
and studied this phenomenon. Further details can be found in [6, 8, 11,
15, 16, 21, 22]. The results obtained cannot be explained using only the
dimension of subalgebras of Lie invariance algebras.
To explain a difference in properties of reduced equations for PDEs
with nontrivial symmetry groups, we investigate the relation between
structure properties of inequivalent subalgebras of the same dimension of
the Lie invariance algebras of those PDEs and properties of the respective
reduced equations. By now, we have studied this relation for the case
of low-dimensional (dimL ≤ 3) inequivalent subalgebras of the same
dimension of the algebra p(1, 4), which is the Lie algebra of the Poincaré
group P (1, 4), and the eikonal equation [8].
This paper is devoted to the study of the relation between structural
properties of low-dimensional (dimL ≤ 3) inequivalent subalgebras of
the same rank of the algebra p(1, 4) and properties of reduced equa-
tions for the Euler–Lagrange–Born–Infeld (ELBI) equation. By now,
this relation has been investigated for three-dimensional subalgebras.
We obtained the following types of reduced equations: identities, linear
ordinary differential equations, nonlinear ordinary differential equations,
partial differential equations. For some subalgebras, it is impossible to
construct ansatzes that reduce the ELBI equation.
We focus our attention on reduction of the ELBI equation to linear
ODEs. More precisely, we only present the results of symmetry reduction
for those types of subalgebras that provide us reductions to linear ODEs.
2. Lie algebra of the Poincaré group P (1, 4) and its nonequi-
valent subalgebras. The group P (1, 4) is the group of rotations and
translations of the five-dimensional Minkowski space M(1, 4). It is the
minimal group that contains, as subgroups, the extended Galilei group
G̃(1, 3) [12] and the Poincaré group P (1, 3), which are underlying groups
of classical and relativistic physics, respectively.
The Lie algebra p(1, 4) of the group P (1, 4) is spanned by 15 basis
elements Mµν = −Mνµ, µ, ν = 0, 1, 2, 3, 4, and Pµ, µ = 0, 1, 2, 3, 4, which
satisfy the commutation relations
[Pµ, Pν ] = 0, [Mµν , Pσ] = gνσPµ − gµσPν ,
On reductions of the Euler–Lagrange–Born–Infeld equation 195
[Mµν ,Mρσ] = gµσMνρ + gνρMµσ − gµρMνσ − gνσMµρ,
where g00 = −g11 = −g22 = −g33 = −g44 = 1, gµν = 0, if µ 6= ν.
We consider the canonical realization [13, 14] of p(1, 4),
P0 =
∂
∂x0
, P1 = − ∂
∂x1
, P2 = − ∂
∂x2
, P3 = − ∂
∂x3
,
P4 = − ∂
∂u
, Mµν = xµPν − xνPµ, x4 ≡ u.
Hereafter we use the following basis elements
G = M04, L1 = M23, L2 = −M13, L3 = M12,
Pa = Ma4 −M0a, Ca = Ma4 +M0a,
X0 = 1
2 (P0 − P4), Xk = Pk, X4 = 1
2 (P0 + P4), a, k = 1, 2, 3.
Subalgebras of the Lie algebra p(1, 4) were studied up to P (1, 4)-con-
jugation in [4, 5, 10], in particular, the classification of subalgebras of
p(1, 4) of dimensions up to three was given in [7]. Note that the Lie
algebra of the extended Galilei group G̃(1, 3) is spanned by L1, L2, L3,
P1, P2, P3, X0, X1, X2, X3 and X4.
3. Classification of symmetry reductions for the Euler–
Lagrange–Born–Infeld equation. Born–Infeld-like equations arise
in fluid dynamics, theory of continuous medium, general relativity, field
theory, theory of minimal surfaces, nonlinear electrodynamics, etc. [1, 2,
3, 18, 26].
We consider the Euler–Lagrange–Born–Infeld (ELBI) equation
2u (1− uνuν) + uµuνuµν = 0, (1)
where u = u(x), x = (x0, x1, x2, x3) ∈M(1, 3), uµ ≡ ∂u
∂xµ
, uµν ≡ ∂2u
∂xµ∂xν
,
uµ = gµνuν , µ, ν = 0, 1, 2, 3, and 2 is the d’Alembert operator.
In 1984, Fushchych and Serov [13] studied symmetry properties of
the multidimensional nonlinear Euler–Lagrange equation. These results
imply that the Lie invariance algebra of the equation (1) contains, as a
subalgebra, the Poincaré algebra p(1, 4).
We carry out Lie symmetry reductions of the ELBI equation to linear
ODEs using subalgebras of p(1, 4) of the following types: 3A1, A2 ⊕
A1, A3,1, A3,2, A3,3, A3,6. The notation of three-dimensional algebras
196 V.M. Fedorchuk, V.I. Fedorchuk
is according to Mubarakzyanov’s classification of low-dimensional Lie
algebras [20].
Among inequivalent subalgebras of the Poincaré algebra p(1, 4) listed
in [7], we select only such subalgebras that do reduce the ELBI equation
to linear ODEs with nonlinear solutions since linear solutions are con-
sidered to be trivial. For each of the selected subalgebras, we construct
an ansatz for u, the corresponding reduced equation, its general solution
and the associated family of invariant solutions of the ELBI equation.
Proposition 1. The Lie algebra p(1, 4) contains 31 three-dimensional
inequivalent subalgebras of the type 3A1.
1. 〈P1〉 ⊕ 〈P2〉 ⊕ 〈X3〉:
the ansatz is x2
0 − x2
1 − x2
2 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 6ωϕ′ + 6ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω
6 + c2ω;
the solution of the ELBI equation is
x2
0 − x2
1 − x2
2 − u2 = c1(x0 + u)6 + c2(x0 + u).
2. 〈P3〉 ⊕ 〈X1〉 ⊕ 〈X2〉:
the ansatz is x2
0 − x2
3 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 4ωϕ′ + 4ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c2ω
4 + c1ω;
the solution of the ELBI equation is
x2
0 − x2
3 − u2 = c2(x0 + u)4 + c1(x0 + u).
3. 〈P1〉 ⊕ 〈P2〉 ⊕ 〈P3〉:
the ansatz is x2
0 − x2
1 − x2
2 − x2
3 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 8ωϕ′ + 8ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω
8 + c2ω;
the solution of the ELBI equation is
x2
0 − x2
1 − x2
2 − x2
3 − u2 = c1(x0 + u)8 + c2(x0 + u).
4. 〈P1〉 ⊕ 〈P2 −X2〉 ⊕ 〈X3〉:
the ansatz is
x2
0−x
2
1−u
2
x0+u − x2
2
x0+u+1 = ϕ(ω), ω = x0 + u;
the reduced equation is (ω+1)5ω5 (ω(ω + 1)ϕ′′−2(2ω + 1)ϕ′) = 0;
the solution of the reduced equation is
On reductions of the Euler–Lagrange–Born–Infeld equation 197
ϕ(ω) = c2ω
3
(
6ω2 + 15ω + 10
)
+ c1;
the solution of the ELBI equation is
x2
0 − x2
1 − u2
x0 + u
− x2
2
x0 + u+ 1
= c2(x0 + u)3
(
6(x0 + u)2
+ 15(x0 + u) + 10
)
+ c1.
5. 〈P1〉 ⊕ 〈P2 − αX2, α > 0〉 ⊕ 〈P3 − γX3, γ 6= 0〉:
the ansatz is 2u+
x2
1
x0+u +
x2
2
x0+u+α +
x2
3
x0+u+γ = ϕ(ω), ω = x0 + u;
the reduced equation is
(ω+γ)5ω5(ω+α)5
[
ω
(
ω2 +(α+γ)ω+αγ
)
ϕ′′−2(3ω2 +2(α+γ)ω+
αγ)(ϕ′ − 1)
]
= 0;
the solution of the reduced equation is
ϕ(ω) = c1
[
1
7ω
4 + 1
3 (α + γ)ω3 + 1
5 (α2 + 4αγ + γ2)ω2 + 1
2αγ(α +
γ)ω + 1
3α
2γ2
]
ω3 + ω + c2;
the solution of the ELBI equation is
2u+
x2
1
x0 + u
+
x2
2
x0 + u+ α
+
x2
3
x0 + u+ γ
= c1
[
1
7 (x0 + u)4 + 1
3 (α+ γ)(x0 + u)3 + 1
5 (α2 + 4αγ + γ2)
× (x0 + u)2 + 1
2αγ(α+ γ)(x0 + u) + 1
3α
2γ2
]
×(x0 + u)3 + x0 + u+ c2.
6. 〈P1〉 ⊕ 〈P2 − αX2, α > 0〉 ⊕ 〈P3〉:
the ansatz is 2u+
x2
1+x2
3
x0+u +
x2
2
x0+u+α = ϕ(ω), ω = x0 + u;
the reduced equation is
(ω + α)5ω5(ω(ω + α)ϕ′′ − 2(3ω + 2α)(ϕ′ − 1)) = 0;
the solution of the reduced equation is
ϕ(ω) = c1
(
1
7ω
2 + α
3ω + α2
5
)
ω5 + ω + c2;
the solution of the ELBI equation is
2u+
x2
1 + x2
3
x0 + u
+
x2
2
x0 + u+ α
= c1
(
1
7 (x0 + u)2 + α
3 (x0 + u) + α2
5
)
(x0 + u)5
+ x0 + u+ c2.
7. 〈P3 − 2X0〉 ⊕ 〈X1〉 ⊕ 〈X2〉:
the ansatz is
198 V.M. Fedorchuk, V.I. Fedorchuk
1
6 (x0 + u)3 + x3(x0 + u) + x0 − u = ϕ(ω), ω = (x0 + u)2 + 4x3;
the reduced equation is 2ωϕ′′ − ϕ′ = 0;
the solution of the reduced equation is
ϕ(ω) = c2ω
3/2 + c1;
the solution of the ELBI equation is
1
6 (x0 + u)3 + x3(x0 + u) + x0 − u
= c2
(
(x0 + u)2 + 4x3
)3/2
+ c1.
8. 〈P3 − 2X0〉 ⊕ 〈X1〉 ⊕ 〈X4〉:
the ansatz is (x0 + u)2 + 4x3 = ϕ(ω), ω = x2;
the reduced equation is ϕ′′ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω + c2;
the solution of the ELBI equation is
u = ε(c1x2 − 4x3 + c2)1/2 − x0, ε = ±1.
Proposition 2. The Lie algebra p(1, 4) contains 10 three-dimensional
inequivalent subalgebras of the type A2 ⊕A1.
1. 〈−(G+ αX2), X4, α > 0〉 ⊕ 〈X1〉:
the ansatz is x2 − α ln(x0 + u) = ϕ(ω), ω = x3;
the reduced equation is ϕ′′ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω + c2;
the solution of the ELBI equation is
x2 − α ln(x0 + u) = c1x3 + c2.
Proposition 3. The Lie algebra p(1, 4) contains 17 three-dimensional
inequivalent subalgebras of the type A3,1.
1. 〈2µX4, P3 − 2X0, X1 + µX3, µ > 0〉:
the ansatz is (x0 + u)2 + 4x3 − 4µx1 = ϕ(ω), ω = x2;
the reduced equation is ϕ′′ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω + c2;
the solution of the ELBI equation is
u = ε (4µx1 + c1x2 − 4x3 + c2)
1/2 − x0, ε = ±1.
Proposition 4. The Lie algebra p(1, 4) contains 3 three-dimensional
nonconjugate subalgebras of the type A3,2.
On reductions of the Euler–Lagrange–Born–Infeld equation 199
1. 〈2βX4, P3, G+ αX1 + βX3, α > 0, β > 0〉:
the ansatz is x1 − α ln(x0 + u) = ϕ(ω), ω = x2;
the reduced equation is ϕ′′ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω + c2;
the solution of the ELBI equation is
x1 − α ln(x0 + u) = c1x2 + c2.
Proposition 5. The Lie algebra p(1, 4) contains five three-dimensional
inequivalent subalgebras of the type A3,3.
1. 〈P3, X4, G+ αX1, α > 0〉:
the ansatz is x1 − α ln(x0 + u) = ϕ(ω), ω = x2;
the reduced equation is ϕ′′ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω + c2;
the solution of the ELBI equation is
u = exp
(
x1 − c1x2 − c2
α
)
− x0.
Proposition 6. The Lie algebra p(1, 4) contains 18 three-dimensional
inequivalent subalgebras of the type A3,6.
1. 〈P1 −X1, P2 −X2,−P3 + L3〉:
the ansatz is
x2
1+x2
2
x0+u+1 +
x2
3
x0+u + 2u = ϕ(ω), ω = x0 + u;
the reduced equation is
ω5(ω + 1)5[ω(ω + 1)ϕ′′ − 2(3ω + 1)(ϕ′ − 1)] = 0;
the solution of the reduced equation is
ϕ(ω) = c1
7 ω
7 + 2
3c1ω
6 + 6
5c1ω
5 + c1ω
4 + c1
3 ω
3 + ω + c2;
the solution of the ELBI equation is
x2
1 + x2
2
x0 + u+ 1
+
x2
3
x0 + u
+ 2u = c1
7 (x0 + u)7 + 2
3c1(x0 + u)6
+ 6
5c1(x0 + u)5 + c1(x0 + u)4 + c1
3 (x0 + u)3 + x0 + u+ c2.
2. 〈P1,−P2,− (L3 + αX3) , α > 0〉:
the ansatz is x2
0 − x2
1 − x2
2 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 6ωϕ′ + 6ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω
6 + c2ω;
the solution of the ELBI equation is
x2
0 − x2
1 − x2
2 − u2 = c1(x0 + u)6 + c2(x0 + u).
200 V.M. Fedorchuk, V.I. Fedorchuk
3. 〈X1,−X2, P3 − L3〉:
the ansatz is x2
0 − x2
3 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 4ωϕ′ + 4ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω
4 + c2ω;
the solution of the ELBI equation is
x2
0 − x2
3 − u2 = c1(x0 + u)4 + c2(x0 + u).
4. 〈P1, P2, L3 − P3〉:
the ansatz is x2
0 − x2
1 − x2
2 − x2
3 − u2 = ϕ(ω), ω = x0 + u;
the reduced equation is ω2ϕ′′ − 8ωϕ′ + 8ϕ = 0;
the solution of the reduced equation is ϕ(ω) = c1ω
8 + c2ω;
the solution of the ELBI equation is
x2
0 − x2
1 − x2
2 − x2
3 − u2 = c1(x0 + u)8 + c2(x0 + u).
5. 〈X1,−X2, P3 − L3 − 2αX0, α > 0〉:
the ansatz is (x0 + u)3 + 6αx3(x0 + u) + 6α2(x0 − u) = ϕ(ω),
ω = (x0 + u)2 + 4αx3;
the reduced equation is 2ωϕ′′ − ϕ′ = 0;
the solution of the reduced equation is ϕ(ω) = c2ω
3/2 + c1;
the solution of the ELBI equation is
(x0 + u)3 + 6αx3(x0 + u) + 6α2(x0 − u)
= c2
(
(x0 + u)2 + 4αx3
)3/2
+ c1.
4. Conclusions. In this paper we focused our attention on Lie
reductions of the ELBI equation to linear ODEs. More precisely, we
presented results for such three-dimensional subalgebras of p(1, 4) that
give reductions of the ELBI equation to linear ODEs with nonlinear
solutions.
It is known [7] that the Lie algebra p(1, 4) contains three-dimensional
inequivalent subalgebras of the following types: 3A1, A2 ⊕ A1, A3,1,
A3,2, A3,3, A3,4, A3,6, Aa3,7, A3,8, A3,9. Results of the paper imply that
all the above Lie reductions of the ELBI equation to linear ODEs can be
obtained using subalgebras of the types 3A1, A2 ⊕ A1, A3,1, A3,2, A3,3
and A3,6. Moreover, all the subalgebras considered in the paper are also
subalgebras of the Lie algebra of the extended Galilei group G̃(1, 3).
On reductions of the Euler–Lagrange–Born–Infeld equation 201
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| id | oai:trim.imath.kiev.ua:article-376 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:22Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
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| resource_txt_mv | trimimathkievua/c3/48831ad008bdd44538864ca4b403b4c3.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3762020-08-13T08:52:53Z On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs Симетрійна редукція рівняння Ейлера-Лагранжа-Борна-Інфельда до лінійних диференціальних рівнянь Fedorchuk, V. M. Fedorchuk, V. I. Федорчук, В. М. Федорчук, В. І. Connections between structure properties of three-dimensional subalgebras of the Poincare algebra ${\mathfrak p}(1,4)$ and Lie reductions of the Euler-Lagrange-Born-Infeld equation are studied. We concentrate our attention on Lie reductions with respect to three-dimensional subalgebras that reduce the Euler-Lagrange-Born-Infeld equation to linear ordinary differential equations. Вивчається зв'язок між структурними властивостями тривимірних підалгебр алгебри Пуанкаре ${\mathfrak p}(1,4)$ і симетрійною редукцією рівняння Ейлера-Лагранжа-Борна-Інфельда. Основну увагу зосереджено на редукцях за тривимірними підалгебрами, що зводять рівняння Ейлера-Лагранжа-Борна-Інфельда до лінійних диференціальних рівнянь.   Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/376 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 193-202 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 193-202 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 193-202 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/376/373 Авторське право (c) 2019 В. М. Федорчук, В. І. Федорчук |
| spellingShingle | Fedorchuk, V. M. Fedorchuk, V. I. Федорчук, В. М. Федорчук, В. І. On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title | On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title_alt | Симетрійна редукція рівняння Ейлера-Лагранжа-Борна-Інфельда до лінійних диференціальних рівнянь |
| title_full | On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title_fullStr | On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title_full_unstemmed | On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title_short | On symmetry reduction of the Euler-Lagrange-Born-Infeld equation to linear ODEs |
| title_sort | on symmetry reduction of the euler-lagrange-born-infeld equation to linear odes |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/376 |
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