Preliminary group classification of the general quasi-linear wave equation
We begin the group classification of quasi-linear second-order wave equations of the most general form. We find the canonical forms for the symmetry operators which generate the invariance group of the equation, as well as the equivalence group, and we describe those equations which admit one- and t...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552875017633792 |
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| author | Basarab-Horwath, P. Lahno, V. Басараб-Хорват, П. Лагно, В. |
| author_facet | Basarab-Horwath, P. Lahno, V. Басараб-Хорват, П. Лагно, В. |
| author_institution_txt_mv | [
{
"author": "P. Basarab-Horwath",
"institution": null
},
{
"author": "V. Lahno",
"institution": null
}
] |
| author_sort | Basarab-Horwath, P. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T10:42:30Z |
| description | We begin the group classification of quasi-linear second-order wave equations of the most general form. We find the canonical forms for the symmetry operators which generate the invariance group of the equation, as well as the equivalence group, and we describe those equations which admit one- and two-dimensional invariance groups. |
| first_indexed | 2026-08-04T01:06:22Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 9–30
УДК 517.912:512.816
Preliminary group classification of
the general quasi-linear wave equation
P. BASARAB-HORWATH †, V. LAHNO ‡
† Linköping University, Linköping, Sweden
E-mail: pehor@mai.liu.se
‡ Poltava State Pedagogical University, Ukraine
E-mail: lvi@pdpu.poltava.ua
Розпочато групову класифiкацiю квазiлiнiйного хвильового рiвняння
другого порядку найбiльш загальної форми. Знайдено канонiчнi фор-
ми операторiв симетрiї, якi генерують групу iнварiантностi рiвнян-
ня, описано перетворення еквiвалентностi та рiвняння, що допускають
одно- та двовимiрнi групи iнварiантностi.
We begin the group classification of quasi-linear second-order wave equa-
tions of the most general form. We find the canonical forms for the
symmetry operators which generate the invariance group of the equation,
as well as the equivalence group, and we describe those equations which
admit one- and two-dimensional invariance groups.
1. Introduction. The group classification of partial differential equa-
tions of mathematical physics occupies an important place among the
fundamental problems of modern group analysis of differential equa-
tions [1–3]. The solution of this problem is of particular interest because
one is able to exploit the powerful methods of Lie groups and algebras
for the analysis and construction of solutions of equations that model
physical processes and which possess non-trivial symmetry properties.
The problem derives its importance from the need to choose a differen-
tial equation from some general class of differential equations modelling
a process, which admit a non-trivial group symmetry. The history of
the solution of the problem of group classification of differential equati-
ons begins with the work of Sophus Lie. The first article of Lie on this
problem was [4]. The modern form of the group classification problem
was formulated by Ovsiannikov in his article [5], in which he proposed
a procedure for its solution (which we shall call the Lie–Ovsiannikov
10 P. Basarab-Horwath, V. Lahno
method) and where he obtained a complete classification of the non-
linear heat conductivity equation. After this, the group classification of
differential equations became the subject of intensive research. A detai-
led survey of the work done in this area up to the beginning of the 1990’s
is given in [6]. In the present article we solve the problem of group classi-
fication for non-linear wave equations of the form
utt = F (t, x, u, ux)uxx +G(t, x, u, ux), (1)
where F 6= 0, G are arbitrary smooth functions, u = u(t, x). We note
that differential equations of the form (1) are of great importance in
mathematical physics and are used in modelling various types of di-
spersion of waves. They have found applications in differential geometry,
hydrodynamics, gas dynamics, as well as in chemical engineering and
superconductivity.
Many articles have been written on the subject of group classification
of quasi-linear equations of the form (1). A complete solution of the
problem of group classification of the general linear equation is given
in [4,7]. The method of Lie–Ovsiannikov has also been applied to give a
full solution of the problem for a number of non-linear wave equations:
utt = uxx + F (u); [8–11]
utt = [f(u)ux]x ; [12, 13]
utt = f(ux)uxx; [14]
utt = f(ux)uxx +G(ux); [15]
utt = umx uxx + F (u); [16]
utt + f(u)ut = [F (u)ux]x ; [17]
utt + f(u)ut = [F (u)ux]x +G(u)ux. [18]
As one can see, the equations listed above involve arbitrary func-
tions of one variable. This is connected with the fact that the standard
method of performing a group classification involves solving a defining
system of equations for the symmetry operators, and for the equations
listed above it is possible to solve these defining equations because the
arbitrary elements are functions of just one variable. However, one is
confronted with a different situation when these arbitrary functions are
functions of two or more variables and the defining equations for the
symmetry operators then involve first order partial derivatives of the
functions which render difficult or even impossible the complete solution
Preliminary group classification 11
of the defining system of equations. It is this which explains why the
classification of the equations
utt = [f(x, u)ux]x ; [19]
utt + λuxx = g(u, ux); [20, 21]
utt = [f(ux)ux + g(x, u)]x ; [22–24]
utt = f(x, ux)uxx + g(x, ux) [25]
is, in the classical sense of Lie, incomplete.
In the differential equations which we study, the arbitrary functions
are functions of four variables, and therefore we shall use the method
for the solution of the problem of the group classification which was
described in [26] and was used in the group classification of the following
equations:
ut = uxx + F (t, x, u, ux); [26]
ut = F (t, x, u, ux)uxx +G(t, x, u, ux); [27]
ut = uxxx + F (t, x, u, ux, uxx); [28]
utt = uxx + F (t, x, u, ux). [29, 30]
A detailed description of the algorithmic method which we use may be
found in [26, 27]. Here, we merely note that it differs from the classical
method of group classification of differential equations in that we exploit
all possible realizations of low-dimensional Lie algebras within the class
of vector fields which are infinitesimal symmetries of the equation under
study, and this also gives us further specification of the arbitrary functi-
ons. We also note that in performing the group classification of equa-
tion (1) we exclude all cases which are equivalent, under local changes
of coordinates, to a linear equation or to
utt = uxx + F (t, x, u, ux). (2)
2. Preliminary results of group classification of equation (1).
Our first task is to determine the form of the vector fields which are
symmetry operators for equation (1), and to determine the equivalence
group of equation (1) (we define this concept later). We seek these
symmetry operators amongst vector fields of the form
Q = τ∂t + ξ∂x + η∂u, (3)
12 P. Basarab-Horwath, V. Lahno
where τ(t, x, u), ξ(t, x, u), η(t, x, u) are smooth functions from V = R2×
R to R. The variables (t, x) ∈ R2 are said to be the independent variables,
and u = u(t, x) ∈ R is said to be the dependent variable. The condition
that such an operator (3) be a symmetry of equation (1) is that
φtt − φxxF − (τFt + ξFx + ηFu + φxFux)uxx −
− (τGt + ξGx + ηGu + φxGux)
∣∣
(1)
= 0. (4)
By a standard but tedious calculation, equation (4) gives us the following
initial information about the coefficients:
τ = a(t, x)u+ b(t, x), ξ = ξ(t, x),
η = at(t, x)u2 + c(t, x)u+ d(t, x)
where the functions a(t, x), b(t, x), c(t, x), d(t, x), ξ(t, x), F , G satisfy
the system of equations
ξt − (axu+ bx + aux)F = 0, 2aF − (axu+ bx + aux)Fux = 0,
2ηtu − τtt − 3τuG+ (τxx + 2uxτxu)F + (τx + uxτu)Gux = 0,
2(ξx − τt)F − (τFt + ξFx + ηFu)− [ηx + (ηu − ξx)ux]Fux = 0,
ηtt − uxξtt − [ηxx + (2ηxu − ξxx)ux + ηuuu
2
x)F − (2τt − ηu)G−
− (τGt + ξGx + ηGu)− [ηx + (ηu − ξx)ux]Gux = 0. (5)
From the first two equations of (5) we find that, since F 6= 0, then a = 0.
Further, we distinguish three cases: (1) Fux 6= 0, (2) Fux = 0, Fu 6= 0,
(3) Fu = Fux = 0.
Case 1: Fux 6= 0. If Fux 6= 0 then it follows from the first two
equations of (5) that ξt = bx = 0 and the third equation then becomes
2ct − btt = 0, from which we find that c = 1
2bt + θ(x). Then in (3) we
have
τ = b(t), ξ = ξ(x), η =
(
1
2
bt + θ(x)
)
u+ d(t, x) (6)
and the functions τ , ξ, η, F , G satisfy the last two equations of (5).
Case 2: Fux = 0, Fu 6= 0. If, in this case, Guxux 6= 0 then we obtain
the same result as in case 1. So, suppose that Guxux = 0, which gives
us G = A(t, x, u)ux + B(t, x, u) with A and B being arbitrary smooth
functions. If A = 0 then equation (1) becomes
utt = Fuxx +B, Fu 6= 0 (7)
Preliminary group classification 13
and in (3) we find
τ = b(t), ξ = ξ(x),
η =
[
1
2 (τt + ξx) + k
]
u+ d(t, x), k ∈ R, k 6= 0.
If, on the other hand, F = λ(x)A, λ(x)Au 6= 0, then equation (1)
becomes
utt = A[λ(x)uxx + ux] +B (8)
and there is then a local change of coordinates t′ = t, x′ = X(x), u =
v(t′, x′), Xx 6= 0 with λXxx + Xx = 0, which transforms (8) to a wave
equation of the form (7). As is well-known (see [1]), from the group-
theoretic point of view, such equations are deemed to be equivalent.
Finally, if F 6= λ(t, x)A, λA 6= 0 or if F = λ(t, x)A, λA 6= 0, λt 6= 0 then
the invariance group of the corresponding wave equation (1) is generated
by the operator (3) with τ , ξ, η satisfying (6).
Case 3: Fux = Fu = 0, F 6= 0. If, in equation (1) we have Guxux 6=
0 or G = A(t, x, u)ux +B(t, x, u) with Au 6= 0 then the invariance group
is generated by operators( 3) with coefficients given by (6). This leaves
us with the case when equation (1) is of the form
utt = F (t, x)uxx +A(t, x)ux +B(t, x, u). (9)
It is well known from the general theory of partial differential equations
that there are invertible changes of coordinates
t′ = α(t, x), x′ = β(t, x),
D(t′, x′)
D(t, x)
6= 0
which transform the equation (9) either to an equation of hyperbolic
type
vt′t′ − vx′x′ = Ã(t′, x′)vx′ + C̃(t′, x′)vt′ + B̃(t′, x′, v) (10)
or to an equation of elliptic type
vt′t′ + vx′x′ = Ã(t′, x′)vx′ + C̃(t′, x′)vt′ + B̃(t′, x′, v). (11)
However from the point of view of the local analytic theory with analytic
coefficients, the elliptic type is equivalent to the hyperbolic type. Therefo-
re there exist corresponding transformations in the complex domain whi-
ch allow us to obtain equations (10) and (11) from each other. Further,
14 P. Basarab-Horwath, V. Lahno
the change of variables y = t′, z = x′, ω(y, z) = Λ(t′, x′)v, Λ 6= 0 where
Λ = exp
(
− 1
2
∫
C̃(t′, x′)dt′
)
transforms equation (10) into an equation of the form
ωyy = ωzz +H(y, z)ωz +R(y, z, ω).
This equation belongs to the class of equations (2) and therefore, as we
noted earlier, we exclude it from our considerations. From the analysis
carried out above, it now follows that, in order to solve our problem of
group classification, we need consider only the following cases:
utt = F (t, x, u, ux)uxx +G(t, x, u, ux), Fux 6= 0, (12)
utt = F (t, x, u, )uxx +G(t, x, u, ux), F 6= 0, Guxux 6= 0, (13)
utt = F (t, x, u)uxx +G(t, x, u, )ux +H(t, x, u), (14)
Fu 6= 0, F 6= λ(t, x)G, λG 6= 0,
utt = F (t, x, u)[H(t, x)uxx + ux] +G(t, x, u), (15)
Fu 6= 0, Ht 6= 0,
utt = F (t, x)uxx +G(t, x, u)ux +H(t, x, u), (16)
F 6= 0, Gu 6= 0,
utt = F (t, x, u)uxx +G(t, x, u), Fu 6= 0. (17)
The following result follows from the above considerations.
Proposition 1. The invariance groups of equations (12)–(16) are gene-
rated by vector fields of the form
Q = τ(t)∂t + ξ(x)∂x +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
∂u. (18)
Further, for equations (12) and (13) the functions τ , ξ, θ, η, F , G satisfy
the system of equations
2(ξx − τt)F − (τFt + ξFx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
)Fu =
=
[
θxu+ ηx + ux
(
1
2τt + θ − ξx
)]
Fux ,
1
2τtttu+ ηtt − [θxxu+ ηxx + ux(2θx − ξxx)]F −
(
3
2τt − θ
)
G−
− (τGt + ξGx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
)Gu =
=
[
θxu+ ηx + ux
(
1
2τt + θ − ξx
)]
Gux . (19)
Preliminary group classification 15
For equation (14) the functions τ , ξ, θ, η, F , G, H satisfy the system
of equations
(ξx − 2τt)G+ (ξxx − 2θx)F −
(
τGt + ξGx +[(
1
2τt + θ(x)
)
u+ η(t, x)
]
Gu
)
= 0,
2(ξx − τt)F −
(
τFt + ξFx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
])
Fu = 0,
1
2τtttu+ ηtt − [θxxu+ ηxx]F − (ηx + uθx)G−
(
3
2τt − θ
)
H −
−
(
τHt + ξHx +
[(
1
2τt + θ(x)
)
u+ η
])
Hu = 0. (20)
For equation (15) the functions τ , ξ, θ, η, F , G, H satisfy the system
of equations
[2(ξx − τt)H − τHt − ξHx]F −
−
{
τFt + ξFx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
Fu
}
H = 0,
(ξxx − 2θx)HF − (2τt − ξx)F −
−
{
τFt + ξFx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
Fu
}
= 0,
1
2τtttu+ ηtt − (ηxx + uθxx)HF − (uθx + ηx)F −
(
3
2τt − θ
)
G−
−
{
τGt + ξGx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
Gu
}
= 0. (21)
For equation (16) the functions τ , ξ, θ, η, F , G, H satisfy the system
of equations
(ξxx − 2θx)F − (2τt − ξx)G−
−
{
τGt + ξGx +
[(
1
2τt + θ(x)
)
u+ η(t, x)
]
Gu
}
= 0,
2(ξx − τt)F − τFt − ξFx = 0,
1
2τtttu+ ηtt − (ηxx + uθxx)F − (uθx + ηx)G−
(
3
2τt − θ
)
H −
−
{
τHt + ξHx +
[(
1
2
τt + θ(x)
)
u+ η(t, x)
]
Hu
}
= 0. (22)
The general infinitesimal operator Q of the invariance group of equation
(17) is given by
Q = τ(t)∂t + ξ(x)∂x +
[(
1
2 (τt + ξx) + k
)
u+ η(t, x)
]
∂u, (23)
where the functions τ , ξ, η, F , G and the constant k satisfy the system
of equations
2(ξx − τt)F −
(
τFt + ξFx +
[(
1
2 (τt + ξx)+
)
u+ η
]
Fu
)
= 0,
16 P. Basarab-Horwath, V. Lahno
ηtt + 1
2τtttu−
(
1
2ξxxxu+ ηxx
)
F −
(
3
2τt −
1
2ξx − k
)
G−
−
(
τGt + ξGx +
[(
1
2 (τt + ξx) + k
)
u+ η(t, x)
]
Gu
)
= 0. (24)
A direct calculation shows that when the equations (12)–(17) contain
arbitrary functions then the equations do not possess any symmetri-
es in the classical sense of Lie. In what follows, we shall carry out a
group classification up to equivalence under the equivalence group (which
we denote by E) of the equation under consideration. The equivalence
group E of a given equation consists of those transformations (of the
space V = R2 × R) t′ = T (t, x, u), x′ = X(t, x, u), v = U(t, x, u) which
preserve the form of the differential equation (1), that is, transformations
of the above type which transform (1) into an equation
vt′t′ = F̃ (t′, x′, v, vx′)vx′x′ + G̃(t′, x′, v, vx′).
In order to determine the transformations of E one may use the infini-
tesimal method [31] or the direct method. These calculations are long
but standard, and we give only their result.
Proposition 2. The equivalence group E of equations (12)–(16) consist
of the transformations
t′ = T (t), x′ = X(x), v = U(x)
√
|Tt|u+ Y (t, x) (25)
with the condition TtXxU 6= 0 and with Y being an arbitrary function.
The transformations of the equivalence group E of equation (17) consist
of the transformations
t′ = T (t), x′ = X(x), v = γ
√
|Tt|
√
|Xx|u+ Y (t, x) (26)
with γTtXx 6= 0, γ ∈ R, and
t′ = T (x), x′ = X(t), v = γ
√
|Tt|
√
|Xx|u+ Y (t, x) (27)
with γTtXx 6= 0, γ ∈ R, and in both cases Y is an arbitrary function.
We now pass to the description of those nonlinear equations which
are invariant under low-dimensional Lie algebras.
3. Invariance of equations under one-dimensional Lie algeb-
ras. As we showed above, when the functions F , G, H in equations (12)–
(17) are allowed to be completely arbitrary, then the equations do not
have any symmetry in the sense of Lie. For this reason we begin our classi-
fication of those equations which admit symmetries by looking at those
Preliminary group classification 17
which are invariant under one-parameter groups of local transformati-
ons (which is equivalent to being invariant under one-dimensional Lie
algebras). In order to do this, we first obtain all inequivalent realizations
of one-dimensional Lie algebras.
3.1. Realizations of one-dimensional Lie algebras.
Theorem 1. There exist transformations of the form (25) which trans-
form the operator (18) into one of the following forms:
Q = ∂t, Q = ∂x, Q = ∂t + ∂x,
Q = f(x)u∂u, Q = g(t, x)∂u, Q = ∂t + f(x)u∂u,
where f, g 6= 0.
Proof. Applying the change of coordinates (25) we transform the ope-
rator (18) into one of the form
Q̃ = τTt∂t′ + ξXx∂x′ +
{[
1
2ετ |T |
−1/2TttU + ξ|Tt|1/2Ux +
+
(
1
2τt + θ
)
|Tt|1/2U
]
u+ τYt + ξYx + η|Tt|1/2U
}
∂v, (28)
where ε = 1 if Tt > 0 and ε = −1 if Tt < 0. We now consider three cases:
(1) τ 6= 0, (2) τ = 0, ξ 6= 0, (3) τ = ξ = 0.
Case 1: τ 6= 0. Putting Tt = τ−1 in (25) we make the coefficient of
∂t′ equal to one. If we also have ξ 6= 0 then we may put Xx = ξ−1. We
also choose U to be a non-trivial solution of ξUx + θU = 0, and Y is
taken as a solution of the equation τYt + ξYx + η|τ |−1/2U = 0. In this
way we transform the operator given in (28) into the operator
Q̃ = ∂t′ + ∂x′ . (29)
If, however, ξ = 0, θ 6= 0 then, putting Y equal to a solution of
τYt + η|τ |−1/2U = 0,
the transformation (25) takes the operator Q into
Q̃ = ∂t′ + θ(x′)v∂v. (30)
If ξ = θ = 0 then, in the same way, we may transform Q into
Q̃ = ∂t′ . (31)
18 P. Basarab-Horwath, V. Lahno
Case 2: τ = 0, ξ 6= 0. Put Xx = ξ−1 in (25) and choose U to be a
non-trivial solution of ξUx + θU = 0 and we choose Y to be a solution
of ξYx + η|Tt|1/2U = 0. This then takes Q into the operator
Q̃ = ∂x′ . (32)
Case 3: τ = ξ = 0. If θ 6= 0 in (18) then we put T = t, Y = θ−1ηU
in (25) and the operator (28) becomes
Q̃ = θ(x′)v∂v. (33)
If we have θ = 0 in (18) then η 6= 0 and we obtain the operator
Q̃ = η̃(t′, x′)∂v. (34)
The forms of the operator Q̃ obtained in (29)–(34) is, apart from the
notation, that given in the statement of the theorem. All that remains
to be done is to verify that these different forms are inequivalent under
the action of transformations (25). We show this for the case of the
operators Q1 = ∂t and Q2 = ∂t + f(x)u∂u, and the other cases are
treated in the same way. First, we assume that there is a transformation
of the form (25) which takesQ1 into Q̃ = ∂t′+f̃(x′)v∂v. Then this implies
that we must have f = 0 which contradicts the requirement for Q2. This
completes the proof.
Theorem 2. There exist transformations of the type (26), (27) which
transform the operator (23) into one of the following:
Q = ∂t + ∂x, Q = ∂t, Q = ∂t + ∂x + u∂u,
Q = ∂t + u∂u, Q = u∂u, Q = g(t, x)∂u, g 6= 0.
The proof is carried out in the same way as that of Theorem 1.
It follows from these two theorems that, in the classes (18) and (23)
of operators, there exist six inequivalent (with respect to the equivalence
group of our partial differential equation) types of one-dimensional Lie
algebras A1 = 〈e1〉. We list these below, using a notation which we shall
use in the rest of this paper.
One-dimensional Lie algebras of operators of type (18).
A1
1 = 〈∂t + ∂x〉, A2
1 = 〈∂t〉, A3
1 = 〈∂x〉, A4
1 = 〈∂t + f(x)u∂u〉,
A5
1 = 〈f(x)u∂u〉, A6
1 = 〈g(t, x)∂u〉, f, g 6= 0.
Preliminary group classification 19
One-dimensional Lie algebras of operators of type (23).
Ã1
1 = 〈∂t + ∂x〉, Ã2
1 = 〈∂t〉, Ã3
1 = 〈∂t + ∂x + u∂u〉,
Ã4
1 = 〈∂t + u∂u〉, Ã5
1 = 〈u∂u〉, Ã6
1 = 〈g(t, x)∂u〉, g 6= 0.
3.2. Equations invariant under one-dimensional Lie algebras.
We now have to determine for each of the above realizations whether the
given operator is an admissible symmetry operator (that is, if there is
a wave equation which is invariant under a given realization). To do
this we use the system of determining equations Proposition 1. Since the
procedure of constructing A1-invariant equations reduces to integrating
systems of first-order partial differential equations, we do not give de-
tails: we merely make some remarks and then give a list of our results.
All the realizations Ai1 (i = 1, . . . , 6) are invariance algebras only for
equations of the form (12). For equations of the form (13), substituting
the values of the coefficients τ , ξ, θ, η in the realizations A5
1 and A6
1
into the system (19) leads to the equality Fu = 0, which contradicts the
condition placed on the equation. We arrive at the same result when we
examine these realizations for the equations (14) and (15). For equati-
on (15) we also find that the condition of its invariance under A2
1 leads to
Ht = 0, which contradicts the condition placed on the equation. Finally,
for equation (16) the invariance under A6
1 leads toGu = 0, as follows from
the second equation in (22). This contradicts the requirements placed on
the equation. The realizations Ã5
1 and Ã6
1 cannot be invariance algebras
of equation (17) since the first equation of the system (24) gives Fu = 0
which is a contradiction. Below we give a list of all A1-invariant equati-
ons, and we give the realization of the algebras A1 which is an invariance
algebra of the corresponding equation, and we give the corresponding
forms of the functions F , G, H.
A1-invariant equations of type (12).
A1
1 : F = F̃ (z, u, ux), G = G̃(z, u, ux), z = t− x, F̃ux 6= 0;
A2
1 : F = F̃ (x, u, ux), G = G̃(x, u, ux), F̃ux 6= 0;
A3
1 : F = F̃ (t, u, ux), G = G̃(t, x, ux), F̃ux 6= 0;
A4
1 : G = uG̃(x, v, ω)− f−1[f ′′u ln |u|+ 2f ′ux ln |u| −
− (f ′)2f−1u ln2 |u|]F̃ , F = F̃ (x, v, ω),
v = u exp (−tf), ω = u−1ux − f−1f ′ ln |u|, F̃ω 6= 0;
20 P. Basarab-Horwath, V. Lahno
A5
1 : G = uG̃(t, x, ω)− f−1[f ′′u ln |u|+ 2f ′ux ln |u| −
− (f ′)2f−1u ln2 |u|]F̃ , F = F̃ (t, x, ω), F̃ω 6= 0,
ω = u−1ux − f−1f ′ ln |u|;
A6
1 : F = F̃ (t, x, ω), G = G̃(t, x, ω)− g−1gxxuF̃ + g−1gttu,
ω = gux − gxu, F̃ω 6= 0.
A1-invariant equations of type (13).
A1
1 : F = F̃ (z, u), G = G̃(z, u, ux), z = t− x, F̃u 6= 0;
A2
1 : F = F̃ (x, u, ), G = G̃(x, u, ux), F̃u 6= 0;
A3
1 : F = F̃ (t, u, ), G = G̃(t, u, ux), F̃u 6= 0;
A4
1 : G = uG̃(x, v, ω)− f−1[f ′′u ln |u|+ 2f ′ux ln |u| −
− (f ′)2f−1u ln2 |u|]F̃ , F = F̃ (x, ω),
ω = u exp (−tf), v = u−1ux − f−1f ′ ln |u|, F̃ω 6= 0.
A1-invariant equations of type (14).
A1
1 : F = F̃ (z, u), G = G̃(z, u), H = H̃(z, u),
z = t− x, F̃u 6= 0, F̃ 6= λ(z)G̃, λ(z)G̃ 6= 0;
A2
1 : F = F̃ (x, u), G = G̃(x, u), H = H̃(x, u),
F̃u 6= 0, F̃ 6= λ(x)G̃, λ(x)G̃ 6= 0;
A3
1 : F = F̃ (t, u), G = G̃(t, u), H = H̃(t, u),
F̃u 6= 0, F̃ 6= λ(t)G̃, λ(t)G̃ 6= 0;
A4
1 : F = F̃ (x, ω), G = G̃(x, ω)− 2f ′f−1 ln |u|F̃ ,
H = uH̃(x, ω) + (f ′)2f−2u ln2 |u|F̃ − f ′f−1u ln |u|G̃−
− f ′′f−1u ln |u|F̃ , ω = u exp (−tf), F̃ω 6= 0;
if f ′ = 0 then F̃ 6= λ(x)G̃, λ(x)G̃ 6= 0.
A1-invariant equations of type (15).
A1
1 : F = F̃ (z, u), G = G̃(z, u), H = H̃(z),
z = t− x, F̃u 6= 0, H̃z 6= 0;
A3
1 : F = F̃ (t, u, ), G = G̃(t, u), H = H̃(t), F̃u 6= 0, H̃t 6= 0;
Preliminary group classification 21
A4
1 : F = F̃ (x, ω)[H̃(x)− 2tf ′], H = [H̃(x)− 2tf ′]−1,
G = etf G̃(x, ω) + uF̃
[
1
4 (H − 2tf ′)2 − tf ′′
]
,
ω = u exp (−tf), F̃ω 6= 0, f ′ 6= 0.
A1-invariant equations of type (16).
A1
1 : F = F̃ (z), G = G̃(z, u), H = H̃(z, u),
z = t− x, G̃u 6= 0;
A2
1 : F = F̃ (x), G = G̃(x, u), H = H̃(x, u), G̃u 6= 0;
A3
1 : F = F̃ (t), G = G̃(t, u), H = H̃(t, u), G̃u 6= 0;
A4
1 : F = F̃ (x), G = G̃(x, ω)− 2f ′f−1 ln |u|F̃ ,
H = uH̃(x, ω) + (f ′)2f−2u ln2 |u|F̃ − f ′f−1u ln |u|G̃−
− f ′′f−1u ln |u|F̃ , ω = u exp (−tf);
G̃ is arbitrary if f ′ 6= 0; G̃ω 6= 0 if f ′ = 0.
A5
1 : F = F̃ (t, x), G = G̃(t, x)− 2f ′f−1 ln |u|F̃ ,
H = uH̃(t, x)− f ′′f−1u ln |u|F̃ + (f ′)2f−2u ln2 |u|F̃ −
− f ′f−1u ln |u|G̃, f ′ 6= 0.
A1-invariant equations of type (17).
Ã1
1 : F = F̃ (z, u), G = G̃(z, u), z = t− x, F̃u 6= 0;
Ã2
1 : F = F̃ (x, u), G = G̃(x, u), F̃u 6= 0;
Ã3
1 : F = F̃ (z, ω), G = etG̃(z, ω),
z = t− x, ω = u exp (−t), F̃ω 6= 0;
Ã4
1 : F = F̃ (x, ω), G = etG̃(x, ω), ω = u exp (−t), F̃ω 6= 0.
We note that in the above lists, f ′ = df
dx , f
′′ = d2f
dx2 . Also, we note
that the Lie algebras given are the maximal invariance algebras of the
corresponding equations when the functions F̃ , G̃, H̃ are arbitrary.
4. Invariance of equations under two-dimensional Lie algeb-
ras. It is well-known (see [32]) that there are, up to isomorphism, only
two Lie algebras A2 = 〈e1, e2〉 of dimension two:
A2.1 : [e1, e2] = 0; A2.2 : [e1, e2] = e2.
22 P. Basarab-Horwath, V. Lahno
In order to obtain a full list of non-linear equations (1) which are invari-
ant under two-dimensional Lie algebras, we must first construct all possi-
ble inequivalent realizations of the Lie algebras A2.1 and A2.2 in the class
of operators (18) and (23). Then we must use the defining system of
equations (19)–(22) and (22) in order to pick out those realizations which
are invariance algebras for equations of the given type (1). To carry out
this construction, we are able to exploit the results of Theorems 1 and 2,
which allow us to choose one of the operators of the two-dimensional Lie
algebras in one of the canonical forms given in these results.
4.1. A2.1-invariant equations. The Lie algebra A2.1 is Abelian, so
we just add one more operator of the form (18) or (23) which commutes
with the first operator chosen from amongst the canonical forms. We do
this for the canonical form A1
1. In this case we put e1 = ∂t + ∂x, and we
let the operator e2 have the form (18). Then the commutation relation
[e1, e2] = 0 gives us
e2 = c1∂t + c2∂x + (c3u+ η(z))∂u, (35)
where c1, c2, c3 ∈ R, z = t−x. To give the operator (35) a canonical form
we use the subgroup Φ of the equivalence group E (given by (25)) which
maps e1 to λe′1with e′1 = ∂t′ + ∂x′ for some arbitrary choice of constant
λ 6= 0. This is allowed because the commutation relation [e1, e2] = 0 is
preserved. A straightforward calculation gives us the following form of
the allowed transformation:
t′ = λt+ λ1, x′ = λx+ λ2, v = λ3
√
|λ|u+ Y (z), (36)
where λ, λ1, λ2, λ3 ∈ R, z = t − x, λ, λ3 6= 0. Applying this transforma-
tion, we obtain the following possible forms for e2:
∂t, ∂x, ∂t + u∂u, ∂x + u∂u, u∂u, g(z)∂u,
with g 6= 0. The extra factor λ gives us flexibility in our calculations so
that no other arbitrary constants arise in the canonical forms for e2. We
then have the following forms for the algebra A2.1:
〈∂t, ∂x〉, 〈∂t + ∂x, u∂u〉, 〈∂t + ∂x, g(z)∂u〉, 〈∂t + ∂x, ∂t + u∂u〉,
where g 6= 0, z = t − x. Note that the two algebras 〈∂t + ∂x, ∂t〉 and
〈∂t+∂x, ∂x〉 are both the same as 〈∂t, ∂x〉. The other cases are treated in
Preliminary group classification 23
the same manner, and we find the following realizations of the algebras
Ai2.1 in the class of operators (18):
A1
2.1 = 〈∂t, ∂x〉; A2
2.1 = 〈∂t + ∂x, u∂u〉;
A3
2.1 = 〈∂t + ∂x, ∂t + u∂u〉;
A4
2.1 = 〈∂t + ∂x, g(z)∂u〉, z = t− x, g 6= 0; A5
2.1 = 〈∂t, ∂u〉;
A6
2.1 = 〈∂x, ∂t + u∂u〉; A7
2.1 = 〈∂x, u∂u〉; A8
2.1 = 〈∂x, ∂u〉;
A9
2.1 = 〈∂t, f(x)u∂u〉, f 6= 0;
A10
2.1 = 〈∂t + f(x)u∂u, e
tf∂u〉, f 6= 0;
A11
2.1 = 〈∂t + f(x)u∂u, h(x)u∂u〉, fh′ − f ′h 6= 0;
A12
2.1 = 〈g(t, x)∂u, h(t, x)∂u〉;
A13
2.1 = 〈f(x)u∂u, h(x)u∂u〉, fh′ − f ′h 6= 0.
In the realization of A12
2.1 the functions h, g are linearly independent
(with respect to at least one of the arguments).
The next step is to check whether these realizations can be invariance
algebras for equations (12)-(16). We find that all the realizations except
A13
2.1 are invariance algebras for equations of the form given in (12).
A2.1-invariant equations of the form (12).
A1
2.1 : F = F̃ (u, ux), G = G̃(u, ux), F̃ux 6= 0;
A2
2.1 : F = F̃ (z, ω), G = uG̃(z, ω), z = t− x,
ω = u−1ux, F̃ω 6= 0;
A3
2.1 : F = F̃ (v, ω), G = ezG̃(v, ω), v = u exp (−z),
ω = u−1ux, z = t− x, F̃ω 6= 0;
A4
2.1 : F = F̃ (z, ω), G = G̃(z, ω)− g−1g′′uF̃ + ug−1g′′,
g = g(z) 6= 0, z = t− x, ω = gux + g′u, F̃ω 6= 0;
A5
2.1 : F = F̃ (x, ux), G = G̃(x, ux), F̃ux 6= 0;
A6
2.1 : F = F̃ (v, ω), G = uG̃(v, ω),
v = u exp (−t), ω = u−1ux, F̃ω 6= 0;
A7
2.1 : F = F̃ (t, ω), G = uG̃(t, ω), ω = u−1ux, F̃ω 6= 0;
A8
2.1 : F = F̃ (t, ux), G = G̃(t, ux), F̃ux 6= 0;
A9
2.1 : F = F̃ (x, ω), ω = u−1ux − f−1f ′ ln |u|,
24 P. Basarab-Horwath, V. Lahno
F̃ω 6= 0, G = uG̃(x, ω)− f−1[f ′′u ln |u|+ 2f ′ux ln |u| −
− (f ′)2f−1u ln |u|]F̃ ;
A10
2.1 : (1) f = 1, F = F̃ (x, ω), G = u+ etG̃(x, ω),
ω = e−tux, F̃ω 6= 0;
(2) f = x, F = F̃ (x, ω), F̃ω 6= 0, ω = [u−1ux −
− x−1 ln |u| − t−2x ln |u| − 2t+ x−1]u exp (−tx),
G = etxG̃(x, ω) + ux2 + u[−4t3x ln |u| − t4x2 ln2 |u| − 5t2−
− 2tu−1ux ln |u|+ 2tx−1 ln2 |u|+ 2t3x ln2 |u|+ 6t2 ln |u| −
− 2tx−1 ln |u|+ 6tx−1 − 2x−2 − 2x−1u−1ux ln |u|+
+ x−2 ln2 |u|]F̃ ;
A11
2.1 : (1) f = 1, h′ 6= 0, F = F̃ (x, ω), F̃ω 6= 0,
ω = u−1ux + (h7 − 1)h′t ln |u|,
G = uG̃(x, ω) + h−1u[th′′ ln |u|+ 2h−1(h′)2u− 2h′ux]F̃ ;
(2) f = x, h 6= 0, λx (λ 6= 0),
F = F̃ (x, ω), ω = u−1ux − (x−1 − txh′h−1 + t) ln |u|,
G = uG̃(x, ω) + h−1u[(2x−1h′ − h−1(h′)2 − hx−2) ln |v| −
− h′′ + 2(x−1h− h′)ω] ln |v|F̃ + [x−2u ln2 |u| −
− 2x−1ux ln |u|]F̃ , v = u exp (−tx), F̃ω 6= 0;
A12
2.1 : g = 1, h = t, F = F̃ (t, x, ux), G = G̃(t, x, ux),
F̃ux 6= 0.
In constructing those A2.1-invariant equations of the form (13) and (14),
we have used the fact that there are no realizations of A5
1 and A6
1 which
can be symmetry algebras. This allows us to shorten the list of reali-
zations of the algebras of type A2.1 to just three: A1
2.1, A3
2.1, A6
2.1. All
these algebras are in fact symmetry algebras of equations of type (13)
and (14).
A2.1-invariant equations of the type (13).
A1
2.1 : F = F̃ (u), G = G̃(u, ux), F̃u 6= 0;
A3
2.1 : F = F̃ (v), G = ezG̃(v, ω), F̃v 6= 0,
v = u exp (−z), z = t− x, ω = u−1ux;
Preliminary group classification 25
A6
2.1 : F = F̃ (ω), G = uG̃(v, ω), F̃ω 6= 0,
ω = ue−t, v = u−1ux.
A2.1-invariant equations of the type (14).
A1
2.1 : F = F̃ (u), G = G̃(u), H = H̃(u),
F̃u 6= 0, F̃ 6= G̃, G̃ 6= 0;
A3
2.1 : F = F̃ (ω), G = G̃(ω), H = ezH̃(ω),
F̃ω 6= 0, F̃ 6= G̃, G̃ 6= 0,
ω = u exp (−z), z = t− x, ω = u−1ux;
A6
2.1 : F = F̃ (ω), G = uG̃(ω), H = uH̃(ω),
F̃ω 6= 0, F̃ 6= G̃, G̃ 6= 0, ω = ue−t.
There are no A2.1-invariant equations of type (15). First, there are none
which are invariant under A2
1, A5
1, A6
1. This then narrows the number of
those which are left in the list of A2.1 algebras down to just two: A3
2.1
and A6
2.1. Then we use the system of defining equations (21) and find
that requiring A3
2.1 invariance leads to H̃z = 0, which contradicts the
conditions of (15); the defining system (21) also leads to f ′ = 0 when
testing the algebra A6
2.1, and this contradicts the condition on f . The
same type of procedure is used for examining the symmetry algebras
for equation (16): there are just five such equations which are invariant
under A2.1.
A2.1-invariant equations of type (16).
A1
2.1 : F = λ, G = G̃(u), H = H̃(u), G̃′ 6= 0, λ ∈ R∗;
A3
2.1 : F = λ, G = G̃(ω), H = ezH̃(ω), G̃′ 6= 0,
λ ∈ R∗, z = t− x, ω = ue−z;
A6
2.1 : F = λ, G = G̃(ω), H = etH̃(ω), G̃′ 6= 0,
λ ∈ R∗, ω = ue−t;
A9
2.1 : F = F̃ (x), G = G̃(x)− 2f ′f−1F̃ ln |u|,
H = uH̃(x)− f ′′f−1u ln |u|F̃ + (f ′)2f−2u ln2 |u|F̃ −
− f ′f−1u ln |u|G̃, F̃ 6= 0, f ′ 6= 0;
A11
2.1 : (1) f = 1, h′ 6= 0; F = F̃ (x) 6= 0,
26 P. Basarab-Horwath, V. Lahno
G = G̃(x) + 2h′h−1t ln |u|F̃ ,
H = uH̃(x) + t2h−2u ln2 |u|F̃ + th′′h−1u ln |u|F̃ +
+ th′h−1u ln |u|G̃;
(2) f = x, h 6= 0, F = F̃ (x) 6= 0,
G = G̃(x)− 2h−1(x−1h− h′)tx ln |u|F̃ − 2x−1 ln |u|F̃ ,
H = uH̃(x) + x−2u ln2 |u|F̃ − x−1u ln |u|G̃+
+ 2th−1(x−1h− h′)u ln2 |u|F̃ + t2u ln2 |u|F̃ −
− tu ln |u|G̃− x2t3h′h−1u ln2 |u|F̃ − txh−1h′′u ln |u|F̃ −
− t2x2(h′)2h−2u ln2 |u|F̃ + txh−1h′u ln |u|G̃.
We are now left with the case of A2.1-invariant equations of type (17).
Having constructed realizations of A2.1 in the class of operators (23),
and then testing them as symmetry algebras for equations of type (17),
we find only three algebras:
Ã1
2.1 = 〈∂t, ∂x〉, Ã2
2.1 = 〈∂t, ∂x + u∂u〉,
Ã3
2.1 = 〈∂t + u∂u, ∂x + u∂u〉.
Ã2.1-invariant equations of type (17).
Ã1
2.1 : F = F̃ (u), G = G̃(u), F̃ ′ 6= 0;
Ã2
2.1 : F = F̃ (ω), G = exG̃(ω), F̃ ′ 6= 0, ω = ue−x;
Ã3
2.1 : F = F̃ (ω), G = e(t+x)G̃(ω), F̃ ′ 6= 0, ω = ue−(t+x).
4.2. A2.2-invariant equations. As in the case of the A2.1-invariant
equations, we must first construct all possible inequivalent algebras A2.2
in the classes of operators (18) and (23) which do not contain Ã5
1 or
Ã6
1 as subalgebras (or algebras equivalent to them). In carrying out our
construction, we begin with the results of Theorems 1 and 2, according
to which we choose one of the basis operators of A2.2 (we choose the basis
operator e2 for this) in one of the canonical forms given in these theorems.
The calculations are similar to those involved in the construction of the
algebras A2.1 so we do not dwell on the calculations for this case, and
we merely give the list of realizations.
Realizations of the algebras A2.2 in the classes of opera-
tors (18).
A1
2.2 = 〈−t∂t − x∂x, ∂t + ∂x〉;
Preliminary group classification 27
A2
2.2 = 〈−t∂t − x∂x + ku∂u, ∂t + ∂x〉;
A3
2.2 = 〈−t∂t + ku∂u, ∂t〉; A4
2.2 = 〈−t∂t + xu∂u, ∂t〉;
A5
2.2 = 〈−t∂t, ∂t〉; A6
2.2 = 〈−t∂t + ∂u, ∂t〉;
A7
2.2 = 〈−t∂t − x∂x, ∂t〉; A8
2.2 = 〈−x∂x + ku∂u, ∂x〉 (k 6= 0);
A9
2.2 = 〈−x∂x + ∂u, ∂x〉; A10
2.2 = 〈−x∂x, ∂x〉;
A11
2.2 = 〈−t∂t − x∂x, ∂x〉;
A12
2.2 = 〈−t∂t − x∂x + ku∂u, ∂x〉 (k 6= 0);
A13
2.2 = 〈−t∂t + x∂x, ∂t + xu∂u〉; A14
2.2 = 〈x∂x, xu∂u〉;
A15
2.2 = 〈t∂t + x∂x, xu∂u〉;
A16
2.2 = 〈∂t + ∂x, e
tg(z)∂u〉, z = t− x, g 6= 0;
A17
2.2 = 〈∂t, et∂u〉; A18
2.2 = 〈∂x, ex∂u〉;
A19
2.2 = 〈∂t + u∂u, e
2t∂u〉; A20
2.2 = 〈∂t + xu∂u, e
(1+x)t∂u〉;
A21
2.2 = 〈−u∂u, ∂t, g(t, x)∂u〉, g 6= 0.
Realizations of the algebras A2.2 in the class of operators
(23). In this list we do not include those realizations which contain one-
dimensional subalgebras equivalent to Ã5
1, Ã6
1. We have:
Ã1
2.2 = 〈−t∂t − x∂x +mu∂u, ∂t + ∂x〉, m ∈ R;
Ã2
2.2 = 〈−t∂t + ∂u, ∂t〉; Ã3
2.2 = 〈−t∂t +mu∂u, ∂t〉;
Ã4
2.2 = 〈−t∂t − x∂x +mu∂u, ∂t〉, m ∈ R.
We remark that when we construct A2.2-invariant equations, we put
k = m + 1 in the system (24) for the realizations of Ã1
2.2, Ã3
2.2, Ã4
2.2.
Further, we use only those algebras which do not contain subalgebras
equivalent to A2
1, A5
1, A6
1. There are eight such algebras: A1
2.2, A2
2.2, A8
2.2,
A9
2.2, A10
2.2, A11
2.2, A12
2.2, A13
2.2. Then, substituting A8
2.2, A9
2.2, A10
2.2 into (21),
we find that HF = 0, which is a contradiction. All the other algebras
are invariance algebras of equations of the form (15).
A2.2-invariant equations of type (15).
A1
2.2 : F = z−1F̃ (ω), H = λz G = |z|−3/2G̃(ω),
F̃ ′ 6= 0, z = t− x, ω = |z|−1/2u, λ 6= 0,
A2
2.2 : F = z−1F̃ (ω), H = λz G = |z|k−3/2G̃(ω),
28 P. Basarab-Horwath, V. Lahno
F̃ ′ 6= 0, z = t− x, ω = |z|k−1/2u, λk 6= 0,
A11
2.2 : F = t−1F̃ (ω), H = λt G = |t|−3/2G̃(ω),
F̃ ′ 6= 0, ω = |t|−1/2u, λ 6= 0,
A12
2.2 : F = t−1F̃ (ω), H = λt G = |t|k−3/2G̃(ω),
F̃ ′ 6= 0, ω = |t|k−1/2u, λk 6= 0,
A13
2.2 : F = x3(λ− 2tx)F̃ (ω), H = x8λ− 2tx]−1,
G = etx|x|3/2G̃(ω) + 1
4x
2u(λ− 2tx)2F̃ ,
F̃ ′ 6= 0, ω = |x|1/2ue−tx, λ ∈ R.
A2.2-invariant equations of type (17).
Ã1
2.2 : F = F̃ (ω), G = |z|−(m+2)G̃(ω),
F̃ ′ 6= 0, z = t− x, ω = |z|mu, m ∈ R,
Ã2
2.2 : F = e2uF̃ (x), G = e2uG̃(x), F̃ 6= 0,
Ã3
2.2 : F = |u|4/(2m+1)F̃ (x), G = |u|(5+2m)/(2m+1)G̃(x),
F̃ 6= 0, m 6= − 1
2 ,
Ã4
2.2 : F = F̃ (ω), G = |x|−(m+2)G̃(ω),
F̃ ′ 6= 0, ω = |x|m, m ∈ R.
Conclusion. It is clear from the above results that the successive in-
crease in dimension of the Lie algebra of invariance of the given equation
leads to a corresponding decrease in arbitrariness in the functions ente-
ring into the equation. This then allows us, at a certain stage, to use the
standard methods to obtain a complete solution of the problem of group
classification of equations of type (1). In particular, for equations of the
form (15) and (17), the A2-invariant equations contain arbitrary func-
tions of one variable, which then allows us to use the Lie–Ovsiannikov
method in order to obtain a complete list of equations of this type, and
whose algebras of invariance are solvable Lie algebras.
V. Lahno thanks the Swedish Research Council (Vetenskapsr̊adet)
for financial support (grant 624-2004-1073) during this research. He
also thanks the Mathematics Department of Linköping University for
its hospitality during his stay.
Preliminary group classification 29
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|
| id | oai:trim.imath.kiev.ua:article-378 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:22Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/1f/e138d47da4bd9adbeedc6b5ce7cd631f.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3782020-08-13T10:42:30Z Preliminary group classification of the general quasi-linear wave equation Попередня групова класифікація загального квазілінійного хвильового рівняння Basarab-Horwath, P. Lahno, V. Басараб-Хорват, П. Лагно, В. We begin the group classification of quasi-linear second-order wave equations of the most general form. We find the canonical forms for the symmetry operators which generate the invariance group of the equation, as well as the equivalence group, and we describe those equations which admit one- and two-dimensional invariance groups. Розпочато групову класифікацію квазілінійного хвильового рівняння другого порядку найбільш загальної форми. Знайдено канонічні форми операторів симетрії, які генерують групу інваріантності рівняння, описано перетворення еквівалентності та рівняння, що допускають одно- та двовимірні групи інваріантності. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/378 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); 9-30 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); 9-30 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); 9-30 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/378/475 Авторське право (c) 2006 П. Басараб-Хорват, В. Лагно http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Basarab-Horwath, P. Lahno, V. Басараб-Хорват, П. Лагно, В. Preliminary group classification of the general quasi-linear wave equation |
| title | Preliminary group classification of the general quasi-linear wave equation |
| title_alt | Попередня групова класифікація загального квазілінійного хвильового рівняння |
| title_full | Preliminary group classification of the general quasi-linear wave equation |
| title_fullStr | Preliminary group classification of the general quasi-linear wave equation |
| title_full_unstemmed | Preliminary group classification of the general quasi-linear wave equation |
| title_short | Preliminary group classification of the general quasi-linear wave equation |
| title_sort | preliminary group classification of the general quasi-linear wave equation |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/378 |
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