Preliminary group classification of general two-dimensional quasi-linear elliptic type equations
In the paper the problem of group classification of quasi-linear elliptic type equations in two-dimensional space is considered. We obtain the list of all equations of this type admitting semisimple Lie algebras of symmetry operators and Lie algebras of symmetry operators with non-trivial Levi decom...
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| author | Spichak, S. Спічак, С. |
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| description | In the paper the problem of group classification of quasi-linear elliptic type equations in two-dimensional space is considered. We obtain the list of all equations of this type admitting semisimple Lie algebras of symmetry operators and Lie algebras of symmetry operators with non-trivial Levi decomposition. |
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Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 284–291
УДК 517.912:512.816
Preliminary group classification of
general two-dimensional quasi-linear
elliptic type equations
S.V. SPICHAK
Institute of Mathematics of NAS of Ukraine, Kyiv
E-mail: spichak@imath.kiev.ua
В статтi розглядається задача групової класифiкацiї квазiлiнiйних рiв-
нянь елiптичного типу в двовимiрному просторi. Отримано перелiки
усiх рiвнянь цього класу, якi допускають напiвпростi алгебри Лi опе-
раторiв симетрiї та алгебри Лi операторiв симетрiї з нетривiальним
розкладом Левi.
In the paper the problem of group classification of quasi-linear elliptic type
equations in two-dimensional space is considered. We obtain the list of
all equations of this type admitting semisimple Lie algebras of symmetry
operators and Lie algebras of symmetry operators with non-trivial Levi
decomposition.
We consider quasilinear two-dimensional equations of elliptic type
∆u = f(x, y, u, ux, uy). (1)
In (1) and below ∆ = ∂xx+∂yy = ∂2
∂x2 + ∂2
∂y2 is a two-dimensional Laplace
operator, u = u(x, y), F is an arbitrary smooth function in some domain
of the space W = R2 × V
1
= 〈x, y〉 × 〈u, ux, uy〉, that is nonlinear, at
least, with respect to one variable u, ux, uy.
Statement 1. The Lie invariance group of equation (1) is generated by
the infinitesimal operator
v = a(x, y)∂x + b(x, y)∂y + c(x, y, u)∂u, (2)
where functions a, b, c, F satisfy the following system of equations:
ay + bx = 0, ax − by = 0,
cxx + cyy + 2uxcxu + 2uycyu + (u2
x + u2
y)cuu + (cu − 2ax)F = (3)
= aFx + bFy + cFu + [cx + ux(cu − ax)− uybx]Fux +
+ [cy + uy(cu − by)− uxay]Fuy .
Preliminary group classification: results 285
It is not difficult to see that the first two equations are Cauchy–
Riemann conditions, that means that functions a and b are harmonic
ones.
The group E is formed by those transformations
x̄ = α(x, y, u), ȳ = β(x, y, u), v = γ(x, y, u),
D(x̄, ȳ, v)
D(x, y, u)
6= 0,
preserving differential structure of equation (1), that transform it to an
equation of the form
vx̄x̄ + vȳȳ = Φ(x̄, ȳ, v, vx̄, vȳ).
Statement 2. The group E of equation (1) is formed by the following
transformations:
x̄ = α(x, y), ȳ = β(x, y), v = γ(x, y, u), (4)
where
αx = εβy, αy = −εβx (ε = ±1),
α2
x + α2
y = β2
x + β2
y 6= 0, γu 6= 0.
Lemma 1. There exist such transformations from the group E that re-
duce operator (2) to one of the following operators:
v = ∂x, v = ∂u. (5)
We start the group classification from the description of equations
invariant with respect to Lie algebras with a non-trivial Levi decomposi-
tion.
First we will consider the following two equations of form (1):
∆u = f(u)(u2
x + u2
y), f 6= 0; (6)
∆u = λeγu, λ, γ ∈ R, λγ 6= 0. (7)
These equations are invariant with respect to groups of infinitesimal
transformation with infinite number of parameters that are generated
by operators of form (2).
Equation (6) can be reduced by the substitution
v =
∫ u
Φ−1(ξ)dξ, Φ(u) = exp
(∫ u
f(η)dη
)
286 S.V. Spichak
to the two-dimensional Laplace equation. Equation (7) is connected to
the Laplace equation by the Bäcklund transformation.
Invariance of equation (1) with respect to Lie algebras of
symmetry operators with non-trivial Levi decomposition. It is
well known in the theory of abstract Lie algebras [1] that arbitrary
Lie algebra with a non-trivial Levi decomposition contains some simple
(semisimple) Lie algebra as a subalgebra. For this reason, first of all, we
will describe equations that are invariant with respect to simple (semi-
simple) Lie algebras of symmetry operators.
Theorem 1. Up to E-equivalence, there are two classes of quasilinear
equations of form (1) admitting Lie algebras of symmetry operators that
are different realizations of the algebra so(3):
I. ∆u = ch−2 yF̃ (u, ω), ω = (u2
x + u2
y) ch2 y :
so1(3) = 〈∂x, sh y cosx∂x − ch y sinx∂y,
− sh y sinx∂x − ch y cosx∂y〉;
II. ∆u = ch−2 yF̃ (ξ ch y, η ch y), ξ = (ux − thy) sinu+ uy cosu,
η = (ux − thy) cosu− uy sinu :
so2(3) = 〈∂x, sh y cosx∂x − ch y sinx∂y + ch y cosx∂u,
− sh y sinx∂x − ch y cosx∂y − ch y sinx∂u〉.
Theorem 2. Up to E-equivalence, there are two classes of quasilinear
equations of form (1) admitting Lie algebras of symmetry operators that
are different realizations of the algebra sl(2,R):
I. ∆u = y−2F̃ (u, ω), ω = y2(u2
x + u2
y) :
sl1(2,R) = 〈2x∂x + 2y∂y,−(x2 − y2)∂x − 2xy∂y, ∂x〉;
II. ∆u = y−2F̃ (v, ω), v = (1− 2yux) cos 2u+ 2yuy sin 2u,
ω = 2yuy cos 2u− (1− 2yux) sin 2u :
sl2(2,R) = 〈2x∂x + 2y∂y,−(x2 − y2)∂x − 2xy∂y + y∂u, ∂x〉.
It is well known from the theory of abstract Lie algebra over the field
R that there exist four types of classical simple algebras:
1) type An−1(n > 1) that contains four real forms of the algebra
sl(n,C): su(n), sl(n,R), su(p, q) (p+ q = n, p ≥ q), su∗(2n);
Preliminary group classification: results 287
2) type Dn (n > 1) that contains three real forms of the algebra
so(2n,C): so(2n), so(p, q) (p+ q = 2n, p ≥ q), so∗(2n);
3) type Bn (n > 1) that contains two real forms of the algebra so(2n+
1, C): so(2n+ 1), so(p, q) (p+ q = 2n+ 1, p > q);
4) type Cn (n ≥ 1) that contains three real forms of the algebra
sp(n,C): sp(n), sp(n,R), sp(p, q) (p+ q = n, p ≥ q).
and the following special cases of semisimple real algebras: G1, F4, E6,
E7, E8.
Theorem 3. Equations of form (1) invariant with respect to a symmetry
algebra with a non-trivial Levi decomposition are exhausted by ones pre-
sented in theorems 1 and 2.
Sketch of proof. 1. so(4) = 〈ei|i = 1, 2, 3〉 ⊕ 〈ēi|i = 1, 2, 3〉. Then in
accordance to the results of theorem 2 the operators ei (i = 1, 2, 3)
represent a basis of realizations so1(3) or so2(3). Direct calculations show
that in such case the operators ēi (i = 1, 2, 3) belong to the class of
operators c(u)∂u, and in accordance to the statement obtained in the
process of proof of theorem 2, there is no realizations of the algebra
so(3) in this class of operators. Thus, there exist no nonlinear equation
of form (1) whose invariance algebra is isomorphic to the algebra so(4).
2. The type G2 contains one compact real form g2 and one non-
compact form g′2. Since g2 ∩ g′2 ∼ su(2) ⊕ su(2) ∼ so(4) and algebra
so(4) does not have realizations in the given class of operators, in this
class of operators the algebras g2 and g′2 also do not have realizations.
3. For the algebras so(3, 1) we will use Cartan’s decomposition:
so(3, 1) = 〈e1, e2, e3〉 ⊕
·
〈N1, N2, N3〉, де 〈e1, e2, e3〉 = so(3), [ei, Nj ] =
εijlNl, [Ni, Nj ] = −εijlel; i, j, l = 1, 2, 3; εijl is antisymmetric tensor of
third rank, ε123 = 1.
It can be proved that the realization has the form:
e1 = ∂x, N1 = ∂y,
e2 = ε1(sh y cosx∂x − ch y sinx∂y) + sh y sinx∂u,
e3 = −ε1(sh y sinx∂x + ch y cosx∂y) + sh y cosx∂u,
N2 = ε1(ch y sinx∂x + sh y cosx∂y)− ch y cosx∂u,
N3 = ε1(sh y cosx∂x − sh y sinx∂y) + ch y sinx.
288 S.V. Spichak
The respective invariant solution having the form
∆u = λe2ε1u, λ ∈ R, λ 6= 0,
is the subcase of equation (7). 2
Invariance with respect to decomposable algebras. In accor-
dance to the above here we study existence of equations of form (1)
invariant with respect to algebras of form S⊕N , i.e. satisfying conditions
[S, S] ⊂ S, [N,N ] ⊂ N, [S,N ] = 0. (8)
Possible realisations so1(3), so2(3), sl1(2,R), sl2(2,R) of semisimple al-
gebras S (Levi factor) have been found in the theorems 1 and 2.
There exist six classes of nonlinear equations of form (1), whose maxi-
mal invariance algebras can be decomposed into the direct sum of the
Levi factor and solvable Lie algebra:
1) ∆u = ch−2 yF̃ (ω), ω = (u2
x + u2
y) ch2 y : so1(3)⊕ 〈∂u〉;
2) ∆u = ch−2 yF̃ (ω), ω = (ux ch y − sh y)2 + u2
y ch2 y :
so2(3)⊕ 〈∂u〉;
3) ∆u = y−2F̃ (ω), ω = (u2
x + u2
y)y2 : sl1(2,R)⊕ 〈∂u〉;
4) ∆u = y−2F̃ (ω), ω = 4y2u2
y + (1− 2yux)2 : sl2(2,R)⊕ 〈∂u〉;
5) ∆u = λ ch−1 y
√
u2
x + u2
y, λ 6= 0 : so1(3)⊕ 〈∂u, u∂u〉;
6) ∆u = λy−1
√
u2
x + u2
y, λ 6= 0 : sl2(2,R)⊕ 〈∂u, u∂u〉.
Here F̃ = F̃ (ω) is an arbitrary smooth function, F̃ωω 6= 0.
Invariance with respect to non-decomposable algebras. Here
we will investigate an existence of equations of form (1) invariant with
respect to algebras of form S ⊂+N , i.e. satisfying conditions
[S, S] ⊂ S, [N,N ] ⊂ N, [S,N ] ⊂ N. (9)
In this investigation we will use results of the paper [2], where classifi-
cation of Lie algebras of dimension not greater then eight, and whose
Levi factor coincides with the algebras so(3) and sl(2,R).
Result. There exist no nonlinear equations of form (1) with invariance
algebras of such structure.
Preliminary group classification: results 289
Invariance with respect to solvable algebras of symmetry
operators. For each two-dimensional and three-dimensional solvable
algebras A2.i = 〈e1, e2〉, (i = 1, 2) and A3.i = 〈e1, e2, e3〉 (i = 1, . . . , 9)
below we adduce realizations and corresponding forms of the functions F .
A2.1 ([e1, e2] = 0):
A1
2.1 = 〈∂x, ∂y〉, F = F̃ (u, ux, uy);
A2
2.1 = 〈∂x, ∂u〉, F = F̃ (y, ux, uy);
A3
2.1 = 〈∂u, g(x, y)∂u〉 (g 6= const)
F =
gxux + gyuy
g2
x + g2
y
(gxx + gyy) +G(x, y, ω),
ω = gyux − gxuy, g2
x + g2
y 6= 0.
A2.2 ([e1, e2] = e2):
A2
2.2 = 〈−x∂x − y∂y, ∂x〉 : F = (u2
x + u2
y)F̃ (u, ω1, ω2),
ω1 = yux, ω2 = yuy;
A1
2.2 = 〈∂x − u∂u, ∂u〉 : F = e−xF̃ (y, ω1, ω2),
ω1 = exux, ω2 = exuy;
A3
2.2 = 〈−u∂u, ∂u〉 : F = (ux + uy)F̃ (x, y, ω), ω = uxu
−1
y .
Let us note that for arbitrary forms of functions F̃ the respective reali-
sations are the maximal invariance algebras of equations.
A3.1 ([ej , el] = 0, j, l = 1, 2, 3):
A1
3.1 = 〈∂x, ∂y, ∂u〉, F = G(ux, uy);
A2
3.1 = 〈∂x, ∂u, f(y)∂u〉, f ′(y) 6= 0, F =
f ′′
f ′
uy +G(y, ux);
A3
3.1 = 〈∂u, f(x, y)∂u, g(x, y)∂u〉.
Further analysis of the determining equations for operators f(x, y)∂u and
g(x, y)∂u shows that either the respective invariant equation is linear or
g(x, y) = λf(x, y) + µ, where λ, µ are constants, and then the case A3
3.1
is reduced to the two-dimensional case.
A3.2 ([e1, e2] = e2, [e1, e3] = [e2, e3] = 0):
A1
3.2 = 〈−x∂x − y∂y, ∂x, ∂u〉, F = u2
xG(yux, yuy);
290 S.V. Spichak
A2
3.2 = 〈−u∂u, ∂u, ∂x〉, F = (ux + uy)G
(
y,
ux
uy
)
;
A3
3.2 = 〈λ∂y − u∂u, ∂u, ∂x〉, λ 6= 0,
F = (ux + uy)G(ey/λux, e
y/λuy);
A4
3.2 = 〈∂x − u∂u, ∂u, f(y)e−x∂u〉, f(y) 6= 0,
F = −f + f ′′
f
ux + e−xG(y, exf ′ux + exfuy).
A3.3 ([e2, e3] = e1, [e1, e2] = [e1, e3] = 0):
A1
3.3 = 〈∂u, ∂x, λ∂y + x∂u〉, λ 6= 0, F = G(λux − y, uy);
A2
3.3 = 〈∂u, (f(y)− x)∂u, ∂x〉, F = −f ′′ux +G(y, uy + f ′ux).
A3.4 ([e1, e3] = e1, [e2, e3] = e1 + e2, [e1, e2] = 0):
A1
3.4 = 〈∂u, ∂x, x∂x + y∂y + (u+ x)∂u〉, F = e−uxG(uy, ye
−ux);
A2
3.4 = 〈∂u, (f(y)− x)∂u, ∂x + u∂u〉,
F =
f ′uy − ux
1 + (f ′)2
f ′′ + exG(y, f ′e−xux + e−xuy).
A3.5 ([e1, e3] = e1, [e2, e3] = e2, [e1, e2] = 0):
A1
3.5 = 〈∂x, ∂y, x∂x + y∂y〉, F = (ux + uy)2G
(
u,
ux
uy
)
;
A2
3.5 = 〈∂x, ∂u, x∂x + y∂y + u∂u〉, F = y−1G(ux, uy);
A3
3.5 = 〈∂u, f(y)∂u, ∂x + u∂u〉, f ′ 6= 0,
F =
uy
f ′
f ′′ + exG(y, f ′e−xux);
A4
3.5 = 〈∂u, f(x, y)∂u.u∂u〉,
The equation that is invariant with respect to A4
3.5 is linear.
A3.6 ([e1, e3] = e1, [e2, e3] = −e2, [e1, e2] = 0):
A1
3.6 = 〈∂x, ∂u, x∂x + y∂y − u∂u〉, F = y−3G(y2ux, y
2uy);
A2
3.6 = 〈∂u, e2xf(y)∂u, ∂x + u∂u〉, f(y) 6= 0,
F =
2fux + f ′uy
4f2 + (f ′)2
(f ′′ + 4f) + exG(y, f ′e−xux − 2fe−xuy);
Preliminary group classification: results 291
A3.7 ([e1, e3] = e1, [e2, e3] = qe2 (0 < |q| < 1), [e1, e2] = 0):
A1
3.7 = 〈∂x, ∂u, x∂x + y∂y + qu∂u〉, F = yq−2G(y1−qux, y
1−quy);
A2
3.7 = 〈∂u, ∂x, qx∂x + qy∂y + u∂u〉,
F = y(1−2q)/qG(y(q−1)/qux, y
(q−1)/quy);
A3
3.7 = 〈∂u, e(1−q)xf(y)∂u, ∂x + u∂u〉,
F =
(1− q)fux + f ′uy
(1− q)2f2 + (f ′)2
(f ′′ + f(1− q)2) +
+ exG(y, f ′e−xux + (q − 1)fe−xuy).
A3.8 ([e1, e3] = −e2, [e2, e3] = e1, [e1, e2] = 0):
A1
3.8 = 〈∂x, ∂y, y∂x − x∂y〉, F = G(u, u2
x + u2
y);
A2
3.8 = 〈∂u, tg(f(y)− x)∂u, ∂x − u tg(f(y)− x)∂u〉,
F =
f ′uy − ux
(f ′)2 + 1
f ′′ + 2(f ′uy − ux) tg(f − x) +
+ (f ′ux + uy)G(y, cos(f − x)(f ′ux + uy)).
A3.9 ([e1, e3] = qe1 − e2, [e2, e3] = e1 + qe2 (q > 0), [e1, e2] = 0):
A1
3.9 = 〈∂x, ∂y, (qx+ y)∂x + (qy − x)∂y〉,
F = (u2
x + u2
y)G
(
u, ln(u2
x + u2
y) + 2q arctan
ux
uy
)
;
A2
3.9 = 〈∂u, tg(f(y)− x)∂u, ∂x + (q − tg(f(y)− x))u∂u〉,
F =
f ′uy − ux
(f ′)2 + 1
f ′′ + 2(f ′uy − ux) tg(f − x) +
+ (f ′ux + uy)G(y, cos(f − x)(f ′ux + uy)e−qx).
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[2] Turkowski P. Low-dimensional real Lie algebras // J. Math. Phys. – 1988. –
29. – P. 2139–2144.
[3] Овсянников Л.В. Групповой анализ дифференциальных уравнений. – Мо-
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|
| id | oai:trim.imath.kiev.ua:article-472 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:56Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/2c/d25b1979279c982c0196ede4c9da6b2c.pdf |
| spelling | oai:trim.imath.kiev.ua:article-4722020-08-13T10:42:30Z Preliminary group classification of general two-dimensional quasi-linear elliptic type equations Попередня групова класифікація загальних двовимірних квазілінійних еліптичних рівнянь Spichak, S. Спічак, С. In the paper the problem of group classification of quasi-linear elliptic type equations in two-dimensional space is considered. We obtain the list of all equations of this type admitting semisimple Lie algebras of symmetry operators and Lie algebras of symmetry operators with non-trivial Levi decomposition. В статті розглядається задача групової класифікації квазілінійних рівнянь еліптичного типу в двовимірному просторі. Отримано переліки усіх рівнянь цього класу, які допускають напівпрості алгебри Лі операторів симетрії та алгебри Лі операторів симетрії з нетривіальним розкладом Леві. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/472 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); 284-291 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); 284-291 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); 284-291 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/472/468 Авторське право (c) 2020 С. Спічак http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Spichak, S. Спічак, С. Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title | Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title_alt | Попередня групова класифікація загальних двовимірних квазілінійних еліптичних рівнянь |
| title_full | Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title_fullStr | Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title_full_unstemmed | Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title_short | Preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| title_sort | preliminary group classification of general two-dimensional quasi-linear elliptic type equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/472 |
| work_keys_str_mv | AT spichaks preliminarygroupclassificationofgeneraltwodimensionalquasilinearelliptictypeequations AT spíčaks preliminarygroupclassificationofgeneraltwodimensionalquasilinearelliptictypeequations AT spichaks poperednâgrupovaklasifíkacíâzagalʹnihdvovimírnihkvazílíníjnihelíptičnihrívnânʹ AT spíčaks poperednâgrupovaklasifíkacíâzagalʹnihdvovimírnihkvazílíníjnihelíptičnihrívnânʹ |