Класифікація симетрійних властивостей системи рівнянь хемотаксису
В данiй роботi проведена повна групова класифiкацiя систем рiвнянь хемотаксису. The full group classification of the systems of chemotaxis equations is performed.
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Інститут прикладної математики і механіки НАН України
2008
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| Cite this: | Класифікація симетрійних властивостей системи рівнянь хемотаксису / М.І. Сєров, О.М. Омелян // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 536 – 562. — Бібліогр.: 26 назв. — укр. |
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| author | Сєров, М.І. Омелян, О.М. |
| author_facet | Сєров, М.І. Омелян, О.М. |
| citation_txt | Класифікація симетрійних властивостей системи рівнянь хемотаксису / М.І. Сєров, О.М. Омелян // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 536 – 562. — Бібліогр.: 26 назв. — укр. |
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| description | В данiй роботi проведена повна групова класифiкацiя систем рiвнянь хемотаксису.
The full group classification of the systems of chemotaxis equations is performed.
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Ukrainian Mathematical Bulletin
Volume 5 (2008), № 4, 529 – 557 UMB
Classification of symmetry properties of a
system of chemotaxis equations
Mykola I. Serov, Oleksandr M. Omelyan
(Presented by A. M. Samoilenko)
Abstract. The full group classification of the systems of chemotaxis
equations is performed.
2000 MSC. 35K57, 58D19.
Key words and phrases. Group classification, invariance algebra,
systems of chemotaxis equations.
1. Introduction
In modern biophysical researches, the processes of symmetric prop-
agation of bacterial population waves, when chemotaxis rings keep a
sharply outlined form and move with a constant speed depending on the
mobility of bacteria and their chemotaxis properties, are well described
by the mathematical models based on the Keller–Segel’s equations [14]
St = DSSxx + k1g(S)b,
bt = −ν∂x[bχ(S)Sx] +Dbbxx + k2g(S)b,
(1.1)
where St = ∂S
∂t ,Sx = ∂S
∂x ,bt = ∂b
∂t , Sxx = ∂2S
∂x2 , bxx = ∂2b
∂x2 ,∂x = ∂
∂x , and
S(t, x) is the concentration of a substrate-attractant which is consumed
by bacteria, b(t, x) is the density of bacteria, g(S) is the specific growth
rate of bacteria, χ(S) is a function of the chemotaxis answer, DS and
Db are diffusion coefficients of a substrate and bacteria, respectively; ν,
k1, and k2 are constants; and t and x are the time and spatial vari-
ables, respectively. The Keller–Segel’s model and its some modifications
Received 10.10.2008
ISSN 1812 – 3309. c© Institute of Mathematics of NAS of Ukraine
530 Classification of symmetry properties...
describe the formation and propagation of Adler’s chemotaxis rings [1]
and different processes of structurization in bacterial colonies at their
interaction [13]. We rewrite system (1.1) in the designations usual in
mathematical researches, having generalized it as follows:
(
u1
u2
)
0
= ∂1
[(
λ1 0
f(u1)u2 λ2
)(
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
. (1.2)
Here, g1(u1, u2), g2(u1, u2), f(u1) are arbitrary smooth functions of their
arguments, and f 6= 0, λ1 > 0, λ2 > 0, ua = ua(x0, x1), a = 1, 2, x0 is the
time variable, x1 is the spatial variable, and the subscripts denote the
differentiation with respect to the corresponding independent variable.
We note that system (1.2) is a special case of the system of nonlinear
equations of a diffusion reaction
(
u1
u2
)
0
= ∂1
[
F (u1, u2)
(
u1
u2
)
1
]
+G(u1, u2), (1.3)
where
F (u1, u2) =
(
f11 f12
f21 f22
)
, G(u1, u2) =
(
g1
g2
)
,
fab = fab(u1, u2), ga = ga(u1, u2), a; b = 1; 2. The symmetry properties
of the equation of a diffusion-convection reaction
u0 = ∂1(f(u)u1) + g(u)u1 + h(u) (1.4)
were considered in a number of works. For example, the symmetry prop-
erties of Eq. (1.4) at g(u) = h(u) = 0 and g(u) = 0 were classified,
respectively, in works [21] and [8]. The full description of symmetries at
arbitrary values of the functions f(u), g(u), and h(u) to within equiv-
alence transformations was done in works [5] and [7]. The symmetry
analysis of the second-order evolutionary equation of a general form
u0 = F (x0, x1, u, u1, u11) (1.5)
was performed in works [15, 16, 25, 26]. The Galilei invariance of system
(1.3) was investigated in works [2,3,10]. The Lie and conditional symme-
try of system (1.3) in the case of a diagonal matrix F was investigated in
work [6].
M. I. Serov, O. M. Omelyan 531
In the given work, we will pose the following problem: to investigate
the symmetry properties of system (1.2) depending on the values of the
functions f(u1), g1(u1, u2), g2(u1, u2) and the constants λ1, λ2. We note
that, at f = 0, the symmetry properties of system (1.2) were investigated
in works [4, 17,18]; therefore, we consider further that f 6= 0.
2. Symmetry kernel and necessary conditions
for its extension
To study the symmetry properties of system (1.2), we will use the Lie
algorithm [9,11,19,22,23].
By acting with the infinitesimal operator extension
X = ξµ∂µ + ηa∂ua , (2.1)
where ξµ = ξµ(x0, x1, u
1, u2), ηa = ηa(x0, x1, u
1, u2), µ = 0, 1, a = 1, 2 on
system (1.2), transiting to a manifold, and splitting the obtained system
by the derivatives of the functions ua, we obtain a determining system to
find coordinates of the infinitesimal operator (2.1) and the functions f ,
g1, and g2. The determining system consists of three subsystems:
S1(ξ, η) = 0, S2(ξ, η, f) = 0, S3(ξ, η, f, g
1, g2) = 0.
The system S1 = 0 is a system of differential equations only for the
functions ξµ and ηa
ξ01 = ξµ
ua = η1
u2 = ηa
ubuc = 0, a, b, c = 1; 2,
ξ00 = 2ξ11 , 2λ1η
1
1u1 = −ξ10 .
(2.2)
The system S2(ξ, η, f) = 0 connects the coordinates of the infinitesimal
operator ξµ, ηa and the function f(u1) with one another and looks like
η1ḟ +
(
η1
u1 − η2
u2 −
1
u2
η2
)
f +
1
u2
(λ2 − λ1)η
2
u1 = 0,
u2η1
1 ḟ +
(
u2η1
1u1 +
1
2
η2
1
)
f + λ2η
2
1u1 = 0,
η1
1f + 2λ2η
2
1u2 = −ξ10 .
(2.3)
The system S3(ξ, η, f, g
1, g2) = 0 consists of two differential equations
532 Classification of symmetry properties...
η1g1
u1 + η2g1
u2 = (η1
u1 − ξ00)g
1 + η1
u2g
2 + η1
0 − λ1η
1
11,
η1g2
u1 + η2g2
u2 = (η2
u2 − ξ00)g
2 + η2
u1g
1 + η2
0 − λ2η
2
11 − u2fη1
11
(2.4)
which connect the functions g1, g2 and the functions ξµ, ηa, f with one
another.
Remark 2.1. Considering f, g1, g2, λ1, λ2 as arbitrary in systems (2.2),
(2.3), (2.4), we obtain
ξ0 = d0, ξ1 = d1, η1 = η2 = 0, (2.5)
where d0, d1 are arbitrary constants. In this case, operator (2.1) looks
like
X = d0∂0 + d1∂1. (2.6)
Operator (2.6) generates the algebra
A0 = 〈∂0, ∂1〉 (2.7)
named as the symmetry kernel of system (1.2).
Let us investigate, for which values of the functions f, g1, and g2, the
symmetry of system (1.2) is wider than that of the algebra A0. The nec-
essary conditions for the symmetry extension are given by the following
proposition.
Theorem 2.1. If system (1.2) admits the extension of the symmetry
kernel A0, then the function f(u1) takes one of the following representa-
tions:
1. f = f(u1),
2. f = λ,
3. f = λ
u1 ,
4. f = λ1−λ2
u1 ,
5. f = 2λ1
u1 ,
where ϕ(u1) is an arbitrary smooth function, and λ is an arbitrary con-
stant.
M. I. Serov, O. M. Omelyan 533
Proof. To prove the theorem, we solve the system of determining equa-
tions which consists of the systems S1 = 0 and S2 = 0.
The general solution of system S1(ξ, η) is the functions
ξ0 = 2A(x0),
ξ1 = Ȧ(x0)x1 +B(x0),
η1 = −
1
2λ1
[1
2
Ä(x0)x
2
1 + Ḃ(x0)x1 + C(x0)
]
u1 + β1(x0, x1),
η2 = α21(x0, x1)u
1 + α22(x0, x1)u
2 + β2(x0, x1),
where A,B,C, α2a, and βa are arbitrary smooth functions of their argu-
ments.
Due to the joint solution of the first and third equations of system
(2.3), the conditions α21
1 = α22
1 = β2
1 = 0 are obtained. Then the system
S2 = 0 takes the form
(α11u1 + β1)ḟ = −α11f,
(α11
1 u
1 + β1
1)f = 2λ1α
11
1 ,
(α21u1 + β2)f = (λ1 − λ2)α
21.
(2.8)
The solution of system (2.8) leads to the appearance of 5 nonequivalent
representations of the function f which are given in the formulation of
the theorem.
Let us consider each of these cases separately. We will show that, at
the specified values of the function f(u1), the extension of the symmetry
of system (1.2) as compared with that of A0 is possible.
1. Let f = f(u1) be an arbitrary smooth function. System (2.8)
yields
ξ10 = αa
1 = βa = 0. (2.9)
In view of (2.9), we obtain
ξ0 = 2c1x0 + d0, ξ1 = c1x1 + d1,
η1 = 0, η2 = α22(x0)u
2,
(2.10)
where α22(x0) is an arbitrary smooth function, c1, d0, and d1 are arbitrary
constants. By comparing formulas (2.5) and (2.10), it is easy to see the
possibility to extend symmetry (2.7).
534 Classification of symmetry properties...
In cases 2–5, the possibility to extend symmetry (2.7) is similarly
proved. Not repeating these reasonings, we present the final form of co-
ordinates of the infinitesimal operator for each of the indicated functions
f(u1).
2. For f = λ, the coordinates of the infinitesimal operator (2.1) look
like
ξ0 = 2c1x0 + d0, ξ1 = c1x1 + d1,
η1 = β1(x0), η2 = α22(x0)u
2,
(2.11)
where β1(x0) is an arbitrary smooth function.
3. For f = λ
u1 (λ is an arbitrary constant), system (2.8) yields
ξ0 = 2c1x0 + d0, ξ1 = c1x1 + d1,
η1 = α1(x0)u
1, η2 = α22(x0)u
2,
(2.12)
where α1(x0) is an arbitrary smooth function.
4. For f = λ1−λ2
u1 , we obtain
ξ0 = 2c1x0 + d0, ξ1 = c1x1 + d1,
η1 = α1(x0)u
1, η2 = α21(x0)u
1 + α22(x0)u
2,
(2.13)
where α21(x0) is an arbitrary smooth function.
5. For f = 2λ1
u1 , we have
ξ0 = 2A(x0), ξ1 = Ȧ(x0)x1 +B(x0),
η1 = α1(x0)u
1, η2 = α22(x0)u
2,
(2.14)
where α1(x0) = − 1
2λ1
[12Ä(x0)x
2
1 + Ḃ(x0)x1 + C(x0)], A(x0), B(x0), and
C(x0) are arbitrary smooth functions. The theorem is proved.
Lemma 2.1. System (1.2) has a group of continuous transformations of
equivalence which are set by the following formulas for the coordinates of
the equivalence operator E:
ξ0 = 2c1x0 + c2, ξ1 = c1x1 + c3,
η1 = c4u
1 + c5, η2 = c6u
1 + c7u
2.
(2.15)
M. I. Serov, O. M. Omelyan 535
Here, c1, c2, c3, c4, c5, c6, and c7 are arbitrary constants which depend on
the form of the function f and take the following values:
1) at f = f(u1), c6 = 0;
2) at f = λ, c4 = c6 = 0;
3) at f = λ
u1 , c5 = c6 = 0;
4) at f = λ1−λ2
u1 , c5 = 0;
5) at f = 2λ1
u1 , c5 = c6 = 0.
Proof. To proof the lemma, we will apply the algorithm of search for the
equivalence transformations (see, e.g., [12, 16,22]).
The form of the equivalence operator E depends on the form of the
function f .
1. If f = f(u1) is an arbitrary smooth function, then we will search
the operator E in the form
E = ξµ∂µ + ηa∂ua + ζ∂f + τa∂ga . (2.16)
Acting by the operator E on system (1.2) and on the additional condi-
tions
∂f
∂xµ
=
∂f
∂u2
=
∂ga
∂xµ
= 0, (2.17)
we obtain a system of determining equations for the coordinates of op-
erator (2.16) ξµ, ηa, ζ, and τa:
ξ01 = ξ10 = ξµ
ua = η1
u2 = ηa
ubuc = ηa
µ = η2
u1 = 0, a, b, c = 1; 2,
ξ00 = 2ξ11 , u2η2
u2 − η2 = 0,
(2.18)
ζ = η1
u1f, τ1 = (η1
u1 − ξ00)g
1, τ2 = (η2
u2 − ξ00)g
2. (2.19)
The general solution of system (2.18) looks like (2.15). Equalities (2.19)
under conditions (2.15) can be written as follows:
ζ = c4f, τ1 = (c4 − 2c1)g
1, τ2 = (c6 − 2c1)g
2.
536 Classification of symmetry properties...
2. If f(u1) is not arbitrary and is set by one of the formulas for f
in cases 2)–5) in the lemma statement, we will search the infinitesimal
operator of equivalence E in the form
E = ξµ∂µ + ηa∂ua + τa∂ga . (2.20)
Acting with the extension of the operator E on system (1.2) and on the
additional conditions
∂ga
∂xµ
= 0 (2.21)
and applying the algorithm [16], we find a system of determining equa-
tions for the coordinates of the operator (2.20) ξµ, ηa, and τa:
ξ01 = ξ10 = ξµ
ua = η1
u2 = ηa
ubuc = ηa
µ = η1
u2 = 0,
a, b, c = 1, 2, µ = 0, 1
ξ00 = 2ξ11 , u2η2
u2 − η2 = 0,
(2.22)
τ1 = (η1
u1 − ξ00)g
1, τ2 = η2
u1g
1 + (η2
u2 − ξ00)g
2,
η1ḟ + fη1
u1 = 0, (u2η2
u2 − η2)f = (λ2 − λ1)η
2
u1 .
(2.23)
The general solution of Eqs. (2.22) are the functions
ξ0 = 2c1x0 + c2, ξ1 = c1x1 + c3,
η1 = c4u
1 + c5, η2 = c6u
1 + c7u
2.
(2.24)
By substituting (2.24) in (2.23), we obtain
τ1 = (c4 − 2c1)g
1, τ2 = c6g
1 + (c7 − 2c1)g
2, (2.25)
(c4u
1 + c5)ḟ + c4f = 0, c6(u
1f − λ1 + λ2) = 0. (2.26)
Solving Eq. (2.26), we come to cases 2)–5) of the lemma. The lemma is
proved.
Remark 2.2. Besides the equivalence transformations which are ob-
tained in Lemma (2.1), the other equivalence transformations we name
additional take place for more exactly specified functions f and g. Addi-
tional equivalence transformations will be presented in what follows for
the function f of a specific form.
M. I. Serov, O. M. Omelyan 537
3. Classification of the symmetry properties
of system (1.2) in the case of
an arbitrary function f(u1)
We now consider the system
(
u1
u2
)
0
= ∂1
[(
λ1 0
u2f(u1) λ2
) (
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
(3.1)
and will classify its symmetry properties depending on the form of the
functions ga(u1;u2) at any function f(u1).
Remark 3.1. It follows from Lemma 2.1 that the basic group of equiv-
alence transformations of system (3.1) looks like
x0 = te2θ2 + θ0, x1 = xeθ2 + θ1,
u1 = w1eθ3 + θ5, u2 = w2eθ4 .
Besides the basic group of equivalence, system (3.1) for specific g admits
some additional equivalence transformations, for example
x0 = at, x1 = bx, u1 = w1, u2 = w2emt,
where a, b, and m are arbitrary constants.
In view of Remark 3.1, we will formulate theorems on the maximal
algebra of invariance of system (3.1) to within the indicated equivalence
transformations.
The following proposition is true.
Theorem 3.1. If system (3.1) admits an extension of the symmetry
kernel A0, the functions g1 and g2 are set by one of the formulas
g1 = ϕ1(u1), g2 = u2[ϕ2(u1) + λ3 lnu2]; (3.2)
g1 = (u2)mϕ1(u1), g2 = (u2)m+1ϕ2(u1), (3.3)
where λ3 and m are arbitrary constants, and ϕ1(u1) and ϕ2(u1) are ar-
bitrary smooth functions.
538 Classification of symmetry properties...
Proof. Taking formulas (2.10) into account, system (2.4) can be written
as follows:
α22(x0)u
2g1
u2 = −2c1g
1,
α22(x0)u
2g2
u2 = (α22(x0) − 2c1)g
2 + α̇22(x0)u
2.
(3.4)
At arbitrary g1 and g2, system (3.4) implies that α22(x0) = c1 = 0. With
regard for formulas (2.10), we obtain that, in this case, the maximal
algebra of invariance of system (3.1) is the algebra A0.
Let us determine now the functions g1 and g2, for which system (3.1)
admits an extension of the symmetry kernel A0. For this purpose, it is
necessary, as follows from (3.4), that the functions g1 and g2 be solutions
of the structural system (see [11])
æu2g1
u2 = mg1,
æu2g2
u2 = (m+ æ)g2 + λ3u
2,
(3.5)
where æ = {0; 1}; m and λ3 are arbitrary constants. System (3.5) at
æ = 1 is connected with system (3.4) by the conditions
mα22(x0) = −2c1, λ3α
22 = α̇22. (3.6)
The solution of system (3.5) at æ = 1 depends on the constant m. Two
essentially different cases are possible.
1. m = 0. The general solution of system (3.5) looks like (3.2), where
λ3 6= 0, ϕ1 and ϕ2 are arbitrary smooth functions.
2. m 6= 0. It follows from the differential consequences of conditions
(3.6) that α̇22 = λ3 = 0. Then the general solution of system (3.5)
looks like (3.3).
At æ = 0, we get from system (3.5) that the extension of a symmetry
kernel A0 occurs only at g1 = g2 = 0, which is a particular case of
formulas (3.2), (3.3). The theorem is proved.
Remark 3.2. Since formulas (3.2), (3.3) coincide at λ3 = m = 0, we
set λ3 6= 0 in formulas (3.2) in order to avoid their coincidence, while
studying the symmetry properties of system (3.1).
M. I. Serov, O. M. Omelyan 539
The conditions of Theorem 3.1 are only necessary conditions for the
extension of the symmetry kernel A0 of system (3.1), but not sufficient.
The classification of the symmetry properties of system (3.1) is presented
by the following theorem.
Theorem 3.2. If system (3.1) admits the extension of the symmetry
kernel A0, its maximal algebras of invariance depending on the functions
g1 and g2 are given in Table 1.
Table 1. Classification of the symmetry properties of system (3.1)
N Kind Operators of maximal
n/n of functions g1, g2 algebra of invariance
1.
g
1 = ϕ
1(u1),
g
2 = u
2(ϕ2(u1) + λ3 ln u
2)
∂0, ∂1, Q1 = eλ3x0u2∂u2
2.
g
1 = (u2)m
ϕ
1(u1),
g
2 = (u2)m+1
ϕ
2(u1)
∂0, ∂1, D = m(2x0∂0 + x1∂1) − 2u2∂u2
3.
g
1 = 0,
g
2 = 0
∂0, ∂1, D = 2x0∂0 + x1∂1,
I = u
2
∂u2
In Table 1, ϕ1(u1) and ϕ2(u1) are arbitrary smooth functions, and λ3 6= 0,
λ4, m are arbitrary constants.
Proof. In Theorem 3.1, it is shown that the extension of the symmetry
kernel A0 of system (3.1) is possible only in the case where the functions
ga are set by formulas (3.2) or (3.3). We will consider each of these
formulas separately.
A. We set that the functions g1, g2 look like (3.2). Substituting (3.2)
in system (2.4), we obtain
c1 = 0, α̇22 − λ3α
22 = 0,
whence α22 = c2e
λ3x0 , where c2 is an arbitrary constant.
From formulas (2.10), we obtain the algebra presented in the first
point of Table 1.
B. If the functions ga are set by formulas (3.3), system (2.4) yields
(mα22 + 2c1)ϕ
1 = 0,
(mα22 + 2c1)ϕ
2 = α̇22(u2)−m.
(3.7)
540 Classification of symmetry properties...
In the case where ϕ1, ϕ2 are arbitrary smooth functions, system (3.7)
yields
α̇22 = 0, mα22 + 2c1 = 0,
that is
α22 = −2c2, c1 = mc2, (3.8)
where c2 is an arbitrary constant. From formulas (2.10) and (3.8), we
obtain the algebra which is presented in the second point of Table 1. The
symmetry extensions of the second point of Table 1 is possible only at
m = 0, ϕ1 = 0, ϕ2 = λ4.
In this case,
α22 = 2λ4c1x0 + c2, (3.9)
where λ4 and c2 are arbitrary constants. By applying the equivalence
transformations presented in Remark 3.1, we obtain the third point of
Table 1 from formulas (2.10) and (3.9). The theorem is proved.
4. Symmetry properties of system (1.2) at f = λ
We mow consider the system
(
u1
u2
)
0
= ∂1
[(
λ1 0
λu2 λ2
) (
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
(4.1)
and will perform the classification of its symmetry properties depending
on the form of the functions ga(u1;u2).
Remark 4.1. It follows from Lemma 2.1 that the basic group of equiv-
alence transformations of system (4.1) looks like
x0 = te2θ2 + θ0, x1 = xeθ2 + θ1,
u1 = w1 + θ3, u2 = w2eθ4 .
(4.2)
Besides the basic group of equivalence, system (4.1) admits additional
equivalence transformations at specific g. For example,
x0 = at, x1 = bx, u1 = w1 + kt, u2 = w2emt, (4.3)
where a, b, k,m are arbitrary constants.
In view of Remark 4.1, we will formulate the theorems on the maxi-
mal invariance algebras of system (4.1) to within the transformations of
equivalence (4.2) and (4.3).
M. I. Serov, O. M. Omelyan 541
The necessary condition for the extension of the symmetry kernel A0
of system (4.1) is given by the following proposition.
Theorem 4.1. If system (4.1) admits the extension of the symmetry
kernel A0, the functions g1, g2 are set by formulas (3.2), (3.3) or one of
the formulas
g1 = ϕ1(ω) + λ3u
1, g2 = u2[ϕ2(ω) + λ4u
1]; (4.4)
g1 = emu1
ϕ1(ω), g2 = u2emu1
ϕ2(ω), (4.5)
where ω = ku1 + lnu2, m, k, λ3, λ4 are arbitrary constants, and ϕ1(ω),
ϕ2(ω) are arbitrary smooth functions.
Proof. Substituting formulas (2.11) in system (2.4), we obtain
β1(x0)g
1
u1 + α22(x0)u
2g1
u2 = −2c1g
1 + β̇1(x0),
β1(x0)g
2
u1 + α22(x0)u
2g2
u2 = (α22(x0) − 2c1)g
2 + α̇22(x0)u
2.
(4.6)
It is obvious that, at arbitrary functions g1, g2, system (4.6) does not
admit the extension of the symmetry kernel A0.
It follows from (4.6) that the functions ga should satisfy the structural
system
æg1
u1 − ku2g1
u2 = mg1 + λ3,
æg2
u1 − ku2g2
u2 = (m− k)g2 + λ4u
2,
(4.7)
where æ = {0; 1}, k,m, λ3, λ4 are arbitrary constants. If æ = 0, sys-
tem (4.7) will coincide with system (3.5), which has been analyzed in
Theorem 3.1, according to which the functions ga are set by formulas
(3.2) and (3.3).
If æ = 1, then system (4.7) is connected with system (4.6) by the
conditions
α22 + kβ1 = 0, mβ1 = −2c1, λ3β
1 = β̇1, λ4α
22 = α̇22. (4.8)
The solution of system (4.7) at æ = 1 depends on the constant m. Two
essentially different cases are possible.
542 Classification of symmetry properties...
1. m = 0. In this case, the general solution of system (4.7) is set by
functions (4.4).
2. m 6= 0. From the differential consequences of the first and second
conditions (4.8), we get
α̇22 = β̇1 = 0, λ3 = λ4 = 0. (4.9)
Under conditions (4.9), the general solution of system (4.7) is functions
(4.5). The theorem is proved.
Remark 4.2. As formulas (4.4) and (4.5) at λ3 = λ4 = m = 0 coincide,
then, in order to avoid their coincidence, we will consider |λ3|+|λ4| 6= 0 in
formulas (4.4), while studying the symmetry properties of system (4.1).
We will classify now the symmetry properties of system (4.1), using
the results of Theorem 4.1.
Theorem 4.2. If system (4.1) admits the extension of the symmetry
kernel A0, its maximal algebras of invariance depending on a kind of the
functions g1, g2 are presented in Tables 1 and 2.
Таблиця 2. Classification of symmetry properties of system (4.1)
№ Kind Operators of maximal
n/n of functions g1, g2 algebra of invariance
1.
g
1 = e
mu1
ϕ
1(ω),
g
2 = u
2
e
mu1
ϕ
2(ω)
∂0, ∂1, D = m(2x0∂0 + x1∂1)
+2(−∂u1 + ku
2
∂u2)
2.
g
1 = ϕ
1(ω) + λ3u
1
,
g
2 = u
2(ϕ2(ω) − kλ3u
1)
∂0, ∂1, Q = eλ3x0(∂u1 − ku2∂u2)
3.
g
1 = λ5e
u1
,
g
2 = λ6e
u1
u
2
∂0, ∂1, Q = u
2
∂u2 ,
D = 2x0∂0 + x1∂1 − 2∂u1
4.
g
1 = λ5(u
2)n
e
mu1
− mλ9,
g
2 = u
2(λ6(u
2)n
e
mu1
+ nλ9)
∂0, ∂1, Q = n∂u1 − mu
2
∂u2 ,
D = n(2x0∂0 + x1∂1)
+ 2nλ9x0Q − 2u
2
∂u2
5.
g
1 = λ3u
1 + λ5 ln u
2 + λ7,
g
2 = u
2(λ4u
1 + λ6 ln u
2 + λ8),
D > 0
∂0, ∂1,
Q1 = e
m1x0(λ5∂u1 + (m1 − λ3)u
2
∂u2),
Q2 = e
m2x0((m2 − λ6)∂u1 + λ4u
2
∂u2)
6.
g
1 = λ3u
1 + λ5 ln u
2 + λ7,
g
2 = u
2(λ4u
1 + λ6 ln u
2 + λ8),
D = 0, |λ3| + |λ5| + |λ6| 6= 0
∂0, ∂1, Q1 = e
αx0 [x0(2λ5∂u1
+ (λ6 − λ3)u
2
∂u2) + u
2
∂u2 ],
Q2 = e
αx0 [2λ5∂u1 + (λ6 − λ3)u
2
∂u2 ]
M. I. Serov, O. M. Omelyan 543
7.
g
1 = λ3u
1 + λ5 ln u
2 + λ7,
g
2 = u
2(λ4u
1 + λ6 ln u
2 + λ8),
D < 0
∂0, ∂1, Q1 = e
αx0 [2λ5 cos βx0∂u1
+ ((λ6 − λ3) cos βx0 − 2β sin βx0)u
2
∂u2 ],
Q2 = e
αx0 [2λ5 sin βx0∂u1
+ (2β cos βx0 + (λ6 − λ3) sin βx0)u
2
∂u2 ]
8.
g
1 = 0,
g
2 = u
1
u
2
∂0, ∂1, Q1 = ∂u1 + x0Q2, Q2 = u2∂u2
9.
g
1 = 0,
g
2 = 0
∂0, ∂1, Q1 = ∂u1 , Q2 = u
2
∂u2 ,
D = 2x0∂0 + x1∂1
In Table 2, m,n, λi, i = 1; 9 are arbitrary constants, ω = lnu2 + ku1,
ϕ1(ω), ϕ2(ω) are arbitrary smooth functions; D = (λ3 − λ6)
2 + 4λ4λ5 is
a discriminant, and m1,m2 are roots of the characteristic equation m2 −
(λ3 + λ6)m+ λ3λ6 − λ4λ5 = 0, α = λ3+λ6
2 , β = 1
2
√
|(λ3 − λ6)2 + 4λ4λ5|.
5. Symmetry properties of system (1.2)
at f = λ
u1
In this subsection, we consider the system
(
u1
u2
)
0
= ∂1
[(
λ1 0
λ
u1u
2 λ2
)(
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
, (5.1)
where λ is an arbitrary constant, and we will perform the classification of
its symmetry properties depending on a kind of the functions ga(u1;u2).
Remark 5.1. It follows from Lemma 2.1 that the basic group of equiv-
alence transformations of system (5.1) looks like
x0 = te2θ2 + θ0, x1 = xeθ2 + θ1,
u1 = w1eθ3 , u2 = w2eθ4 .
(5.2)
In addition to the basic group of equivalence, system (5.1) at specific g
admits additional equivalence transformations, for example,
x0 = at, x1 = bx, u1 = w1ekt, u2 = w2emt, (5.3)
where a, b, k,m are arbitrary constants. Therefore, we will formulate the
theorems on the maximal algebra of invariance of system (5.1) to within
the transformations of equivalence (5.2) and (5.3).
The necessary condition for the extension of the symmetry kernel A0
of system (5.1) is given by the following proposition.
544 Classification of symmetry properties...
Theorem 5.1. If system (5.1) admits the extension of the symmetry
kernel A0, then the functions g1, g2 are set by formulas (3.2), (3.3) or
one of the following formulas:
g1 = u1(ϕ1(ω) + λ3 lnu1), g2 = u2(ϕ2(ω) + λ4 lnu2); (5.4)
g1 = (u1)m+1ϕ1(ω), g2 = u2(u1)mϕ2(ω), (5.5)
where ω = u2
(u1)k ; ϕ1(ω), ϕ2(ω) are arbitrary smooth functions, m, k, λ3,
λ4 are arbitrary constants.
Proof. Substituting formulas (2.12) which set the coordinates of the in-
finitesimal operator (2.1) for system (5.1) in system (2.4), we obtain
α1(x0)u
1g1
u1 + α22(x0)u
2g1
u2 = (α1(x0) − 2c1)g
1 + α̇1(x0)u
1,
α1(x0)u
1g2
u1 + α22(x0)u
2g2
u2 = (α22(x0) − 2c1)g
2 + α̇22(x0)u
2.
(5.6)
It is obvious that, at arbitrary functions g1, g2, system (5.6) does not
admits the extension of the symmetry kernel A0.
System (5.6) admits the widest class of functions ga, at which the
extension of the symmetry kernel A0 is possible, if they satisfy the struc-
tural system
æu1g1
u1 + ku2g1
u2 = (m+ æ)g1 + k1u
1,
æu1g2
u1 + ku2g2
u2 = (m+ k)g2 + k2u
2,
(5.7)
where æ = {0; 1}; k,m, k1, k2 are arbitrary constants. The general solu-
tion of system (5.7) at æ = 0 is set by formulas (3.2), (3.3). System (5.7)
at æ = 1 is connected with system (5.6) by the conditions
α22 − kα1 = 0, mα1 = −2c1, k1α
1 = α̇1, k2α
1 = α̇22.
The solution of system (5.7) depends on the constant m.
If m = 0, the general solution of system (5.7) looks like (5.4). At
m 6= 0, it is set by formulas (5.5). The theorem is proved.
Remark 5.2. If we set λ3 = λ4 = 0 in the representations of the func-
tions ga given by formulas (5.4) the obtained form of the functions ga will
be a special case of the representation of functions ga given by formu-
las (5.5) under the condition m = 0. Hence, the classes of systems (5.1),
(5.4) and (5.1), (5.5) will have a nonempty crossing. To avoid the consid-
M. I. Serov, O. M. Omelyan 545
eration of equivalent systems in the subsequent researches of symmetry
properties, we impose restrictions on the parameters of representations
of the functions ga in formulas (5.4): |λ3| + |λ4| 6= 0.
Let’s classify now the symmetry properties of system (5.1), by using
Theorem 5.1.
Theorem 5.2. If system (5.1) admits the extension of the symmetry
kernel A0, its maximal algebras of invariance depending on a kind of the
functions g1, g2 are presented in Tables 1 and 3.
Таблиця 3. Classification of the symmetry properties of system (5.1)
№ Kind Operators of maximal
n/n of functions g1, g2 algebra of invariance
1.
g
1 = u
1(ϕ1(u2) + λ3 ln u
1),
g
2 = u
2(ϕ2(u2) + λ4 ln u
2)
∂0, ∂1, Q = eλ3x0u1∂u1
2.
g
1 = u
1(ϕ1(ω) + λ3 ln u
1),
g
2 = u
2(ϕ2(ω) + λ3 ln u
2)
∂0, ∂1, Q = eλ3x0(u1∂u1 + ku2∂u2)
3.
g
1 = (u1)m+1
ϕ
1(ω),
g
2 = u
2(u1)m
ϕ
2(ω)
∂0, ∂1, D = m(2x0∂0 + x1∂1)
− 2(u1
∂u1 + ku
2
∂u2)
4.
g
1 = u
1(λ5(u
1)n(u2)m + mλ7),
g
2 = u
2(λ6(u
1)n(u2)m − nλ7)
∂0, ∂1, Q = mu
1
∂u1 − nu
2
∂u2 ,
D = m(2x0∂0 + x1∂1 + 2λ7x0Q)
− 2u
2
∂u2
5.
g
1 = λ5(u
1)n+1
,
g
2 = λ6(u
1)n
u
2
∂0, ∂1, D = 2x0∂0 + x1∂1 −
2
n
u
1
∂u1 ,
Q = u
2
∂u2
6.
g
1 = u
1(λ3 ln u
1 + λ5 ln u
2 + λ7),
g
2 = u
2(λ4 ln u
1 + λ6 ln u
2 + λ8),
D > 0
∂0, ∂1, Q1 = e
m1x0(λ5u
1
∂u1
+ (m1 − λ3)u
2
∂u2),
Q2 = e
m2x0((m2 − λ6)u
1
∂u1
+ λ4u
2
∂u2)
7.
g
1 = u
1(λ3 ln u
1 + λ5 ln u
2 + λ7),
g
2 = u
2(λ4 ln u
1 + λ6 ln u
2 + λ8),
D = 0, |λ3| + |λ5| + |λ6| 6= 0
∂0, ∂1, Q1 = e
αx0 [x0(λ5u
1
∂u1
+ (α − λ3)u
2
∂u2) + u
2
∂u2 ],
Q2 = e
αx0 [λ5u
1
∂u1 + (α − λ3)u
2
∂u2 ]
8.
g
1 = λ7u
1
,
g
2 = λ4u
2 ln u
1
∂0, ∂1, Q1 = u
2
∂u2 ,
Q2 = u
1
∂u1 + λ4x0u
2
∂u2
9.
g
1 = u
1(λ3 ln u
1 + λ5 ln u
2 + λ7),
g
2 = u
2(λ4 ln u
1 + λ6 ln u
2 + λ8),
D < 0
∂0, ∂1, Q1 = e
αx0 [λ5 cos βx0u
1
∂u1
+ ((α − λ3) cos βx0 − β sin βx0)u
2
∂u2 ],
Q2 = e
αx0 [λ5 sin βx0u
1
∂u1
+ (β cos βx0 + (α − λ3) sin βx0)u
2
∂u2 ]
10.
g
1 = 0,
g
2 = 0
∂0, ∂1, Q1 = u
1
∂u1 , Q2 = u
2
∂u2 ,
D=2x0∂0 + x1∂1 + 2x0(λ7Q1 + λ8Q2)
546 Classification of symmetry properties...
In Table 3, m,n, λi, i = 1; 9 are arbitrary constants, ω = u2
(u1)k , ϕ1(ω),
ϕ2(ω) are arbitrary smooth functions, D = (λ3 − λ6)
2 + 4λ4λ5 is a dis-
criminant, and m1,m2 are roots of the characteristic equation m2−(λ3 +
λ6)m+ λ3λ6 − λ4λ5 = 0, α = λ3+λ6
2 , β = 1
2
√
|D|.
6. Symmetry properties of system (1.2)
at f = λ1−λ2
u1
We now consider the system
(
u1
u2
)
0
= ∂1
[(
λ1 0
λ1−λ2
u1 u2 λ2
)(
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
(6.1)
and will perform the classification of its symmetry properties depending
on a kind of the functions ga(u1;u2).
Remark 6.1. It follows from Lemma 2.1 that the basic group of equiv-
alence transformations of system (6.1) looks like
x0 = te2θ2 + θ0, x1 = xeθ2 + θ1,
u1 = w1eθ3 , u2 = w2eθ4 + θ5w
1.
(6.2)
Besides the basic group of equivalence transformations (6.2), system (6.1)
at specific ga admits additional transformations of equivalence of the form
(5.3). Therefore, we will formulate theorems on the maximal algebras of
invariance of system (6.1) to within the equivalence transformations (5.3)
and (6.2).
The necessary condition for the extension of the symmetry kernel A0
of system (6.1) is given the following proposition.
Theorem 6.1. If the system (6.1) admits the extension of the symmetry
kernel A0, then the functions g1, g2 are set by formulas (3.2), (3.3), (5.4),
(5.5) or, to within the equivalence transformations (5.3) and (6.2) look
like
g1 = u1(ϕ1(ω) + λ3), g2 = u1ϕ2(ω) + u2ϕ1(ω), (6.3)
where ω = u1, λ3 is an arbitrary constant;
g1 = u1e
u2
u1 ϕ1(ω), g2 = e
u2
u1 [u1ϕ2(ω) + u2ϕ1(ω)], (6.4)
where ω = u1;
M. I. Serov, O. M. Omelyan 547
g1 = (u1)m+1ϕ1(ω), g2 = (u1)m[u1ϕ2(ω) + u2ϕ1(ω)], (6.5)
where ω = u2
u1 + k lnu1, k 6= 0 and m are arbitrary constants;
g1 = u1(ϕ1(ω) + λ3) + λ4u
2,
g2 = u1ϕ2(ω) + u2ϕ1(ω) + λ4
(u2)2
u1
,
(6.6)
where ω = u2
u1 + k lnu1, λ3, λ4, k are arbitrary constants, k 6= 0, |λ3| +
|λ4| 6= 0, and, in formulas (6.3)–(6.6), ϕ1(ω), ϕ2(ω) are arbitrary smooth
functions.
Proof. As was already mentioned at the beginning of the present article,
the solution of determining systems S1(ξ, η) = 0 and S2(ξ, η, f) = 0 at
f = λ1−λ2
u1 are the coordinates of operator (2.1) given by formulas (2.13).
In view of these formulas, the determining system S3(ξ, η, f, g) = 0 can
be written as follows:
α1(x0)u
1g1
u1 + (α21(x0)u
1 + α22u2)g1
u2
= (α1(x0) − 2c1)g
1 + α̇1u1,
α1(x0)u
1g2
u1 + (α21(x0)u
1 + α22u2)g2
u2
= (α22(x0) − 2c1)g
2 + α21(x0)g
1 + α̇21u1 + α̇22u2.
(6.7)
It is obvious that, at arbitrary functions g1, g2, system (6.7) does not
admits the extension of the symmetry kernel A0.
The widest class of functions ga such that they satisfy system (6.7)
and allow the symmetry kernel A0 to be extended is possible under the
conditions
α1 = k1ϕ(x0), α21 = k0ϕ(x0), α22 = k2ϕ(x0),
α̇1 = k4ϕ(x0), α̇21 = k5ϕ(x0), α̇22 = k6ϕ(x0),
− 2c1 = k3ϕ(x0),
(6.8)
where ϕ(x0) are arbitrary smooth functions, k0, k1, . . . , k6 are arbitrary
constants. With regard for (6.8) and (6.7), we obtain the structural
system for the functions ga:
k1u
1g1
u1 + (k0u
1 + k2u
2)g1
u2 = (k1 + k3)g
1 + k4u
1,
k1u
1g2
u1 + (k0u
1 + k2u
2)g2
u2 = (k2 + k3)g
2 + k0g
1 + k5u
1 + k6u
2.
(6.9)
548 Classification of symmetry properties...
Let us analyze this system and its influence on solutions of system (6.7).
The solution of system (6.9) essentially depends on the ratios between the
constants k0, k1, k2. If we set k0 = 0 in the structural system (6.9), then
system (6.9) coincides with system (5.7). If k0 6= 0, and k1 6= k2, we can
use the equivalence transformations (6.2) at θ5 = −k0
k2
and θi = 0, i = 0, 4
and reduce system (6.9) to system (5.7) investigated in Theorem 5.1,
according to which the functions ga are set by formulas (5.5), (5.4) or
(3.2), (3.3).
If k0 6= 0 (without loss generality, it is possible to consider k0 = 1)
and k1 = k2, then formulas (6.8) yield k4 = k6. Then system (6.9) takes
the form
k1u
1g1
u1 + (u1 + k1u
2)g1
u2 = (k1 + k3)g
1 + k4u
1,
k1u
1g2
u1 + (u1 + k1u
2)g2
u2 = (k1 + k3)g
2 + g1 + k4u
2 + k5u
1.
(6.10)
The solution of system (6.10) depends on the parameters k1, k3. We
obtain 4 nonequivalent cases:
1) k1 = 0, k3 = 0,
2) k1 = 0, k3 6= 0,
3) k1 6= 0, k3 = 0,
4) k1 6= 0, k3 6= 0.
1) Let k1 = 0, k3 = 0. If k1 = 0, Eqs. (6.8) imply that k4 = 0. Then
system (6.10) takes the form
u1g1
u2 = 0, u1g2
u2 = g1 + k5u
1. (6.11)
By solving Eq. (6.11), we obtain the representation of functions ga of
form (6.3), where λ3 = −k5.
2) Consider the case where k1 = 0, k3 6= 0. Without loss of generality,
it is possible to consider that k3 = 1. It follows from Eqs. (6.8) that
k4 = k5 = 0. Then system (6.10) takes the form
u1g1
u2 = g1, u1g2
u2 = g2 + g1,
whose general solution is functions (6.4).
M. I. Serov, O. M. Omelyan 549
3) If k1 6= 0, k3 = 0, system (6.10) takes the form
k1u
1g1
u1 + (u1 + k1u
2)g1
u2 = k1g
1 + k4u
1,
k1u
1g2
u1 + (u1 + k1u
2)g2
u2 = k1g
2 + g1 + k5u
1 + k4u
2.
(6.12)
The general solution of system (6.12) looks like (6.6), where k = − 1
k1
,
λ3 = −k5, λ5 = k4.
4) If k1 6= 0, k3 6= 0, it follows from (6.8) that k4 = k5 = 0. In this
case, system (6.10) becomes
k1u
1g1
u1 + (u1 + k1u
2)g1
u2 = (k1 + k3)g
1,
k1u
1g2
u1 + (u1 + k1u
2)g2
u2 = (k1 + k3)g
2 + g1.
(6.13)
By solving system (6.13), we obtain the representation of the functions ga
which is set by formulas (6.5), where m = k3
k1
. The theorem is proved.
Let us classify the symmetry properties of system (6.1), by using
Theorem 6.1.
Remark 6.2. In formulas (6.5) and (6.6), the restrictions are imposed
to avoid their crossing.
Theorem 6.2. If system (6.1) admits the extension of the symmetry
kernel A0, its maximal algebras of invariance depending on a kind of the
functions g1, g2 are given in Tables 1, 3, and 4.
Таблиця 4. Classification of the symmetry properties of system (6.1)
№ Kind Operators of maximal
n/n of functions g1, g2 algebra of invariance
1.
g
1 = u
1(ϕ1(u1) + λ3),
g
2 = u
1
ϕ
2(u1) + u
2
ϕ
1(u1)
∂0, ∂1, Q1 = e−λ3x0Q
2.
g
1 = u
1
e
u
2
u
1 ϕ
1(u1),
g
2 = e
u
2
u
1 (u1
ϕ
2(u1) + u
2
ϕ
1(u1))
∂0, ∂1, Q
3.
g
1 = (u1)m+1
ϕ
1(ω),
g
2 = (u1)m(u1
ϕ
2(ω) + u
2
ϕ
1(ω)),
ω =
u2
u1
+ k ln u
1
∂0, ∂1, D = m(2x0∂0 + x1∂1)
− 2I + 2kQ
4.
g
1 = u
1(ϕ1(ω) + k) + u
2
,
g
2 = u
1
ϕ
2(ω) + u
2
ϕ
1(ω) +
(u2)2
u1
,
ω =
u2
u1
+ k ln u
1
, k 6= 0
∂0, ∂1, Q = e−kx0(I − kQ)
550 Classification of symmetry properties...
5.
g
1 = (u1)m+1
,
g
2 = u
1((u1)n + λ8) + u
2(u1)m
,
m 6= 0, n 6= 0
∂0, ∂1, Q,
D = m(2x0∂0 + x1∂1 − 2u
1
∂u1)
+ 2n(λ8x0Q − u
2
∂u2)
6.
g
1 = λ5(u
1)m+1
,
g
2 = (u1)m(λ6u
2 + u
1),
|λ5| + |λ6| 6= 0
∂0, ∂1, D = m(2x0∂0 + x1∂1) − 2I,
Q1 = Q + (λ6 − λ5)u
2
∂u2
7.
g
1 = u
1((u1)m + λ6),
g
2 = u
2((u1)m + λ7) + λ8u
1
,
m 6= 0, λ6 6= 0, λ7 6= λ6
∂0, ∂1, Q1 = e
(λ7−λ6)x0Q,
Q2 = λ8Q + (λ7 − λ6)u
2
∂u2
8.
g
1 = u
1((u1)m + λ7),
g
2 = u
2((u1)m + λ7) + λ8u
1
,
m 6= 0, λ7 6= 0
∂0, ∂1, Q, Q1 = λ8x0Q − u2∂u2
9.
g
1 = (u1)m+1
,
g
2 = (u1)m(λ8(u
1)m+1 + u
2),
m 6= 0
∂0, ∂1, Q,
D=m(2x0∂0 + x1∂1) − 2(I + mu
2
∂u2)
10.
g
1 = λ5u
1
,
g
2 = (u1)n+1 + λ7u
2
,
n 6= 0, λ7 6= λ5
∂0, ∂1, Q1 = e
(λ7−λ5)x0Q,
Q2 = I + nu
2
∂u2
11.
g
1 = u
1
,
g
2 = u
1((u1)n + λ8) + u
2
,
n 6= 0
∂0, ∂1, Q, Q1 = I + nu2∂u2 − nλ8x0Q
12.
g
1 = 0,
g
2 = (u1)n + λ6u
1 + λ7,
n 6= 1, λ7 6= 0
∂0, ∂1, Q, D = 2x0∂0 + x1∂1 + 2u2∂u2
13.
g
1 = λ3u
2 + λ5u
1
,
g
2 = u
1
(
λ4
(u2
u1
)2
+
(λ6 − λ5)
2
4(λ4 − λ3)
)
+ λ6u
2
,
|λ3| + |λ4| 6= 0, λ4 6= λ3, λ6 6= λ5
∂0, ∂1, I, D = (λ4 − λ3)
×[(2x0∂0 + x1∂1) − 2u
2
∂u2 ]
−(λ6 − λ5)Q + x0(2λ5(λ4 − λ3)
−λ3(λ6 − λ5))I
14.
g
1 = u
2
,
g
2 = u
1
((u2
u1
)2
+ λ8
)
± u
2
∂0, ∂1, I, Q1 = e±x0(I ± Q)
15.
g
1 = e
n u
2
u
1 (u1)p+1
,
g
2 = e
n u
2
u
1 (u1)p(u2 + λ4u
1),
p 6= 0, n 6= 0
∂0, ∂1, Q1 = nI − pQ,
D = n(2x0∂0 + x1∂1) − 2Q
16.
g
1 = [λ3e
n u
2
u
1 (u1)p + λ5]u
1
,
g
2 = e
n u
2
u
1 (u1)p(λ3u
2 + λ4u
1)
+ λ5u
2 −
p
n
λ5u
1
,
n 6= 0, |λ3| + |λ4| 6= 0
∂0, ∂1, Q1 = I −
p
n
Q,
D = 2x0∂0 + x1∂1 + 2λ5x0Q1 −
2
n
Q
17.
g
1 = (u1)m+1
,
g
2 = u
1(ln u
1 + λ8) + u
2(u1)m
,
m 6= 0
∂0, ∂1, Q,
D = m(2x0∂0 + x1∂1) − 2I − 2x0Q
M. I. Serov, O. M. Omelyan 551
18.
g
1 = u
1(λ5 ln u
1 + λ3) + λ4u
2
,
g
2 = (λ6u
1 + λ5u
2) ln u
1
+ λ8u
1 + λ7u
2 + λ4
(u2)2
u1
,
|λ5| + |λ6| 6= 0, ∆ > 0
∂0, ∂1, Q1 = e
m1x0 [λ4I + (m1 − λ5)Q],
Q2 = e
m2x0 [λ4I + (m2 − λ5)Q]
19.
g
1 = u
1(λ5 ln u
1 + λ3) + λ4u
2
,
g
2 = (λ6u
1 + λ5u
2) ln u
1
+ λ8u
1 + λ7u
2 + λ4
(u2)2
u1
,
|λ3| + |λ5| + |λ6| 6= 0, ∆ = 0
∂0, ∂1, Q1 = e
αx0 [λ4I + (α − λ5)Q],
Q2 = e
αx0Q + x0Q1
20.
g
1 = u
1(λ5 ln u
1 + λ3) + λ4u
2
,
g
2 = (λ6u
1 + λ5u
2) ln u
1
+ λ8u
1 + λ7u
2 + λ4
(u2)2
u1
,
∆ < 0
∂0, ∂1, Q1 = e
αx0 [λ4 cos βx0I
+ ((α − λ5) cos βx0 − β sin βx0)Q],
Q2 = e
αx0 [λ4 sin βx0I
+ ((α − λ5) sin βx0 + β cos βx0)Q]
21.
g
1 = u
1(λ5 ln u
1 + λ3),
g
2 = u
1 ln u
1 + u
2(λ5 ln u
1 + λ7),
λ3 6= λ7 − λ5
∂0, ∂1, Q1 = e
(λ7−λ3)x0Q,
Q2 = e
λ5x0 [(λ3 + λ5 − λ7)I + Q]
22.
g
1 = u
1(λ5 ln u
1 + λ7 − λ5),
g
2 = u
1 ln u
1 + u
2(λ5 ln u
1 + λ7)
∂0, ∂1, Q1 = e
λ5x0Q,
Q2 = e
λ5x0I + x0Q1
23.
g
1 = 0,
g
2 = ln u
1
∂0, ∂1, Q,
D = 2x0∂0 + x1∂1 + 2u
2
∂u2
24.
g
1 = (u1)m+1
,
g
2 = u
2(u1)m + λ8u
1
,
m 6= 0
∂0, ∂1,
D = m(2x0∂0 + x1∂1) + 2λ8x0Q − 2I,
Q, Q1 = λ8x0Q − u
2
∂u2
25.
g
1 = λ5u
1
,
g
2 = u
1((u1)n + λ8)
+ (n + 1)λ5u
2
,
n 6= 0
∂0, ∂1,
D = 2x0∂0 + x1∂1 + 2x0Q2 −
2
n
I,
Q1 = e
nλ5x0Q,
Q2 = λ5(I + nu
2
∂u2) + λ8Q
26.
g
1 = 0,
g
2 = u
1((u1)n + λ8),
n 6= 0
∂0, ∂1, Q,
D = 2x0∂0 + x1∂1 −
2
n
I + 2λ8x0Q,
Q1 = I + nu
2
∂u2 − nλ8x0Q
27.
g
1 = u
2
,
g
2 = u
1
((u2
u1
)2
+ λ8
)
∂0, ∂1, I, Q1 = x0I + Q,
D = 2x0∂0 + x1∂1 + 2λ8x
2
0I
+ 4λ8x0Q − 2u
2
∂u2
28.
g
1 = 0,
g
2 = λ4u
1 + λ5
∂0, ∂1, D = 2x0∂0 + x1∂1 + 2u
2
∂u2 ,
Q, Q1 = u
1
∂u1 + λ4x0Q
29.
g
1 = u
1 ln u
1
,
g
2 = u
2 ln u
1 + λ8u
1
∂0, ∂1, Q, Q1 = e
x0I,
Q2 = u
2
∂u2 − λ8x0Q
552 Classification of symmetry properties...
30.
g
1 = λ7u
1
,
g
2 = u
1 ln u
1 + λ7u
2
∂0, ∂1, Q, Q1 = I + x0Q,
D = 2x0∂0 + x1∂1 + 2λ7x0Q1 − λ7x
2
0Q + 2u
2
∂u2
31.
g
1 = 0,
g
2 = u
1 ln u
1
∂0, ∂1, D = 2x0∂0 + x1∂1 + 2u
2
∂u2 ,
Q, Q1 = I + x0Q
32.
g
1 = 0,
g
2 = u
2
∂0, ∂1, I, Q1 = e
x0Q, Q2 = u
1
∂u1 ,
D = 2x0∂0 + x1∂1 + 2x0u
2
∂u2
33.
g
1 = 0,
g
2 = λ8u
1
∂0, ∂1, I, Q, D = 2x0∂0 + x1∂1 − 2u
1
∂u1 ,
Q3 = u
1
∂u1 + λ8x0Q
In Table 4, m,n, λi, i = 1; 9 are arbitrary constants, ω = u2
u1 + k lnu1,
ϕ1(ω), ϕ2(ω) are arbitrary smooth functions, I = u1∂u1 + u2∂u2 ,
Q = u1∂u2 , ∆ = (λ3 + λ5 − λ7)
2 + 4λ4λ6 is a discriminant, and m1,m2
are roots of the characteristic equation
∣
∣
∣
∣
λ5 −m λ4
λ6 λ7 − λ3 −m
∣
∣
∣
∣
= 0,
α = λ7+λ5−λ3
2 , β = 1
2
√
|∆|.
7. Invariance under the Galilei algebra
In the present subsection, we will comprehensively study the symme-
try properties of system (1.2) for f(u1) = 2λ1
u1 .
So, we consider the system
(
u1
u2
)
0
= ∂1
[(
λ1 0
2λ1
u2
u1 λ2
)(
u1
u2
)
1
]
+
(
g1(u1, u2)
g2(u1, u2)
)
. (7.1)
Remark 7.1. It follows from Lemma 2.1 that the basic group of equiv-
alence transformations of system (7.1) looks like
x0 = te2θ2 + θ0, x1 = xeθ2 + θ1,
u1 = w1eθ3 , u2 = w2eθ4 .
(7.2)
Besides the basic group of equivalence transformations (7.2), system (7.1)
admits the additional equivalence transformations of form (5.3) at spe-
cific g. Therefore, we will formulate the theorems on the maximal invari-
ance algebras of system (7.1) to within the specified equivalence trans-
formations (5.3), (7.2).
The following proposition is valid.
M. I. Serov, O. M. Omelyan 553
Theorem 7.1. If system (7.1) admits the extension of the symmetry
kernel A0, the functions g1, g2 are set by formulas (3.2), (3.3), (5.4), or
by the formulas
g1 = u1[(u1)mϕ1(ω) + λ3], g2 = u2[(u1)mϕ2(ω) + λ4], (7.3)
where ω = u2
(u1)k , m,λ3, λ4, k are arbitrary constants, and ϕa(ω) are ar-
bitrary smooth functions.
Proof. As has been shown in the proof of Theorem 2.1, the solution of
systems S1(ξ, η) = 0 and S2(ξ, η, f) = 0 for f = 2λ1
u1 are the coordinates
of the infinitesimal operator X set by formulas (2.14).
In view of values of ξ0, ξ1, ηa in formulas (2.14), system (2.4) can be
written as
α1u1g1
u1 + α22u2g1
u2 = (α1 − 2Ȧ)g1 + (α1
0 − λ1α
1
11)u
1,
α1u1g2
u1 + α22u2g2
u2 = (α22 − 2Ȧ)g2 + (α22
0 − 2λ1α
1
11)u
2.
(7.4)
It is obvious that, at arbitrary functions g1, g2, system (7.4) does not
admits the extension of the symmetry kernel A0.
Since the functions α1, α22, A depend only on the variables x0, x1, and
the functions ga do on the variables u1, u2, the widest class of the func-
tions g1, g2 such that they satisfy system (7.4) and allow the symmetry
kernel A0 to be extended is a solution of the structural system
æu1g1
u1 + ku2g1
u2 = (m+ æ)g1 + k1u
1,
æu1g2
u1 + ku2g2
u2 = (m+ k)g2 + k2u
2.
(7.5)
Moreover, α1 = æψ(x0, x1), α
22 = kψ(x0, x1), −2Ȧ = mψ(x0, x1),
α1
0−λ1α
1
11 = k1ψ(x0, x1), α
22
0 −2λ1α
1
11 = k2ψ(x0, x1), where æ = {0, 1};
k,m, k1, k2 are arbitrary constants which we will call structural constants
for the functions ga, and ψ(x0, x1) is an arbitrary smooth function.
1. If æ = 0, system (7.5) takes form (3.4), whose solutions are formulas
(3.2) and (3.3), as it has been shown in Theorem 3.1.
2. If æ = 1, the general solution of system (7.5) is expressed through
the first integrals of the system of ordinary differential equations
554 Classification of symmetry properties...
du1
u1
=
du2
ku2
=
dg1
(m+ 1)g1 + k1u1
=
dg2
(k +m)g2 + k2u2
. (7.6)
One of the first integrals of system (7.6) is J1 = ω = u2
(u1)k , and two
other ones, J2, J3, depend on the constant m.
The following nonequivalent cases are possible:
2.1) m = 0. In this case,
J2 =
g1
u1
− k1 lnu1, J3 =
g2
u2
− k2 lnu1.
By constructing the general solution of system (7.5) in the
standard way (see, for example, [24]), we obtain formulas
(5.4), where λ3 = −k1, λ4 = −k2 are arbitrary constants,
and ϕa(ω) are arbitrary smooth functions, ω = u2
(u1)k .
2.2) m 6= 0. In this case, by calculating the first integrals of system
(7.6),
J2 = (u1)−m(
g1
u1
+
k1
m
), J3 = (u1)−m(
g2
u2
+
k2
m
),
we obtain the general solution of system (7.5) which looks like
(7.3), where λ3 = −k1
m , λ4 = −k2
m are arbitrary constants, and
ϕa(ω) are arbitrary smooth functions, ω = u2
(u1)k .
So, by solving system S3(ξ, η, f, g) = 0 for g1, g2 at f = 2λ1
u1 , we have
obtained the nonequivalent forms (3.2), (3.3), (5.4), (7.3) of the functions
g1, g2, what proves the theorem.
Conditions of Theorem 7.1, as well as those of Theorem 2.1, are only
necessary conditions for the extension of the symmetry kernel of system
(1.2). To obtain sufficient conditions, it is necessary to substitute each
representation of the functions ga of forms (3.2), (3.3), (5.4), (7.3) in the
system S3 = 0 and to solve the obtained system for functions A(x0),
B(x0), C(x0), α(x0) with regard for a kind of the functions ϕa(ω) and
values of the constants m, k, λ3, λ4. The following statement is a result
of such researches.
Theorem 7.2. The maximal invariance algebras of system (7.1) depend-
ing on values of the functions g1, g2 are given in Tables 1, 3, and 5.
M. I. Serov, O. M. Omelyan 555
Таблиця 5. Classification of the symmetry properties of system (7.1)
№ Kind Operators of maximal
n/n of functions g1, g2 algebra of invariance
1.
g
1 = u
1
ϕ
1(u2),
g
2 = u
2
ϕ
2(u2)
∂0, ∂1, G = x0 −
x1
2λ1
u1∂u1 , I1 = u1∂u1
2.
g
1 = λ6u
1 ln u
2
,
g
2 = λ8u
2 ln u
2
∂0, ∂1, G, I1, Q1 = eλ8x0(λ6I1 + λ8I2)
3.
g
1 = λ6u
1 ln u
2
,
g
2 = λ9u
2
∂0, ∂1, G, I1, λ6x0I1 + I2
4.
g
1 = u
1[λ6(u
2)n + λ3],
g
2 = λ5(u
2)n+1
∂0, ∂1, G, I1,
D1 = 2x0∂0 + x1∂1 + 2λ3x0I1 −
2
n
I2
5.
g
1 = u
1[λ6(u
2)2 + λ3],
g
2 = λ5(u
2)3
∂0, ∂1, G, I1,
D2 = 2x0∂0 + x1∂1 + 2λ3x0I1 − I2,
Π1 = x
2
0∂0 + x0x1∂1
−
1
2λ1
(x2
1
2
− 2λ1λ3x
3
0 − λ1x0
)
I1 − x0I2
6.
g
1 = 0,
g
2 = 0
∂0, ∂1, G, I1, I2,
D3 = 2x0∂0 + x1∂1 −
1
2
I1 − I2,
Π2 = x
2
0∂0 + x0x1∂1 −
(x2
1
2
+ λx0
)
I1 − x0I2
7.
g
1 = u
1(ϕ1(u2) + λ3 ln u
1),
g
2 = u
2
ϕ
2(u2)
∂0, ∂1,G = e
λ3x0
(
∂1 −
λ3
2λ1
x1I1
)
,
M = e
λ3x0I1
8.
g
1 = u
1(λ3 ln u
1 + λ6 ln u
2)
g
2 = λ8u
2 ln u
2
,
λ8 6= λ3
∂0, ∂1,G, M, Q2 = eλ8x0(λ6I1 + (λ8 − λ3)I2)
9.
g
1 = u
1(λ3 ln u
1 + λ6 ln u
2),
g
2 = λ3u
2 ln u
2
∂0, ∂1,G, M, Q3 = eλ3x0(λ6x0I1 + I2)
In Table 5, λ3, . . . , λ9 are arbitrary constants, ϕa = ϕa(u2) are arbitrary
smooth functions, and I2 = u2∂u2 .
Remark 7.2. Theorems 4.2, 5.2, 6.2 are proved similarly to Theo-
rem 3.2. The proof of Theorem 7.2 is given in work [20].
Conclusions
The nonrelativistic movement of any macroobject is satisfied with
transformations of shift and stretching and the Galilei law of the move-
ment relativity. Therefore, it is obvious that the models of movement
investigated in the given work, being invariant under the Galilei algebra
556 Classification of symmetry properties...
and the algebras setting the transformation of shift and stretching, claim
for the reliability of the description of the movement of objects within
the Keller-Segel’s model. In addition, the maximal algebras of invariance
of systems established in the present work can considerably facilitate the
work on the establishment of trajectories of movement of the objects
whose movement is investigated within the mentioned model.
The authors are grateful to R. M. Cherniha for the problem statement
and the discussion of the results of studies.
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Contact information
Mykola I. Serov,
Oleksandr M.
Omelyan
Yu. Kondratyuk Poltava National
Technical University,
24, Pershotravnenyi Av.,
36000, Poltava,
Ukraine
E-Mail: k26@pntu.edu.ua
|
| id | nasplib_isofts_kiev_ua-123456789-10951 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Ukrainian |
| last_indexed | 2025-12-07T16:32:33Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Сєров, М.І. Омелян, О.М. 2010-08-10T10:20:22Z 2010-08-10T10:20:22Z 2008 Класифікація симетрійних властивостей системи рівнянь хемотаксису / М.І. Сєров, О.М. Омелян // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 536 – 562. — Бібліогр.: 26 назв. — укр. 1810-3200 https://nasplib.isofts.kiev.ua/handle/123456789/10951 В данiй роботi проведена повна групова класифiкацiя систем рiвнянь хемотаксису. The full group classification of the systems of chemotaxis equations is performed. uk Інститут прикладної математики і механіки НАН України Класифікація симетрійних властивостей системи рівнянь хемотаксису Classification of symmetry properties of a system of chemotaxis equations Article published earlier |
| spellingShingle | Класифікація симетрійних властивостей системи рівнянь хемотаксису Сєров, М.І. Омелян, О.М. |
| title | Класифікація симетрійних властивостей системи рівнянь хемотаксису |
| title_alt | Classification of symmetry properties of a system of chemotaxis equations |
| title_full | Класифікація симетрійних властивостей системи рівнянь хемотаксису |
| title_fullStr | Класифікація симетрійних властивостей системи рівнянь хемотаксису |
| title_full_unstemmed | Класифікація симетрійних властивостей системи рівнянь хемотаксису |
| title_short | Класифікація симетрійних властивостей системи рівнянь хемотаксису |
| title_sort | класифікація симетрійних властивостей системи рівнянь хемотаксису |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/10951 |
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