On equations of Korteweg-de Vries type with highest symmetry properties
We present results on group classification of one class of third-order nonlinear evolution equations admitting four-dimensional solvable Lie algebras of symmetry operators.
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| author | Lahno , H. Smalij , V. Лагно, Г. Смалій, В. |
| author_facet | Lahno , H. Smalij , V. Лагно, Г. Смалій, В. |
| author_institution_txt_mv | [
{
"author": "H. Lahno ",
"institution": null
},
{
"author": "V. Smalij ",
"institution": null
}
] |
| author_sort | Lahno , H. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T10:42:30Z |
| description | We present results on group classification of one class of third-order nonlinear evolution equations admitting four-dimensional solvable Lie algebras of symmetry operators. |
| first_indexed | 2026-08-04T01:07:39Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 182–186
УДК 517.9
On equations of Korteweg–de Vries
type with highest symmetry
properties
H. LAHNO †, V. SMALIJ ‡
† Poltava State Pedagogical University
E-mail: laggo@poltava.bank.gov.ua
‡ National Agrarian University, Kyiv
Представлено результати групової класифiкацiї одного класу нелiнiй-
них еволюцiйних рiвнянь третього порядку, що допускать чотирьохви-
мiрнi алгебри Лi операторiв симетрiї.
We present results on group classification of one class of third-order nonli-
near evolution equations admitting four-dimensional solvable Lie algebras
of symmetry operators.
The standard Korteweg–de Vries equation ut = uxxx + uux belongs
to the family of evolution equations
ut = uxxx + F (t, x, u, ux, uxx), (1)
where u = u(t, x), ut = ∂u
∂t , ux = ∂u
∂x , uxx = ∂2u
∂x2 , uxxx = ∂3u
∂x3 .
The problem of group classification of equation (1) was solved by
F. Güngor, V. Lahno and R. Zhdanov [1]. But their result of group
classification is not complete. They obtained all classes of nonlinear equa-
tions of the form (1) that admit one-, two-, three- and four-dimensional
solvable Lie algebras.
Here we investigate the symmetry properties of nonlinear equations
of the form (1) whose invariance algebras are isomorphic to solvable
Lie algebras 2A2.2 = 〈e1, e2〉⊕〈e3, e4〉 ([e1, e2] = e2, [e3, e4] = e4), A2.2⊕
2A1 = 〈e1, e2〉⊕〈e3〉⊕〈e4〉 ([e1, e2] = e2) and A3.3⊕A1 = 〈e1, e2, e3〉⊕〈e4〉
([e2, e3] = e1, [e1, e2] = [e1, e3] = 0).
According to [1] the complete list of such equations contains following
nine equations:
1) ut = uxxx + u3
x − 3uxuxx + x−2uxF̃ (ω),
On equations of Korteweg–de Vries type 183
ω = x(u−1
x uxx − ux);
2) ut = uxxx + λ
3tω1 ln |ω1|+ ω1
t F̃ (ω), ω1 = t
1
3ux,
ω = t
1
3u−1
x uxx, λ ∈ R;
3) ut = uxxx − λxux − λux ln |ux|+ uxF̃ (ω),
ω = u−1
x uxx, λ 6= 0;
4) ut = uxxx − (1 + λ−1)ux + e−xF̃ (ω),
ω = ex(ux + uxx), λ 6= 0;
5) ut = uxxx − γ−1(1 + γ3)ux + e(γ−β−1)x−tF̃ (ω),
ω = et+(β−1−γ)x(γux − uxx), γβ 6= 0;
6) ut = uxxx − ux + e−xF̃ (ω), ω = ex(ux + uxx);
7) ut = uxxx + uxF̃ (ω), ω = uxxu
−1
x ;
8) ut = uxxx − (λ3 + 1)λ−1ux + e−t+λxF̃ (ω),
ω = et−λx(λux − uxx), λ 6= 0;
9) ut = uxxx + λ−1x− βux + F̃ (uxx), λ > 0, β ∈ R. (2)
Using the standard Lie approach we prove that the maximal invari-
ance group of equations (1) is generated by the operator
v = τ(t)∂t +
(
1
3 τ̇x+ ρ(t)
)
∂x + η(t, x, u)∂u, (3)
where the functions τ , ρ, η and F are arbitrary solutions of a single
partial differential equation
−3uxρ̇− xuxτ̈ − 9uxuxxηuu − 3u3
xηuuu + 3ηt − 9uxxηxu −
− 9u2
xηxuu − 9uxηxxu − 3ηxxx + 3(ηu − τ̇)F + (2uxxτ̇ −
− 3uxxηu − 3u2
xηuu − 6uxηxu − 3ηxx)Fuxx + (uxτ̇ − 3uxηu −
− 3ηx)Fux − 3ηFu − 3τFt − (3ρ+ xτ̇)Fx = 0. (4)
Here the dot over a symbol stands for the time derivative.
The equations (2) contain arbitrary functions of one variable. The-
refore we utilize the Lie–Ovsyannikov method [2,3] of group classificati-
on of differential equations. We consider in more detail first and sixth
equations (2).
In first equation (2)
F = u3
x − 3uxuxx + x−2uxF̃ (ω), ω = x(u−1
x uxx − ux).
184 H. Lahno, V. Smalij
From the equation (4) we find that the functions τ , ρ, η in the opera-
tor (3) and the function F̃ satisfy following system of equations:
ηuuu − ηu = 0;
x−1[ηuu − ηu]F̃ω + 3x−1[ηuu − ηu]ω + 3[ηxuu − ηxu] = 0;
x−1[x−2ρω − 2ηx + 2ηxu]F̃ω − 2x−3ρF̃ =
= 3x−1(ηx − ηxu)ω + 3(ηxx − ηxxu)− 1
3xτ̈ − ρ̇;
[x−2ηxω − x−1ηxx]F̃ω − x−2ηxF̃ = ηxxx − ηt. (5)
If F̃ is arbitrary function, then from (5) we obtain that corresponding
operator v has following form:
v = (C1t+ C2)∂t + 1
3C1x∂x + C3e
u∂u + C4∂u,
where C1, C2, C3, C4 ∈ R.
The corresponding invariance algebra is isomorphic to solvable Lie
algebra 2A2.2: e1 = −t∂t 1
3x∂x, e2 = ∂t, e3 = ∂u, e4 = eu∂u.
The extension of symmetry properties of first equation (2) takes place
in two cases:
(1) F̃ = λω2
(
λ 6= 0,− 3
2
)
: here
τ = C1t+ C2, ρ = C3, η = C4e
u + C5,
Ci ∈ R (i = 1, 2, . . . , 5);
(2) F̃ = − 3
2ω
2 : here
τ = C1t+ C2, ρ = C3, η = C4e
u + C5e
−u + C6,
Ci ∈ R (i = 1, 2, . . . , 6).
In sixth equation (2) F = −ux+e−xF̃ (ω), ω = ex(ux+uxx), F̃ωω 6= 0,
and from the equation (4) we find that the functions τ , ρ, η, F̃ satisfy
following system of equations:
ηuuu = 0; ηuu(1− F̃ω)− ηxuu = 0;
(τ̇ + 6ηxu)F̃ω = −9e−xωηuu − 3ρ̇− xτ̈ + 9ηxu − 9ηxxu;
[ex(2τ̇ − 3ηu − 3ρ− xτ̇)ω − 3ηx − 3ηxx]F̃ω +
+ e−x(3ηu − 3τ̇ + 3ρ+ xτ̇)F̃ − 9e−xωηxu +
+ 3ηt − 3ηx − 3ηxxx = 0. (6)
On equations of Korteweg–de Vries type 185
From second equation (6) we obtain the condition
ηuuF̃ωω = 0,
consequently ηuu = 0.
From third equation (6) we obtain the condition
(τ̇ + 6ηxu)F̃ωω = 0,
consequently
τ̇ + 6ηxu = 0, −3ρ̇− xτ̈ + 2τ̇ + 9ηxu − 9ηxxu = 0.
From obtained relations we obtain following values of the functions
τ , ρ, η :
τ = C1t+ C2, ρ = 1
6C1t+ C3,
η =
[
− 1
6C1x+ γ(t)
]
u+ β(t, x), C1, C2, C3 ∈ R.
Fourth equation (6) transforms into following system:
1
2C1F̃ω = −3γ̇ + 1
2C1,[
e−x
(
2C1 − 1
2xC1 − 3γ − 1
2C1t− 3C3
)
ω − 3βx − 3βxx
]
F̃ω +
+ e−x
(
−3C1 + 1
2xC1 + 1
2C1t+ 3C3 + 3γ
)
F̃ =
= − 3
2e
−xC1ω − 3βt − 3βx + 3βxxx. (7)
From first equation (7) we obtain condition
C1F̃ωω = 0,
consequently C1 = 0, γ = C4, C4 ∈ R. Second equation (7) reduces to
equation
[(C3 + C4)ω + βx + βxx]F̃ω − (C3 + C4)F̃ = ex(βt + βx − βxxx),
from which we obtain condition
[(C3 + C4)ω + βx + βxx]F̃ωω = 0.
Consequently, C3 + C4 = 0, βx + βxx = 0, βt + βx − βxxx = 0, and the
operator v (3) has following form:
v = C2∂t + C3∂x + (−C3u+ C5 + e−xC6)∂u,
186 H. Lahno, V. Smalij
C2, C3, C4, C5 ∈ R. The corresponding invariance algebra is isomorphic
to solvable Lie algebra A2.2 ⊕ 2A1: e1 = ∂x − u∂u, e2 = ∂u, e3 = ∂t,
e4 = e−x∂u.
So we have obtained that sixth equation (2) does not suppose the
extension of symmetry properties. Analogous results we have obtained
for 3, 4, 5 and 8 equations (2). The rest equations (2) suppose the
extensions of symmetry properties. We give these equations with the
corresponding invariance algebras:
1) ut = uxxx + u3
x − 3uxuxx + λux(u−1
x uxx − ux)2, λ 6= 0,− 3
2 :
〈t∂t + 1
3x∂x, ∂t, ∂x, e
u∂u, ∂u〉;
2) ut = uxxx − 3
2u
−1
x u2
xx − 1
2u
3
x :
〈t∂t + 1
3x∂x, ∂t, ∂x, e
u∂u, e
−u∂u, ∂u〉;
3) ut = uxxx + λ−1x+m ln |uxx| − βux, λ ·m 6= 0, β ∈ R :
〈t∂t +
(
1
3x+ 2
3βt
)
∂x +
[
u+ 1
3 t
(
λ−1x+ 1
2βλ
−1t+m
)]
∂u,
∂x + λ−1t∂u, (x− βt)∂u, ∂t, ∂u〉;
4) ut = uxxx + λ−1x− βux +m|uxx|p, λm 6= 0, p 6= 0, 1, β ∈ R :
〈t∂t +
(
1
3x+ 2
3βt
)
∂x +
[
2p−3
3(p−1)u+ 2p−1
3λ(p−1) tx+
+ β
6λ(1−p) t
2
]
∂u, ∂x + λ−1t∂u, (x− βt)∂u, ∂t, ∂u〉;
5) ut = uxxx + λ−1x− βux +menuxx , λmn 6= 0, β ∈ R :
〈t∂t +
(
1
3x+ 2
3βt
)
∂x +
[
2
3u+ 1
6nx
2 +
(
2
3λ −
β
3n
)
tx+
+ β2
6n t
2
]
∂u, ∂x + λ−1t∂u, (x− βt)∂u, ∂t, ∂u〉.
[1] Güngör F., Lahno V., Zhdanov R. Symmetry classification of KdV-type nonlinear
evolution equations // J. Math. Phys. – 2004. – 45. – P. 2280–2113.
[2] Ovsyannikov L.V. Group analysis of differential equations. – New York: Aca-
demic, 1982.
[3] Olver P.J. Applications of Lie groups to differential equations. – New York: Sprin-
ger-Verlag, 1986.
|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
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| language | English |
| last_indexed | 2026-08-04T01:07:39Z |
| publishDate | 2006 |
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| spelling | oai:trim.imath.kiev.ua:article-4552020-08-13T10:42:30Z On equations of Korteweg-de Vries type with highest symmetry properties Про рівняння Кортевега-де Фріза з найвищим симетрійними властивостями Lahno , H. Smalij , V. Лагно, Г. Смалій, В. We present results on group classification of one class of third-order nonlinear evolution equations admitting four-dimensional solvable Lie algebras of symmetry operators. Представлено результати групової класифікації одного класу нелінійних еволюційних рівнянь третього порядку, що допускать чотирьохвимірні алгебри Лі операторів симетрії. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/455 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 3 No. 2 (2006): Symmetry and Integrability of Equations of Mathematical Physics (Dedicated to the 70-th Anniversary of Professor W.I. Fushchych); 182-186 Сборник Трудов Института математики НАН Украины; Том 3 № 2 (2006): Симетрія та інтегровність рівнянь математичної фізики (До 70-річчя від дня народження Вільгельма Ілліча Фущича); 182-186 Збірник Праць Інституту математики НАН України; Том 3 № 2 (2006): Симетрія та інтегровність рівнянь математичної фізики (До 70-річчя від дня народження Вільгельма Ілліча Фущича); 182-186 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/455/452 Авторське право (c) 2020 Г. Лагно, В. Смалій http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Lahno , H. Smalij , V. Лагно, Г. Смалій, В. On equations of Korteweg-de Vries type with highest symmetry properties |
| title | On equations of Korteweg-de Vries type with highest symmetry properties |
| title_alt | Про рівняння Кортевега-де Фріза з найвищим симетрійними властивостями |
| title_full | On equations of Korteweg-de Vries type with highest symmetry properties |
| title_fullStr | On equations of Korteweg-de Vries type with highest symmetry properties |
| title_full_unstemmed | On equations of Korteweg-de Vries type with highest symmetry properties |
| title_short | On equations of Korteweg-de Vries type with highest symmetry properties |
| title_sort | on equations of korteweg-de vries type with highest symmetry properties |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/455 |
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