Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix
We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion...
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2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509380592533504 |
|---|---|
| author | Nikitin, A. G. Нікітін, А. Г. |
| author_facet | Nikitin, A. G. Нікітін, А. Г. |
| author_sort | Nikitin, A. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:00Z |
| description | We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives. |
| first_indexed | 2026-03-24T02:40:11Z |
| format | Article |
| fulltext |
UDC 517.95
A. G. Nikitin (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
GROUP CLASSIFICATION OF SYSTEMS
OF NONLINEAR REACTION-DIFFUSION EQUATIONS
WITH TRIANGULAR DIFFUSION MATRIX
HRUPOVA KLASYFIKACIQ SYSTEM
NELINIJNYX RIVNQN| REAKCI}-DYFUZI}
Z TRYKUTNOG MATRYCEG DYFUZI}
Group classification of systems of two coupled nonlinear reaction-diffusion equations with a general
diffusion matrix started in previous author’s works completed in present paper where all nonequivalent
equations with triangular diffusion matrix are classified. In addition, symmetries of diffusion systems
with nilpotent diffusion matrix and additional first order derivative terms are described.
Zaverßeno hrupovu klasyfikacig system dvox zaçeplenyx rivnqn\ reakci]-dyfuzi] z zahal\nog
matryceg dyfuzi], rozpoçatu v poperednix robotax avtora. A same, proklasyfikovano vsi neek-
vivalentni rivnqnnq z trykutnog matryceg dyfuzi]. Krim c\oho, opysano symetri] dyfuzijnyx
system z nil\potentnog matryceg dyfuzi] ta dodatkovymy çlenamy z poxidnymy perßoho po-
rqdku.
1. Introduction. Group classification is a corner stone of qualitative analysis of
differential equations. Such classification makes it possible to specify equivalence
relations in a considered class of equations. In addition, it opens a way to application
of powerful group-theoretical tools for searching for conservation laws, construction of
exact solutions, group generation of families of solutions starting with known ones, etc.
In the present paper we complete the group classification of systems of reaction-
diffusion equations with general diffusion matrix performed in papers [1, 2]. These
equations can be written in the following form:
ut – ∆m ( A11
u + A12
v ) = f 1 ( u, v ),
(1)
vt – ∆m ( A21
u + A22
v ) = f 2 ( u, v ),
where u and v are function of t, x1 , x2 , … , xm , A11, A12 , A21 and A22 are
elements of diffusion matrix (which are supposed to be constant) and ∆ m is the
Laplace operator in Rm.
Equations (1) form the grounds for a great many models of mathematical physics,
biology, chemistry, etc., which makes their group classification to be especially
relevant. A survey of investigation of symmetries of systems (1) can be found
in [1, 2].
Up to linear transformation of dependent variables there exist three a priori
nonequivalent versions of equations (1), corresponding to a diagonal, triangular and
square diffusion matrix. The equations with diagonal and square diffusion matrix have
been classified in papers [1] and [2] correspondingly.
Here we consider the remaining case when the diffusion matrix is triangular and
system (1) is reduced to the following form:
ut – a ∆m u = f 1 ( u, v ),
(2)
vt – ∆m u – a ∆m v = f 2 ( u, v ),
where a is a real constant.
© A. G. NIKITIN, 2007
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3 395
396 A. G. NIKITIN
In the case when a = 0 the diffusion matrix is nilpotent and it is naturally to
generalize (2) to the following system:
ut – pµ vxµ
= f 1 ( u, v ),
(3)
vt – ∆ u = f 2 ( u, v ),
were summation from 1 to m is imposed over the repeated indices, pµ are arbitrary
constants. Up to linear transformations of independent variables
p1 = p2 = … = pm – 1 = 0, pm = p, (4)
where p = p p pm1
2
2
2 2+ + … + .
We notice that in addition to applications in mathematical biology equations (3) can
serve as a potential ones for the nonlinear D’Alembert equation.
2. Symmetries and determining equations. To describe symmetries of
equations (2) and (3) with respect to continuous groups of transformations we use the
Lie infinitesimal approach. Using the standard Lie algorithm we find the determining
equations for coordinates η, ξ a , πb of generator X of the symmetry group:
X = η ∂t + ξν
∂xν
– π1
∂u – π2
∂v . (5)
The first set of determining equations describes dependence of η, ξν and πb on u
and v:
ηu = ηv = 0, ξ ξν ν
u = v = 0, π = π = πuu
a
u
a a
v vv = 0. (6)
Here and in the following subscripts denote derivatives with respect to the
corresponding variables, i.e., ηv =
∂
∂
η
v
, etc.
In accordance with (6) η and ξν are functions of t and xµ only, and
πa = N
a1
u + N
a2
v + Ba, a = 1, 2, (7)
where N
ab, Ba are functions of t and xν
. The remaining determining equations in
the case a ≠ 0 are [1]:
2A A A Nx m tξ δ η
µ
ν µν= − + [ ]( ), , ηxν t
= 0, (8)
ξ ξν ν
νt x maN a− −2 11 ∆ = 0, N12 = 0, N aNx xν ν
21 11= − , (9)
ηt
k kb b
t
kb
m
ks sb
b t
k
m
kc cf N f N A N u B A B+ + ( − ) + −∆ ∆ = ( + )B N u fa ab
b u
k
a
. (10)
Here N and A are matrices whose elements are N
ab and Aab, δa b is the Kronecker
symbol, and we use the temporary notation u = u1
, v = u2
.
Using (7) – (10) we find the general form of symmetry (5):
X = Ψµν
µ µ µ µ µ µν µ
ν ρ λ σ ω µx K G G Dx t x∂ + ∂ + ∂ + + + +ˆ –
– C u C u B Bu u
1 2 1 2( ∂ + ∂ ) − ∂ − ∂ − ∂v v v v , (11)
where the Greek letters denote arbitrary constants, B1, B2 are functions of t, x, and
C1, C2 are functions of t,
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GROUP CLASSIFICATION OF SYSTEMS … 397
K =
2
2
12
t t x x
a
u u tm ut x u u( ∂ + ∂ ) − ( ∂ + ∂ ) − ∂
− ( ∂ + ∂ )µ µ
v vv v v ,
Gµ =
t x
a
u
a
ux u∂ + ( ∂ + ∂ ) − ∂
µ µ
1
2
1 1
2v v v ,
(12)
Ĝ e x
a
u
a
ut
x uµ
γ
µµ
γ= ∂ + ( ∂ + ∂ ) − ∂
1
2
1 1
2v v v ,
D = t xt x∂ + ∂1
2 µ µ
.
For a = 0 symmetry X again has the form (11) where however λ = σµ = ωµ = 0.
In addition, B2 can depend not on t, x only, but also on u.
In accordance with (5), (7) and (10) equation (2) admits symmetry (11) iff the
following classifying equations are satisfied:
λ µ λ σ γ ωµ µ
γ
µ µ( + ) + + + +
+
m t
a
x x e x C ft4 1 1
2
2 1 1 +
+ C u B a B B B C u C ut t u u
1 1 1 1 2 1 2+ − =
∂ + ∂ + ( ∂ + ∂ ) + ∂∆ v v vv +
+ λ λ σ γ ωµ µ
γ
µ µmt u x x e x
a
u
a
u fu
t
u( ∂ + ∂ ) + + +
( ∂ + ∂ ) − ∂
v vv v v
1
2
1 12
2
1,
(13)
( )( + ) + + + + + +
−
λ µ λ σ γ ωµ µ
γ
µ µm t C f C f x x e x
a
f ft4 1
2
11 2 2 1 2 2 1 +
+
C C u B B a B B B C u C ut t t u u
1 2 2 2 1 1 2 1 2v vv v v+ + − − =
∂ + ∂ + ( ∂ + ∂ ) + ∂∆ ∆ +
+
λ λ σ γ ωµ µ
γ
µ µmt u x x e x
a
u
a
u fu
t
u( ∂ + ∂ ) + + +
( ∂ + ∂ ) − ∂
v vv v v
1
2
1 12
2
2 .
Equations (13) are nothing but a special case of the generic classifying
equations (15), ref. [1] which are valid for arbitrary invertible diffusion matrix.
Equation (3) needs a particular analysis. For p ≠ 0 the related symmetry
operator (5) reduces to the form
X = X X0 + ˜ , (14)
where
X0 = λ να α
∂ + ∂ + ∂t x
kl
k lxΨ ,
(15)
X̃ t x F u B Bt u u= ( ∂ + ∂ − ∂ ) − ( ∂ + ∂ ) − ∂ − ∂µ ν ν3 2 1 2v vv v v .
Here Ψµν is an antisymmetric tensor and summation is imposed over the repeated
Greek and Latin indices from 1 to m and from 1 to m – 1 correspondingly.
Symmetries X0 are admitted by equations (3), (4) with arbitrary nonlinearities f 1
and f 2 while the classifying equations generated by symmetries X̃ are
( + ) + + −3 1 1 2µ F f Fu B pBt t xm
= ( )∂ + ∂ + ∂ + ( + ) ∂B B Fu F fu u
1 2 1
v vvµ ,
(16)
( + ) + + −4 2 2 1µ F f F B Bt tv ∆ = ( )∂ + ∂ + ∂ + ( + ) ∂B B Fu F fu u
1 2 2
v vvµ ,
where F and B1, B2 are unknown functions of t and t, x respectively.
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398 A. G. NIKITIN
The determining equations for symmetries of equation (3) with p = 0 are
qualitatively different for the cases, when the number m of spatial variables x1,
x2 , … , xm is m = 1, m = 2 and m > 2. The related generator (5) has the form
X = α αD N M dt mH N m H ut x x
a
ua a
+ ( − )( )∂ + ∂ − + ( − ) ∂∫ ( )2 2 –
–
( )+ ( + ) ∂ − ∂ − ∂ − ∂M m H B B B ux u2 1 2 3
α
α v v v v , (17)
where summation from 1 to m is imposed over repeating indices, the Greek letters
denote arbitrary parameters, M, N are functions of t, B1 , B3 are functions of t, x,
B2 are functions of t, x, u, and
H x x xa
b b a a= −2 2λ λ if m > 2.
For m = 2 Ha are arbitrary functions satisfying the Cauchy – Riemann conditions:
H Hx x1 2
1 2= , H Hx x2 1
1 2= − .
In the case when m = 1 H1 is a function of x and the sums with respect to a in
(17) are reduced to single terms.
The related classifying equations have the form
( ) (+ − + ( − ) + + = ∂ + ∂α 2 2 1 1 1 2N M m H f N u B B Bx
a
t t ua v +
+
B u N m H u M m H fx
a
u x
a
a a
3 12 2∂ + + ( − ) ∂ + + ( + ) ∂( ) ( ) )v vv ,
(18)
( )+ + ( + ) + + + + −α N m H f B f M B u B Bx
a
t t ta
2 2 3 1 3 2
1v ∆ +
+
( − ) = ∂ + ∂ + ∂ + + ( − ) ∂( ( )2 21 2 3m H u B B B u N m H ux
a
u x
a
ua a
∆ v v +
+ ( ) )+ ( + ) ∂M m H fx
a
a
2 2v v .
We notice that in this case symmetry classification appears to be rather complicated
and cumbersome. Nevertheless, the classifying equations can be effectively solved
using the approach outlined in the following sections.
3. Classification of symmetries. Following [1] we specify basic, main and
extended symmetries for the analyzed systems of reaction-diffusion equations.
Basic symmetries form the kernel of the main symmetry group and so are admitted
by equation (2) with arbitrary nonlinearities f 1 and f
2. They are generated by the
following infinitesimal operators:
P0 = ∂t
, Pλ = ∂xλ
, J x xx xµν µ νν µ
= ∂ − ∂ (19)
and correspond to shifts of independent variables and rotations of variables xν.
Main symmetries X̃ form an important subclass of general symmetries (11) which
correspond to λ = σν = ων = 0 and have the following form:
X̃ = µ D + Y, Y = − ( ∂ + ∂ ) − ∂ − ∂ − ∂C u C u B Bu u
1 2 1 2v v v v . (20)
To describe all Lie symmetries admitted by equation (2) we follow the procedure
outlined in [1] which includes the following steps:
Finding all main symmetries (20), i.e., solving equations (13) for Ψµν = ν = pν =
= σν = ων = 0:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GROUP CLASSIFICATION OF SYSTEMS … 399
( + ) + + −µ C f C u B a Bt t
1 1 1 1 1∆ = ( )( ∂ + ∂ ) + ∂ + ∂ + ∂C u C u B B fu u
1 2 1 2 1v v v v ,
(21)
( + ) + + + − −µ C f C u C B a B Bt t t
1 2 2 1 2 2 1v ∆ ∆ =
= ( )( ∂ + ∂ ) + ∂ + ∂ + ∂C u C u B B fu u
1 2 1 2 2v v v v .
Specifying all cases when the main symmetries can be extended, i.e., at least one of
the following systems is satisfied:
af a u u fu
1 1= ( ∂ + ∂ ) − ∂( )v v v ,
(22)
af f a u u fu
2 1 2− = ( ∂ + ∂ ) − ∂( )v v v ;
a f u a u u fu( + ) = ( ∂ + ∂ ) − ∂( )1 1γ v v v ,
(23)
a f u a u u fu( + ) − = ( ∂ + ∂ ) − ∂( )2 2γ γv v v v
or if equation (22) is satisfied together with the following condition:
( + ) = ( ∂ + ∂ )m f m u fa
u
a4 v v , a = 1, 2. (24)
If relations (22), (23) or (24) are valid then the system (2) admits symmetry Ga , Ĝα
or K correspondingly.
When classifying equations (3) or (2) with a = 0 the second step in not needed in
as much as in accordance with (17) and (15) these equations admit basic and main
symmetries only.
The first step of the described procedure is rather complicated. In the next section
we present specific tools to solve equations (21).
4. Algebras of main symmetries for equation (2). In accordance with the plane
outlined in Section 3, to make symmetry classification of equations (2) we first
describe the main symmetries generated by operators (20) and then indicate extensions
of these symmetries.
First we note that for any f 1 and f 2 equation (2) admits the following equivalence
transformations
u → K1 u + b1, v → K1 v + K2 u + b2,
f 1 → λ2 K1 f 1, f 2 → λ2 ( K1 f 2 + K2 f 1 ), (25)
t → λ– 2 t, xb → λ– 1 xb ,
where K1, K 2 , λ , b 1 and b 2 are constants, λ K1
K2 ≠ 0. Equivalence
transformations keep the general form of equation (2) but can change the concrete
realization of its right-hand side. For some nonlinearities f 1 and f
2 there exist
additional equivalence transformation which will be specified in the following.
To solve rather complicated classifying equations (21), we use the main algebraic
property of the main symmetries, i.e., the fact that they should form a Lie algebra
(which we denote by A ). In other words, instead of going throw all nonequivalent
possibilities arising via separation of variables in the classifying equations we first
specify all nonequivalent realizations of algebra A for our equations up to arbitrary
constants and arbitrary functions. Then we easily solve classifying equations (21) with
known functions C
k and Ba.
Consider consequently one-, two-, … , n-dimensional algebras of operators (20).
Let X̃ be a basis element of a one-dimensional algebra A then commutators of X̃
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
400 A. G. NIKITIN
with P0 and Pa are equal to a linear combination of X̃ and operators (19). This can
happen in the following cases:
C
a = µa, Ba = νa,
C
a = eλ t µa, Ba = eλ t νa, (26)
C
a = 0, Ba = eλ t + ω ⋅ x νa, µ = 0,
where µab, µa, λ, and ω = ( ω1 , ω2 , … , ωm
) are constants, ω ⋅ x = ων
xν and ω ≠ 0.
To classify all nonequivalent symmetries (20), (26) we use isomorphism of the
related operators Y with 3 × 3 matrices of the following form:
g =
0 0 0
01 1
2 2 1
ν µ
ν µ µ
. (27)
Equivalence transformations (25) generate the following transformation for
matrix (27):
g → g′ = U g U
– 1, (28)
where
U =
1 0 0
01 1
2 2 1
b K
b K K
, U
– 1 = 1
0 0
1 0
1
1
1
1
1 2 2
2
1
K
K
b
b K b K
K
−
− −
. (29)
Up to transformations (28) there exist six nonequivalent matrices g, i.e.,
g1 =
0 0 0
0 1 0
0 0 1
, g2 =
0 0 0
0 0
1 0 0
λ
, g3 =
0 0 0
1 0 0
0 0 0
,
(30)
g4 =
0 0 0
0 1 0
0 1 1
, g5 =
0 0 0
0 0 0
0 1 0
, g6 =
0 0 0
1 0 0
0 1 0
.
Let us denote
̂g g u g g u g gu u= ∂ + ∂ + ∂ + ∂ + ∂22 33 32 21 31v v v v ,
where gks are elements of matrix g. Then in accordance with (26) the related
symmetries (20) have the form
˜ ˆX D g= +µ for g = g1
, g4
, g5
, g6
,
˜ ˜X e gt x= + ⋅λ ω for g = g1
, g2
, (31)
˜ ˆX e gt= λ for any g Eq. (30).
Formulae (30), (31) give the principal description of all possible one-dimension
algebras A which can be admitted by equation (2).
To describe two-dimension algebras A we classify matrices g (27) forming two-
dimension Lie algebras. Up to equivalence transformations (25) there exist six such
algebras:
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GROUP CLASSIFICATION OF SYSTEMS … 401
A2,
1 = { }g g3 2, ˜ , A2,
2 = { }g g1 5, , A2,
3 = { }g g5 2, ˜ , A2,
4 = { }g g6 2, ˜ , (32)
A2,
5 = { }g g1 2, , A2,
6 = { }g g1 3, , (33)
where g̃2 is matrix g2 (30) with λ = 0.
Algebras (32) are Abelian while (33) are characterized by the following
commutation relations:
[ e1 , e2
] = e2
. (34)
Two-dimension algebras A generated by (32) and (33) are spanned on the following
basis elements:
〈 + + 〉µ νD e te eˆ ˆ , ˆ1 2 2 , 〈 + + 〉µ νD e te eˆ ˆ , ˆ2 1 1 ,
(35)
〈 − − 〉µ νD e D eˆ , ˆ1 2 , 〈 + + 〉F e G e F e G e1 1 1 2 2 1 2 2ˆ ˆ , ˆ ˆ
and
〈 − 〉µD e eˆ , ˆ1 2 , 〈 + + 〉µ νD e te eˆ ˆ , ˆ1 2 2 (36)
respectively, where { }F G1 1, and { }F G2 2, are fundamental solutions of the following
system:
Ft = λ F + α G, G = σ F + γ G (37)
with arbitrary parameters λ , α , σ , γ. Arbitrary parameters µ and ν in particular
can be equal to zero.
In addition, there exist two dimension algebras A which are induced by one-
dimension algebras of matrices g (27), namely
〈 〉Fg Ggˆ, ˆ , 〈 + 〉+ ⋅ + ⋅µ λ ν ω ν ωD e g e gt x t xˆ, ˆ (38)
with F and G satisfying (37). Such algebras correspond to incompatible classifying
equations (21).
Up to transformations (28) there exist four three-dimension algebras A3,
1 – A3,
4
of matrices (27) and the only four-dimension algebra of such matrices which we denote
as A4 (Table 1).
Table 1. Three- and four-dimension algebras of matrices (27)
Algebra Basis elements Nonzero commutators
A3,
1 e1 = g1
, e2 = g3
, e3 = g̃2 [ e1 , e2
] = e2
, [ e1 , e3
] = e3
A3,
2 e1 = g5
, e2 = g1
, e3 = g̃2 [ e2 , e3
] = e3
A3,
3 e1 = g̃2, e2 = g5
, e3 = g6 [ e2 , e3
] = e1
A3,
4 e1 = g̃2, e2 = g3
, e3 = g4 [ e1 , e2
] = e2
, [ e1 , e3
] = e2 + e3
A4 e1 = g1
, e2 = g3
, e3 = g̃2 [ e1 , e2
] = e2
, [ e1 , e3
] = e3
e4 = g5 [ e4 , e2
] = e3
Using commutation relations present in the table we come to the following related
three-dimension algebras A:
〈 − 〉µD e e eˆ , ˆ , ˆ1 2 3 , 〈 + + 〉ˆ , ˆ ˆ , ˆ ˆe F e G e F e G e1 1 2 1 3 2 2 2 3
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
402 A. G. NIKITIN
with ea belonging to A3,
1
;
〈 − − 〉µ νD e D e e2 21 2 3ˆ , ˆ , ˆ
with ea belonging to A3,
2
;
〈 − − 〉µ νD e D e e2 22 3 1ˆ , ˆ , ˆ , 〈 + + 〉′ˆ , ˆ , ˆe D e te e1 12 2α αν ,
〈 〉+ ⋅ + ⋅
′e e e e et x t xν ω ν ω
α αˆ , ˆ , ˆ1 ,
where α, α′ = 2, 3, α′ ≠ α and ea belong to A3,
3
;
〈 − 〉µD e e e2 1 2 3ˆ , ˆ , ˆ , 〈 〉+ ⋅ + ⋅ˆ , ˆ , ˆe e e e et x t x
1 2 3
ν ω ν ω
with ea belonging to A3,
4
.
The four-dimensional algebra A4 induces algebras A given below:
〈 − − 〉µ νD e D e e e2 21 4 2 3ˆ , ˆ , ˆ , ˆ ,
〈 〉+ ⋅ + ⋅e e e e e et x t xν ω ν ωˆ , ˆ , ˆ , ˆ1 4 2 3 .
Thus we had specified algebras of main symmetries which can be admitted by
equation (2).
5. Solution of classifying equations for the case of invertible diffusion matrix.
Applying results of the previous section we can easily classify main symmetries of
equation (2). Such classification reduces to solving equations (21) with their known
coefficients C1, C2 and B1, B2 which can be found comparing (20) with the found
realizations of algebras A. To complete the group classification of equations (2) we
will specify all cases when relations (22), (23) are satisfied, i.e., when the main
symmetries can be extended.
Solving of equations (21) with known C1, C2 and B1, B 2 is a rather routine
procedure. We restrict ourselves to presentation of an example of such solution in the
following. Note that asking for invariance of (2) with respect to one-dimension algebra
A we fix the nonlinearities f 1 and f
2 up to arbitrary functions while an invariance
with respect to a two-dimension algebra A usually fixes these nonlinearities up to
arbitrary parameters.
We will solve classifying equations (21) up to equivalence transformations u →
→ ũ = U ( u, v, t, x ), v → ṽ = V ( u, v, t, x ), t → t̃ = T ( u, v, t, x ), x → x̃ = X ( u, v, t,
x ) and f a → f̃ a = Fa
( u, v, t, x, f a ) which keep the general form of equations (2) but
can change functions f 1 and f 2.
The group of equivalence transformations for equation (2) can be found using the
classical Lie approach and treating f 1 and f
2 as additional dependent variables. In
accordance with their definition, equivalence transformations include all symmetry
transformations (i.e., transformations generated by operators (19) and other symmetries
which will be found in the following) and also transformations (25). In addition, for
some particular nonlinearities f 1 and f
2 there exist additional equivalence
transformations, whose list is given in formulae:
1. u → exp ( ω t ) u, v → exp ( ω t ) v,
2. u → u + ω t, v → v,
3. u → u, v → v + ρ t,
4. u → u, v → v + ρ t u,
5. u → u, v → v – ρ t u + ρδ t2
2
,
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GROUP CLASSIFICATION OF SYSTEMS … 403
6. u → e2ρ
t
u, v → e2ρ
t
( v + ω t u + ρ t2 u ),
7. u → u + ρ t, v → v + ρ t u + ρ2
2
2
t , (39)
8. u → eκ
t
u, v → eκ
t
( v – ν κ η t u ) ,
9. u → u + 3ω t, v → v + 3ω t2 u + 6ω2
t3 + ρ t u + 3ω ρ t u2,
10. u → eν
ω
t
u, v → eν
ω
t
( v – σ ω t u ) ,
11. u → eλ
ω
t2
u, v → eλ
ω
t2
( v + 2ω t u ) ,
12. u → u + ρ t, v → v + η ρ t,
13. u → u, v → v + ρ xm
.
In the following we specify additional equivalence transformations (39) admitted by
some of equations (2).
Let us present an example of solving of classifying equations. Consider the first of
algebras A given by relation (35) with e1
, e2 belonging to algebra A2,
2 (32). It
includes two basis elements
X1 = µ D – u ∂u – v ∂v
, X2 = ν D – u ∂v
.
Comparing X1 with X̃ (20) we conclude that in this case C1 = 1, C2 = B1 = B2 =
= 0 and so the classifying equations (21) are reduced to the following ones:
( µ + 1 ) f a = ( u ∂u + v ∂v ) f a, a = 1, 2.
General solutions of this system have the form
f 1 = uµ
+
1
F1
, f 2 = vµ
+
1
F2 (40)
where F1 and F2 are arbitrary functions of
u
v
.
Asking for symmetry of equation (2) with respect to transformations generated by
X2 we come to the following classifying equations (21):
ν f 1 = u ∂v f
1,
(41)
ν f 2 + f 1 = u ∂v f
2,
Substituting (40) into (41) we come to the equation whose general solution for µ ≠
≠ 0 is
f 1 = λ µ νu e u+1 v / ,
(42)
f 2 = e u uuν µλ σv v/ ( + ) .
In the special case µ = 0 the solution has the form
f 1 = λ ωνue uuv / + ,
(43)
f 2 = e uuν λ σ ωv v v/ ( + ) + .
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404 A. G. NIKITIN
However, in this case there exist the additional equivalence transformation (39), Item 1
which reduces parameter ω in (43) to zero. So without loss of generality we can
restrict ourselves to solutions (42) for any µ.
Thus equation (2) admits the two-dimension algebra of main symmetries spanned
on X1
, X2 provided f 1 and f 2 have the form (42). This symmetry can be extended
if functions (42) and f 2 satisfy one of conditions (22), (23) or both the conditions (22),
(24).
Equation (23) is incompatible with (42). In order equation (22) be satisfied we
have to impose the condition µ = – a ν on parameters µ, ν and a . The related
equation (2), (42) has the form
ut – a ∆ u = λ ν νu ea u1− v / ,
(44)
ut – ∆ u – a ∆ v = e u uu aν νλ σv v/ ( + ) −
and admits the Galilei generators Gµ (12). Finally, asking for equation (24) be
satisfied we obtain one more condition ν = − 4
a
m which guaranties invariance of
equation (44) with respect to the conformal generator K of (12).
The obtained classification results are presented in Table 2, Item 3.
In analogous way we solve classifying equations for other algebras A and specify
the cases when the main symmetry can be extended.
6. Classification results for equations (2) with invertible diffusion matrix. The
classification results are given in the following Tables 1 – 5 when we also indicate the
additional equivalence transformations (AET) (39) which are admitted by some
particular equations (2). The symbols D, Ĝν , Gν and K denote operators given by
Eq. (12).
In the following tables F1
, F2 and F are arbitrary functions whose arguments are
specified in the third column, Ψ ( x ) is an arbitrary function of x1 , x2 , … , xm and
ψν ( x ) is a solution of the linear heat equation ( ∂t – a ∆ ) ψν = ν ψν . In addition, we
denote by Ψµ ( x ) a solution of the Laplace equation ∆ Ψµ ( x ) = µ Ψµ ( x ).
The letters ε, η and δ denote parameters which can take the values ε = ± 1, ν =
= 0, 1 and δ = 0, ± 1; the other Greek letters denote parameters which can take any
(including zero) real values.
Table 2. Nonlinearities with arbitrary functions,
symmetries and AET for equations (2)
No. Nonlinear terms
Arguments
of F1
, F2
, F
Symmetries and AET
Eq. (39)
1 f 1 = δ u + F1 2v – u2 ψδ ( u ∂v + ∂u )
f 2 = δ u2 + F1 u + F2
2 f 1 = eη
u
F1 2v – u2 η D – u ∂v – ∂u
f 2 = eη
u
( F2 + F1 u ) [AET 7 if η = 0]
3 f 1 = F1 u ψδ
∂v
f 2 = F2 + δ v [AET 3 if δ = 0]
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GROUP CLASSIFICATION OF SYSTEMS … 405
Continuation of Table 2
No. Nonlinear terms
Arguments
of F1
, F2
Symmetries and AET
4 f 1 = α u + η u ψδ
∂v
, e(
δ
–
α
)
t
( u – η t ) ∂v
f 2 = δ v + F, α η = 0 [AET 3 if δ = 0;
AET 4 if δ = α, η = 0]
5 f 1 = u2 u eδ
t
u ∂v
f 2 = ( u + δ ) v + F eδ
t
( ∂v + t u ∂v )
6 f 1 = u2 – 1 u e(
ν
+
1
)
t
( u ∂v + ∂v )
f 2 = ( u + ν ) v + F e(
ν
–
1
)
t
( u ∂v – ∂v )
7 f 1 = u2 + 1 u eν
t
( cos t u ∂v – sin t ∂v )
f 2 = ( u + ν ) v + F eν
t
( sin t u ∂v + cos t ∂v )
8 f 1 = e Fuuv /
1 u D – u ∂v
f 2 = e F Fuv v/ ( + )1 2
9 f 1 = ev
F1 , f 2 = ev
F2 u D – ∂v
10 f 1 = eu
F1 , f 2 = eu
F2 v D – ∂u
11 f 1 = δ u + F1 v e(
δ
+
a η
)
t
Ψη ( x ) ∂u
f 2 = – η u + F2
12 f 1 = u ( F1 + ε ln u )
u
v
eε
t
( u ∂u + v ∂v )
f 2 = v ( F2 + ε ln u )
13 f 1 = u F1 + δ η v ue u− ηv / eδ
t
( u ∂v + η ( u ∂u + v ∂v ) )
f 2 = δ u
v
( u + η v ) + & Ĝα if a = – η, δ η ≠ 0
+ u F2 + v F1 or Gα if a = – η ≠ 0, δ = 0
14 f 1 = uε
+
1
F1 ue uv / ε D + u ∂v – u ∂u – v ∂v
f 2 = uε
( F2 u – F1 v )
15 f 1 = uµ
+
1
F1
v
u
µ D – u ∂u – v ∂v
f 2 = uµ
+
1
F2 [AET 1 if µ = 0]
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406 A. G. NIKITIN
Table 3. Nonlinearities with arbitrary parameters
and symmetries for equations (2) with a ≠≠≠≠ 0
No. Nonlinear terms Symmetries AET (39)
1 f 1 = δ, f 2 = uν ψ0
∂v
, ( u – δ t ) ∂v for any ν, δ 3, 5 & 9
δ = 0 if ν = 2 & u ∂u + ν v ∂v for δ = 0 if δ = 0,
& ∂u + 2t u ∂v for ν = 2 ν = 2
2 f 1 = u, f 2 = ε uν ψ0
∂v
, e– t u ∂v for any ν 3
ν ≠ 0, 1 & et ( u ∂v + ε ∂u ) for ν = 2
3 f 1 = δ eu D – ∂u , ψ0
∂v 3 & 4 if
f 2 = η eu & u ∂v if δ = 0 δ = 0
4 f 1 = δ uν
+
1
e
uηv / η D – u ∂v
, ν D – u ∂u – v ∂v 8 & 4 if
f 2 = e u uuη νδ σv v/ ( + ) & Gα if ν = η a η = 0
( ν – 1 )
2 + η + σ2 ≠ 0 & K if ν = 4
m
5 f 1 = u2, f 2 = u v D – u ∂u – v ∂v , u ∂v , ( 1 + t u ) ∂v 4
6 f 1 = ε uµ
+
1, f 2 = 0 µ D – u ∂u – v ∂v , ψ0
∂v 3
7 f 1 = η vν
+
1, f 2 = δ vν
+
1 ν D – u ∂u – v ∂v , Ψ0 ( x ) ∂u 2
8 f 1 = η ev, f 2 = δ ev D – ∂v , Ψ0 ( x ) ∂u 2
9 f 1 = δ eu, f 2 = ε u eu δ ( D – ∂u ) – u ∂v , ψ0
∂v 3 & 4 if
& u ∂v if δ = 0 δ = 0
10 f 1 = ηe u2 2v− 2D – ∂v , ∂v + u ∂v 7
f 1 = ( + ) −η µu e u2 2v
11 f 1 = η ln v Ψ0 ( x ) ∂u
, D + u ∂u + v ∂v + 2
f 2 = δ ln v + ( − ) −
∂η δ δa t
m
x u2
2
12 f 1 = δ D + u ∂u + v ∂v + ε t ∂v 3, 5 & 4
f 2 = ε ln u ψ0
∂v
, ( u – δ t ) ∂v if δ = 0
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GROUP CLASSIFICATION OF SYSTEMS … 407
Continuation of Table 3
No. Nonlinear terms Symmetries AET (39)
13 f 1 = ε u ln v eε t ( u ∂u + v ∂v ), v ∂v 4
f 2 = ε v ln u
14 f 1 = ε uν
+
1, ν ≠ – 1 ν D – u ∂u – v ∂v – ε u ∂v 3
f 2 = uν
+
1
ln u ψ0
∂v
15 f 1 = ε u D – u ∂u – v ∂v – t u ∂v
, u ∂v 4
f 2 = ε v + u ln u
16 f 1 = 2 v – u2 X1 = eε t ( 2∂u + 2u ∂v + ε ∂v ) —
f 2 =
( + )( − ) −ε u u u2
2
2v 2t X1 + eε t ∂v
17 f 1 = 2 v – u2 X± = e ut
u
( ± ) ( ∂ + ∂µ 1 2 2 v + 7 if µ2 = 1
f 2 = ( + )( − )µ u u2 2v + + ( ± )∂ )µ 1 v
+
1
2
2− µ
u
18 f 1 = 2 v – u2 e t ut
u
µ ( (∂ + ∂ )2cos v + —
f 2 = ( + )( − )µ u u2 2v – + ( − )∂ )2µ cos sint t v
–
1
2
2+ µ
u e t ut
u
µ ( (∂ + ∂ )2sin v +
+ ( + )∂ )2µsin cost t v
In the following table we present symmetries of a special subclass of equations (2)
whose right-hand side is given in the table title. The following notations are used here:
∆ = 1
4
2( − ) +µ ν λσ , Ω = 1
2
( + )µ ν , ω± = Ω ± 1 where µ , ν , λ and σ are
parameter used in definition of the related nonlinearities f 1, f 2.
Table 4. Symmetries of equations (2) with nonlinearities
f
1 = λλλλ vvvv + µµµµ u ln u, f
2 =
λλ vv2
u
+ (((( σσσσ u + µµµµ vvvv )))) ln u + νννν vvvv and a ≠≠≠≠ 0
No.
Conditions for
coefficients
Symmetries
and AET Eq. (39) Additional symmetries
1 λ = 0 e utδ ∂v , e R tut
R
δ σ( ∂ + ∂ )v Gα if σ = δ = 0
µ = ν = δ [AET 6 if µ = 0] Ĝα if σ = 0, δ ≠ 0
̃ψ0∂v if µ = 0
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408 A. G. NIKITIN
Continuation of Table 4
No.
Conditions for
coefficients
Symmetries
and AET Eq. (39) Additional symmetries
2 λ = 0, µ ≠ ν e R ut
R
µ µ ν σ( )( − ) ∂ + ∂v , e utν ∂v Ĝα if µ ≠ 0
µ2 + ν2 = 1 [AET 10 if µ ν = 0] σ = a ( ν – µ )
ψν
∂v if µ = 0
& Gα if a ν = σ
3 ∆ = 0 X4 = e R ut
R
Ω ( )∂ + ( − ) ∂2ε ν µ v Gα if Ω = 0
λ = ε 2
4e u tXtΩ ∂ +v [AET 10 if ν = – aε ≠ 0
µ + ν = 2Ω µ = – ν & 1, 11 if Ĝα if Ω ≠ 0
µ = ν = 0] 2aε = ( µ – ν ) ≠ 0
4 λ ≠ 0 e R ut
R
ω λ ω µ+ ( )∂ + ( − ) ∂+ v Gα if µ = a λ
∆ = 1 e R ut
R
ω λ ω µ− ( )∂ + ( − ) ∂− v ω+ ω– = 0
ω± = Ω ± 1 [AET 10 if µ ν = λ σ Ĝα if ω+ ≠ 0
& 1 if µ = σ = 0] aλ = – ( ω+ – µ )
5 ∆ = – 1 e tRt
R
Ω [ ∂2λ cos + none
+ ( ]( − ) − ) ∂ν µ cos sint t u2 v
e tRt
R
Ω [ ∂2λsin +
+ ( ]( − ) + ) ∂ν µ sin cost t u2 v
7. Group classification of reaction-diffusion equations with nilpotent diffusion
matrix. Consider now equations (3), (4) and specify their Lie symmetries. We restrict
ourselves to the case p ≠ 0 when generators of admitted Lie group have the general
form (14) while the related classifying equations are given by formula (16). Moreover
without loss of generality we put p = 1.
The results of group classifications are presented in Table 5. All the corresponding
equations (3) admit the additional equivalence transformations
t → ρ t, x → ρ2 3/ x , v → ω v, u → ωρ1 3/ u. (45)
Table 5. Nonlinearities and symmetries for equations (3), (4) with p = 1
No. Nonlinearities Arguments
of F1
, F2
Symmetries
1 f 1 = u1
+
3µ
F1 vu− −µ 1 2µD̃ – u ∂u – v ∂v
f 2 = u1
+
4µ
F2 [AET 1 if µ = 0]
2 f 1 = v3
F1 u – η ln v D̃ – η ∂u
f 2 = v4
F2
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GROUP CLASSIFICATION OF SYSTEMS … 409
Continuation of Table 5
No. Nonlinearities Arguments
of F1
, F2
Symmetries
3 f 1 = u ( F1 + η ln u ) v
u
eη
t
( u ∂u + v ∂v )
f 2 = v ( F2 + η ln u ) [AET 1 if η = 0]
4 f 1 = u– 2
F1 v – η ln u D̃ + u ∂u + v ∂v + η ∂v
f 2 = u– 3
F2
5 f 1 = ν v + F1 u e xt xmη ν− ( )∂Ψ ˜ v
f 2 = η v + F2 [AET 3, 13 if η = ν = 0]
6 f 1 = η̃ u + F1 η u – v e x x tt
m u
λ
µ η ηΨ ( + )(∂ + ∂ )˜, ˜
v
f 2 = λ u + F2 µ = η̃η – λ [AET 12
if λ = η̃ = 0]
7 f 1 = µ u– 2
v3 D̃ , u ∂u + v ∂v [AET 1]
f 2 = ν u– 3
v4
8 f 1 = η e3u D̃ – ∂u
, Ψ( )∂x̃ v [AET 3]
f 2 = ν e4u
9 f 1 = α e– 2v D̃ + u ∂u + v ∂v + ∂v
, Ψ0( )∂x u
f 2 = η e– 3v [AET 2]
10 f 1 = η u3µ
+
1 µD̃ – u ∂u – v ∂v
, Ψ( )∂x̃ v
f 2 = ν u4µ
+
1 [AET 3, 13]
11 f 1 = α v2ν
+
1 νD̃ – u ∂u – v ∂v
, Ψ0( )∂x u
f 2 = η v3ν
+
1 [AET 2]
12 f 1 = α u1
/
4 1
4
D̃ + v ∂v + u ∂u + δ t ∂v
, Ψ( )∂x̃ v
f 2 = δ ln u [AET 3, 13]
13 f 1 = ν ln v
1
3
D̃ + u ∂u + v ∂v + η u t ∂u
, Ψ0( )∂x u
f 2 =
η
v
[AET 2]
Here Ψµ ( x ) and Ψ( + )˜,x x tm are arbitrary solutions of the Laplace equation
∆ Ψµ = µ Ψµ in m-dimensional space, ˜ ˜Ψµ( )x is a solution of the Laplace equation in
( m – 1 )-dimensional space, x̃ = ( x1 , x2 , … , xm – 1 ). Finally, we denote D̃ = 3t ∂t +
+ 2xν
∂ν – v ∂v
.
We solve the classifying equations using the technique developed in Sections 5 and
6. The general analysis of admissible algebras A can be carried out in complete
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
410 A. G. NIKITIN
analogy with Section 4. Moreover, the results present in Section 4 can be extended to
the case of equations (3), (4) provided we make a formal change D → D̃ = 3t ∂t +
+ 2xν
∂ν – v ∂v in all formulae where the operator D appeared, and exclude all
algebras A where matrices g4
, g5 and g6 (30) appear. Of course it is necessary to
take into account that in contrast with D operator D̃ does not commute with ∂ v
. As
a result we come to the following one-dimension algebras
˜ ˜( )X D u u1
1 = − ∂ − ∂µ v v ,
˜ ˜( )X D u1
2 = − ∂ν ,
˜ ( )X e ut
u2
ν ν= ( ∂ + ∂ )v v ,
(46)
˜ ˜( )X D u u1
3 = + ∂ + ∂ + ∂v v vν ,
˜ ( )X e t x
u3
3 3 3= (∂ + ∂ )+σ ρ
v ,
˜ ( )X e t x
u3
1 3 1= ∂+ ⋅σ ρ ,
˜ ( )X e t x
3
2 2 2= ∂+ ⋅σ ρ
v
and two-dimension algebras
˜ ˜ , ˜ ( )A D X1 2
0= 〈 〉 , ˜ ˜ , ˜( ) ( )A X X2 1
2
3
3= 〈 〉, ˜ ˜ , ˜( ) ( )A X X3 1
3
3
1= 〈 〉,
˜ ˜ , ˜( ) ( )A X X4 1
1
3
1= 〈 〉 ,
˜ ˜ , ˜ ( )A D u t Xu6 3
24= 〈 + ( ∂ + ∂ ) + ∂ 〉v v v , (47)
˜ ˜ , ˜( ) ( )A X X5 1
1
3
2= 〈 〉 ,
˜ ˜ , ˜ ( )A D u t Xu7 3
13= 〈 + ( ∂ + ∂ ) + ∂ 〉v v v .
In this way the problem of group classification of the equations with the first order
derivative terms reduces to solving the classifying equations (16) which their known
coefficients B1, B2, F and specifying the case when these equations have nontrivial
solutions. Solving the related classifying equations (16) we easily find the related
nonlinearities f 1, f 2 which are given in Table 5.
In accordance with the results present in Table 7 equations (3), (4) with p ≠ 0 can
admit neither Galilei nor conformal symmetry transformations. This result follows
directly from formulae (14), (15).
8. Discussion. We complete the group classification of systems of reaction-
diffusion equations started in papers [1, 2], where systems with a diagonal and square
diffusion matrix are studied. To save a room we do not present the classification
results for equations (3) with p = 0 which can be found in preprint [3].
The case of triangular diffusion matrix considered in the present paper appears to be
rather complicated due to very large number of versions with different symmetries.
In Table 2 equations defined up to arbitrary functions are presented.
Tables 3 and 4 include the results of classification of equations (2) which are
defined up to arbitrary parameters.
Note that we did not consider linear equations whose group classification is a rather
trivial problem.
Among the classified equations there are only seven of them being invariant with
respect to the Galilei transformations. The general form of Galilei-invariant
equations (2) is
ut – a ∆ u = u F1 ,
(48)
vt – ∆ u – a ∆ v = u F2 – v F1
,
where F1 and F2 are arbitrary functions of variable ξ = ue uv / .
The only system of equations (2) which admits the extended Galilei group including
the dilatation and conformal transformations is given by formula (44).
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
GROUP CLASSIFICATION OF SYSTEMS … 411
Thus we have completed the group classification of systems of coupled reaction-
diffusion equations (1) started in paper [4] and continued with varying success in [5 –
10] and [1, 2]. The number of nonequivalent equations of this type appears to be very
large ( > 300 ). Nevertheless, using the approach presented in [1] it was possible to
make an effective classification of pairs of coupled reaction-diffusion equations with
general diffusion matrix. Moreover, we classify the equations with arbitrary number of
independent variables.
A survey of results of group classification of systems of reaction-diffusion
equations with general diffusion matrix can be found in [11]. Essentially new point of
the present paper (in comparison with [11] is completed description of equivalence
transformations and their systematical use to simplify the classified equations.
We notice that group analysis of reaction-diffusion equations is being intensively
developed in many lines including equations with arbitrary elements depending on t, x,
u, ut , ux , … , refer to [12] for a survey. Exact solutions generated by conditional
symmetry of multidimensional diffusion equations where found in paper [13]. Thus
our analysis of systems of diffusion equations is nothing but a part of general study of
diffusion models which is currently rather popular.
1. Nikitin A. G. Group classification of systems of non-linear reaction-diffusion equations with
general diffusion matrix. I. Generalized Ginzburg – Landau equations // J. Math. Anal. and Appl. –
2006. – 324, Issue 1. – P. 615 – 628.
2. Nikitin A. G. Group classification of systems of non-linear reaction-diffusion equations with
general diffusion matrix. II. Generalized Turing systems // J. Math. Anal. and Appl. – 2007,
doi:10.1016/j.jmaa.2006.10.032.
3. Nikitin A. G. Group classification of systems of non-linear reaction-diffusion equations with
general diffusion matrix. III. Triangular diffusion matrix / math-ph/0502048.
4. Danilov Yu. A. Group analysis of the Turing systems and of its analogues. – 1980. – (Preprint /
Kurchatov Inst. Atomic Energy, IAE-3287/1).
5. Nikitin A. G., Wiltshire R. Symmetries of systems of nonlinear reaction-diffusion equations // Proc.
Inst. Math. Nat. Acad. Sci. Ukraine. – 2000. – 30. – P. 47 – 59.
6. Cherniha R. M., King J. R. Lie symmetries of nonlinear multidimensional reaction-diffusion
systems. I // J. Phys. A: Math. and Gen. – 2000. – 33. – P. 267 – 282.
7. Cherniha R. M., King J. R. Lie symmetries of nonlinear multidimensional reaction-diffusion
systems. I. Addendum // Ibid. – P. 7839 – 7841.
8. Nikitin A. G., Wiltshire R. Systems of reaction diffusion equations and their symmetry properties //
J. Math. Phys. – 2001. – 42. – P. 1667.
9. Nikitin A. G., Popovych R. O. Group classification of nonlinear Schrödinger equations // Ukr. Mat.
Zh. – 2001. – 53, # 8. – P. 1053 – 1060.
10. Cherniha R. M., King J. R. Lie symmetries of nonlinear multidimensional reaction-diffusion
systems. II // J. Phys. A: Math. and Gen. – 2002. – 36. – P. 405 – 425.
11. Nikitin A. G. Group classification of systems of nonlinear reaction-diffusion equations // Ukr.
Math. Bull. – 2005. – 2, # 2. – P. 153 – 204.
12. Popovych R. O., Ivanova N. M. New results on group classification of nonlinear diffusion-
convection equations // J. Phys. A: Math. and Gen. – 2004. – 37, # 30. – P. 7547 – 7565.
13. Barannyk T. A. Conditional symmetries and exact solutions for multidimensional diffusion
equation // Ukr. Mat. Zh. – 2002. – 54, # 10. – P. 1416 – 1460.
Received 02.10.2006
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 3
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| resource_txt_mv | umjimathkievua/a8/464ec9bf8a433473eb236a6809819da8.pdf |
| spelling | umjimathkievua-article-33152020-03-18T19:51:00Z Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix Групова класифікація систем нелінійних рівнянь реакції-дифузії з трикутною матрицею дифузії Nikitin, A. G. Нікітін, А. Г. We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives. Завершено групову класифікацію систем двох зачеплених рівнянь реакції-дифузії з загальною матрицею дифузії, розпочату в попередніх роботах автора. A саме, прокласифіковано всі нееквівалентні рівняння з трикутною матрицею дифузії. Крім цього, описано симетрії дифузійних систем з нільпотентною матрицею дифузії та додатковими членами з похідними першого порядку. Institute of Mathematics, NAS of Ukraine 2007-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3315 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 3 (2007); 395–411 Український математичний журнал; Том 59 № 3 (2007); 395–411 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3315/3375 https://umj.imath.kiev.ua/index.php/umj/article/view/3315/3376 Copyright (c) 2007 Nikitin A. G. |
| spellingShingle | Nikitin, A. G. Нікітін, А. Г. Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title | Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title_alt | Групова класифікація систем нелінійних рівнянь реакції-дифузії з трикутною матрицею дифузії |
| title_full | Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title_fullStr | Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title_full_unstemmed | Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title_short | Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| title_sort | group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3315 |
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