Classification of admissible transformations of differential equations
The framework of group classification is modified and extended to classification of admissible transformations in classes of differential equations. For this purpose, existing notions of group analysis are revised. Recently introduced notions (conditional equivalence group, normalized class of diffe...
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| author | Popovych, R. Попович, Р. |
| author_facet | Popovych, R. Попович, Р. |
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| description | The framework of group classification is modified and extended to classification of admissible transformations in classes of differential equations. For this purpose, existing notions of group analysis are revised. Recently introduced notions (conditional equivalence group, normalized class of differential equations) are described and their properties are investigated. |
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Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 239–254
УДК 517.912:512.816
Classification of admissible
transformations of differential
equations
R.O. POPOVYCH
Institute of Mathematics of NAS of Ukraine, Kyiv
Fakultät für Mathematik, Universität Wien
E-mail: rop@imath.kiev.ua
Концепцiю групової класифiкацiї модифiковано i поширено до класи-
фiкацiї допустимих перетворень у класах диференцiальних рiвнянь.
З цiєю метою переглянуто iснуючi поняття групового аналiзу. Описано
недавно введенi поняття (умовна група еквiвалентностi, нормалiзова-
ний клас диференцiальних рiвнянь) та дослiджено їх властивостi.
The framework of group classification is modified and extended to classi-
fication of admissible transformations in classes of differential equations.
For this purpose, existing notions of group analysis are revised. Recently
introduced notions (conditional equivalence group, normalized class of di-
fferential equations) are described and their properties are investigated.
1. Introduction. The beginnings of the theory of Lie groups and Lie
algebras were inseparably linked with group analysis of differential equa-
tions and, in particular, with group classification problems. Inspired by
the idea of creating a universal theory of integration of ordinary di-
fferential equations similar to the Galois theory of solving algebraic
equations, S. Lie developed the theory of continuous transformation
groups, classified such locally non-singular groups acting on the complex
and real planes, described their differential invariants and then carried
out group classification of second-order ordinary differential equations.
At present there is a substantial number of papers devoted to studying
important classes of differential equations of theoretical and mathemati-
cal physics, biology, financial mathematics and other sciences from the
Lie symmetry point of view (see e.g. all references in this paper and citati-
on therein). The group classification in a class of (systems of) differen-
tial equations is reduced to integration of a complicated overdetermi-
ned system of partial differential equations with respect to both coeffici-
ents of infinitesimal symmetry operators and arbitrary elements. That is
240 R.O. Popovych
why it is a considerably more complicated problem than finding the Lie
symmetry group of a single differential equation. Whereas programs for
solving the latter problem had been created for most existing symbolic
calculations packages a significant progress in computer realization of
the group classification algorithm was achieved only recently.
Classes of differential equations are usually chosen based on their
importance for applications without any mathematical background, al-
though such choice is an important step in successful and exhaustive
classification. It is a well-established fact that in the presence of certain
properties with respect to point transformations the implementation of
group classification is simplified and final results can be formulated in
a clear and complete form. As is often the case, before mathematical
notions are defined in a rigorous and precise form, they can be implicitly
used for a long time. This commonplace is particularly true for the noti-
on of a normalized class of differential equations, which was introduced
recently [6, 18, 19] and may become a cornerstone of the framework of
classification problems of group analysis. Knowledge that a class of dif-
ferential equations is normalized allows to reduce the group classification
problem in this class to subgroup analysis of the corresponding equi-
valence group.
Development of techniques of group analysis also allows to formulate
and to solve new classification problems concerning transformational
properties of differential equations. In particular, the notions of condi-
tional equivalence group and normalized class of differential equations
gives a language to describe complete sets of admissible transformations
for classes of differential equations.
In this paper we outline extension of the framework of group classi-
fication to classification of admissible transformations in classes of diffe-
rential equations. For this purpose, existing notions of group analysis
(class of differential equations, equivalence group, gauge equivalence group
[15], form-preserving transformation [8–10]) are discussed and revised.
Recently introduced notions (conditional equivalence group [20], norma-
lized class of differential equations [6,19]) are described and their proper-
ties are investigated.
Note 1. All functions are assumed to be smooth (e.g. analytical) and
defined on certain subsets of their variables. A point transformation in
the space of the variables z = (z1, . . . , zk) is a smooth function ϕ: z̃ =
ϕ(z) which is invertible at least locally.
Classification of admissible transformations 241
2. Classes of systems of differential equations. Let Lθ be a
system L(x, u(p), θ(x, u(p))) = 0 of l differential equations form unknown
functions u = (u1, . . . , um) of n independent variables x = (x1, . . . , xn).
Here u(p) denotes the set of all the derivatives of u with respect to x of
order no greater than p, including u as the derivatives of the zero order.
L = (L1, . . . , Ll) is a tuple of l fixed functions depending on x, u(p)
and θ. θ denotes the tuple of arbitrary (parametric) functions θ(x, u(n)) =
(θ1(x, u(p)), . . . , θ
k(x, u(p))) running the set S of solutions of the auxiliary
system S(x, u(p), θ(q)(x, u(p))) = 0. This system consists of differential
equations with respect to θ, where x and u(p) play the role of independent
variables and θ(q) stands for the set of all the partial derivatives of θ of
order no greater than q. Sometimes the set S is additionally constrained
by the non-vanish condition S′(x, u(p), θ(q)(x, u(p))) 6= 0 with another
tuple S′ of differential functions. In what follows we call the functions θ
as arbitrary elements. Denote the class of systems Lθ with the arbitrary
elements θ running S as L|S .
Let Lkθ denote the set of all algebraically independent differential
consequences of Lθ, which have, as differential equations, orders no
greater than k. We identify Lkθ with the manifold determined by Lkθ
in the jet space J (k). In particular, Lθ is identified with the manifold
determined by Lpθ in J (p).
It should be noted that the above definition of a class of systems of di-
fferential equations is not complete. The problem is that correspondence
θ → Lθ between arbitrary elements and systems (treated not as formal
algebraic expressions but as real systems of differential equations or
manifolds in J (p)) may be not one-to-one. Namely, the same system may
correspond to different values of arbitrary elements. A reason of this
indeterminancy is that different values θ and θ̃ of arbitrary elements can
result after substitution of them to L in the same expression in x and u(p).
Moreover, it is enough for Lpθ and L
p
θ̃
to coincide if the associated system
completed with independent differential consequences differ each from
other with a nonsingular matrix being a function in the variables of J (p).
The values θ and θ̃ of arbitrary elements are called gauge-equivalent
(θ g∼ θ̃) if Lθ and Lθ̃ are the same system of differential equations. For the
correspondence θ → Lθ to be one-to-one, the set S of arbitrary elements
should be factorized with respect to the gauge equivalence relation. We
formally consider Lθ and Lθ̃ as different representations of the same
system from L|S . It is often possible to realize gauge informally via
changing the chosen representation of the class under consideration with
242 R.O. Popovych
replacement of the number k of arbitrary elements and the differential
functions L and S although then this may result in more complicated
calculations.
Definition 1. The classes L|S and L′|S′ are called similar if n =
n′, m = m′, p = p′, k = k′ and there exists a point transformation
Ψ: (x, u(p), θ)→ (x′, u′(p), θ
′) which is projectible on the space of (x, u(q))
for any 0 ≤ q ≤ p, and Ψ|(x,u(q)) being the q-th order prolongation of
Ψ|(x,u), ΨS = S ′ and Lθ′ = Ψ|(x,u)Lθ.
Hereafter the action of a such point transformation Ψ in the space of
(x, u(p), θ) on arbitrary elements from S as pth-order differential functi-
ons is given by the formula:
θ̃ = Ψθ if θ̃(x, u(p)) = Ψθ
(
Θ(x, u(p)), θ
(
Θ(x, u(p))
))
,
where Θ = (prpΨ|(x,u))
−1 and prp denotes the operation of standard
prolongations of a point transformations to the derivatives of orders not
greater than p.
The set of transformations used in definition 1 can be extended via
admitting different kinds of dependence on arbitrary elements in the
ways as it is made for equivalence groups below.
Similar classes of systems have similar properties with the group
analysis point of view.
Subclasses are singled out in the class L|S with additional auxiliary
systems (or non-vanish conditions) which are attached to the main auxi-
liary system and the set of non-vanish conditions. Note that unions and
intersections of subclasses of L|S also are subclasses of L|S :
L|S′∪ L|S′′ = L|S′∪S′′, L|S′∩ L|S′′ = L|S′∩S′′, S ′,S ′′⊂ S.
3. Admissible transformations. For θ, θ̃ ∈ S we call the set of
point transformations which maps the system Lθ into the system Lθ̃
as the set of admissible transformations from Lθ into Lθ̃ and denote it
by T(θ, θ̃). The maximal point symmetry group Gθ of the system Lθ
coincides with T(θ, θ). If the systems Lθ and Lθ̃ are equivalent with
respect to point transformations then T(θ, θ̃) = Gθ ◦ ϕ0 = ϕ0 ◦ Gθ̃,
where ϕ0 is a fixed transformation from T(θ, θ̃). Otherwise, T(θ, θ̃) = ∅.
The set T(θ,L|S) = { (θ̃, ϕ) | θ̃ ∈ S, T(θ, θ̃) 6= ∅, ϕ ∈ T(θ, θ̃) }
is called the set of admissible transformations of the system Lθ in the
class L|S .
Classification of admissible transformations 243
Analogously, T(L|S) = {(θ, θ̃, ϕ) | θ, θ̃ ∈ S, T(θ, θ̃) 6= ∅, ϕ ∈ T(θ, θ̃)}
is called the set of admissible transformations in L|S .
First the set of admissible transformations was described by Kingston
and Sophocleous for a class of generalised Burgers equations. These
authors call transformations of such type form-preserving ones [8–10].
Notions and results adduced in this and the next sections can be re-
formulated in the infinitesimal terms by means of using the notions of
vector fields, Lie algebras instead of point transformations, Lie groups
etc. For instance, see [3] for the definition of “cones of tangent equiva-
lences”, which is the infinitesimal analogue of the definition of T(θ,L|S).
Ibid a non-trivial example of semi-normalized classes of differential equa-
tions (see definition 7) is investigated in the framework of the infinitesi-
mal approach.
In the case of one dependent variable (m = 1) we can extend above
and below notions to contact transformations.
An element (θ, θ̃, ϕ) from T(L|S) is called a gauge admissible trans-
formations in L|S if θ g∼ θ̃ and ϕ is the identical transformation.
Proposition 1. Similar classes have similar sets of admissible trans-
formations. Namely, a similarity transformation Ψ from the class L|S
into the class L′|S′ generates a one-to-one mapping ΨT from T(L|S) into
T(L′|S′) via the rule (θ′, θ̃′, ϕ′) = ΨT(θ, θ̃, ϕ) if θ′ = Ψθ, θ̃′ = Ψθ̃ and
ϕ′ = Ψ|(x,u)
−1 ◦ϕ◦Ψ|(x,u). Here (θ, θ̃, ϕ) ∈ T(L|S), (θ′, θ̃′, ϕ′) ∈ T(L′|S′).
Proposition 2. T(L|S′) ⊂ T(L|S) for any subclass L|S′ of the class L|S .
If L|S′′ is another subclass of L|S then T(L|S′) ∩ T(L|S′′) = T(L|S′∩S′′).
A number of notions connected with admissible transformations in
classes of systems of differential equations can be reformulated in terms
of the category theory [23].
4. Equivalence groups. The usual equivalence group of the class
L|S is defined in a rigorous way via the notion of admissible transformati-
ons. Namely, any element Φ from the usual equivalence group G∼ =
G∼(L|S) of the class L|S is a point transformation in the space of
(x, u(p), θ), which is projectible on the space of (x, u(p′)) for any 0 ≤
p′ ≤ p, and Φ|(x,u(p′))
being the p′-th order prolongation of Φ|(x,u), and
∀θ ∈ S: Φθ ∈ S and Φ|(x,u) ∈ T(θ,Φθ).
Let us remind that the point transformation ϕ: z̃ = ϕ(z) in the space
of the variables z = (z1, . . . , zk) is called projectible on the space of
the variables z′ = (zi1 , . . . , zik′ ), where 1 ≤ i1 < · · · < ik′ ≤ k, if the
244 R.O. Popovych
expressions for z̃′ depend only on z′. We denote the restriction of ϕ on
the space of z′ as ϕ|z′ : z̃′ = ϕ|z′(z′).
If the arbitrary elements θ explicitly depend on x and u only (one
always can do it formally, assuming derivatives as new dependent vari-
ables), we can admit dependence of transformations of (x, u) on θ and
consider the generalized equivalence group G∼gen = G∼gen(L|S) [16]. Any
element Φ from G∼gen is a point transformation in the space of (x, u, θ)
such that ∀θ ∈ S: Φθ ∈ S and Φ(·, ·, θ(·, ·))|(x,u) ∈ T(θ,Φθ).
The action of Φ ∈ G∼gen on arbitrary elements as functions of (x, u) is
given by the formula: θ̃ = Φθ if θ̃(x, u) = Φθ(Θ(x, u), θ(Θ(x, u))), where
Θ = (Φ(·, ·, θ(·, ·))|(x,u))
−1.
Roughly speaking, G∼ is the set of admissible transformations which
can be applied to any θ ∈ S and G∼gen is formed by the admissible
transformations which can be separated to classes parameterized with θ
running whole S.
It is possible to consider other generalizations of equivalence groups,
e.g. groups with transformations which are point with respect to inde-
pendent and dependent variables and include nonlocal expressions with
arbitrary elements [7,24]. Let us give definitions of some generalizations.
Definition 2. The extended equivalence group Ḡ∼ = Ḡ∼(L|S) of the
class L|S is formed by the transformations each of which are composi-
tions Φ1 ◦ Φ2, where ∀θ ∈ S: (Φ1 ◦ Φ2)θ ∈ S and Φ1|(x,u) ∈ T(θ, (Φ1 ◦
Φ2)θ). Here Φ1 is a point transformation in the space of (x, u(p), θ),
which is projectible on the space of (x, u(p′)) for any 0 ≤ p′ ≤ p, and
Φ1|(x,u(p′))
being the p′-th order prolongation of Φ1|(x,u). Φ2 is an inverti-
ble transformation in the space of arbitrary elements assumed as functi-
ons of (x, u(p)), and Φ2 having certain special properties.
Definition 3. A transformation Φ is called to belong to the extended
generalized equivalence group Ḡ∼gen = Ḡ∼gen(L|S) of the class L|S iff ∀θ ∈
S: Φθ ∈ S and, after fixing θ, Φ becomes a point transformation from
T(θ,Φθ).
The classes of chosen transformations with respect to arbitrary ele-
ments should be specified depending on the investigated classes of sys-
tems of differential equations. We do not point out a fixed kind of equi-
valence group where it is possible, implying any of the above kind.
Similar classes of systems of differential equations have similar equi-
valence groups.
Classification of admissible transformations 245
The equivalence group generates an equivalence relations on the set
of admissible transformations. Namely, the admissible transformations
(θ1, θ̃1, ϕ1) and (θ2, θ̃2, ϕ2) from T(L|S) are called G∼-equivalent if there
exist Φ ∈ G∼ such that θ2 = Φθ1, θ̃2 = Φθ̃1 and ϕ2 = Θ−1 ◦ ϕ1 ◦ Θ,
where Θ = Φ|(x,u) (or Θ = Φ(·, ·, θ(·, ·))|(x,u) in case of G∼gen).
5. Group classification problems. Let G∩ = G∩(L|S) =
⋂
θ∈S Gθ
be the common part of Gθ, θ ∈ S, which is called the kernel of the maxi-
mal point symmetry groups of systems from the class L|S . Note that G∩
can be naturally embedded into G∼ via trivial (identical) prolongation
of the kernel transformations to the arbitrary elements. The associated
subgroup of G∼ is normal.
The group classification problem for the class L|S is to describe all
G∼-inequivalent values of θ ∈ S together with the corresponding groups
Gθ, for which Gθ 6= G∩. The solution of the group classification problem
is the list of pairs (Sγ , {Gθ, θ ∈ Sγ}), γ ∈ Γ. Here {Sγ , γ ∈ Γ} is a
family of subsets of S,
⋃
γ∈Γ Sγ contains only G∼-inequivalent values of
θ with Gθ 6= G∩, and for any θ ∈ S with Gθ 6= G∩ there exists γ ∈ Γ
such that θ ∈ Sγ mod G∼. Structures of Gθ are similar for different
values of θ ∈ Sγ under fixed γ. In particular, Gθ, θ ∈ Sγ , have the same
arbitrariness of group parameters.
Group classification problems in the above formulation are very com-
plicated and, in the general case, are impossible to be solved since they
leads to systems of functional differential equations. That is why, one
usually considers only the connected component Gp
θ of unity for each θ
instead of the whole group Gθ. G
p
θ is called the principal (symmetry)
group of the system Lθ. The generators of one-parametric subgroups of
Gp
θ form a Lie algebra Aθ of vector fields in the space of (x, u), which is
called the maximal Lie invariance (or principal) algebra of infinitesimal
symmetry operators of Lθ. The kernel of principal groups of the class L|S
is the group G∩p = G∩p(L|S) =
⋂
θ∈S G
p
θ for which the Lie algebra is
A∩ = A∩(L|S) =
⋂
θ∈S Aθ.
Knowing Aθ, one can reconstruct Gθ. Then the problem of group
classification is reformulated in finding all possible inequivalent cases of
extensions for Aθ, i.e. in listing allG∼-inequivalent values of the arbitrary
parameters θ together with Aθ satisfying the condition Aθ 6= A∩ [1, 17].
6. Gauge equivalence groups. The equivalence group G∼ of the
class L|S can contain transformations which act only on arbitrary ele-
ments and do not really change systems, i.e. which generate gauge admi-
246 R.O. Popovych
ssible transformations. In general, transformations of such type can be
considered as trivial [15] (gauge) equivalence transformations and form
the gauge subgroup Gg∼ = {Φ ∈ G∼ | Φx = x, Φu = u, Φθ
g∼ θ} of the
equivalence group G∼. Moreover, Gg∼ is a normal subgroup of G∼.
Application of gauge equivalence transformations is equivalent to
rewriting systems in another form. In spite of regular equivalence trans-
formations, their role in group classification comes not to choice of rep-
resentatives in equivalence classes but to choice of form of these repre-
sentatives. It is quite common that the gauge equivalence relation on the
set of arbitrary elements of a class of differential equations is generated
by its gauge equivalence group.
We use the name “gauge equivalence transformation” since there exist
really trivial equivalence transformations which do not transform even
arbitrary elements. Such transformations arise if the auxiliary system
implies functional dependence of arbitrary elements. They form normal
subgroups in the corresponding equivalence groups and in the correspon-
ding gauge equivalence groups. We will neglect these transformations and
assume that equivalence groups coincide if they have the same factor
group with respect to the trivial equivalence subgroups.
7. Conditional equivalence groups. The concept of conditional
equivalence arises as an extension of the notion of conditional symmetry
transformations of a single system of differential equations [4] to equiva-
lence transformations in classes of systems. It is even more natural
than the concept of conditional symmetry since description of any class
includes, as a necessary element, an auxiliary system (a condition) for
arbitrary elements. Imposing additional constraints on arbitrary elements,
we may single out a subclass in the class under consideration, the equi-
valence group of which is not contained in the equivalence group of the
whole class.
Let L|S∩S′ denote the subclass of the class L|S , which is singled out
with the additional constrained system S′(x, u(p), θ(q)(x, u(p))) = 0. Here
S ∩ S ′ is the set of solutions of the united system S = 0, S′ = 0.
We assume that the united system is compatible for the subclass to be
nonempty.
Definition 4. The equivalence group G∼(L|S∩S′) of the subclass L|S∩S′
is called a conditional equivalence group of the whole class L|S under the
condition S′ = 0. The conditional equivalence group is called nontrivial
iff it is not a subgroup of the equivalence group G∼(L|S).
Classification of admissible transformations 247
The equivalence group G∼(L|S) generates an equivalence relation on
the set of pairs of additional auxiliary conditions and the correspondi-
ng conditional equivalence groups. Namely, if a transformation from
G∼(L|S) transforms the system S′ = 0 to the system S′′ = 0 then the
conditional equivalence groups G∼(L|S∩S′) and G∼(L|S∩S′′) are similar
with respect to this transformation and will be called G∼-equivalent.
Basing on the concept of conditional equivalence, we can formulate
the problem of description of T(L|S) similarly to the group classifi-
cation problem. Nontrivial additional auxiliary conditions for arbitrary
elements naturally arise under studying T(L|S). Steps of investigation
could be the following:
1. Construction of G∼(L|S) (or G∼gen(L|S) etc).
2. Description of conditional equivalence transformations in L|S , i.e.
searching for a complete family of G∼-inequivalent additional auxi-
liary conditions Sγ , γ ∈ Γ, such that any Sγ determines the set Sγ
of arbitrary elements, for which G∼(L|S∩Sγ ) 6⊂ G∼(L|S).
3. Finding admissible transformations which belong to no conditional
equivalence groups.
Actually, the proposed procedure is wide of optimality. We return to
discussion of it after presentation of a more developed technique.
8. Normalized classes of differential equations. Solving group
classification problems is essentially simpler if the class L|S of system
differential equations under consideration has an additional property of
normalization with respect to point transformations. The procedure of
investigation of T(L|S) can also be additionally enhanced with consi-
deration of conditional equivalence groups for subclasses possessing this
property.
Definition 5. The class L|S is called normalized if ∀(θ, θ̃, ϕ)∈T(L|S)
∃Φ∈G∼: θ̃ = Φθ and ϕ = Φ|(x,u).
The class L|S is called normalized in generalized sense if ∀(θ, θ̃, ϕ)∈
T(L|S) ∃Φ∈G∼gen: θ̃ = Φθ and ϕ = Φ(·, ·, θ(·, ·))|(x,u).
Proposition 3. If the class L|S is normalized (in usual or generali-
zed sense) then for any θ0 ∈S the point symmetry group Gθ0 coincides
with restriction, on the space of (x, u), of the subgroup of G∼ (or G∼gen)
preserving the value θ = θ0(x, u(p)).
248 R.O. Popovych
Definition 6. The class L|S is called strongly normalized if it is normali-
zed and G∼|(x,u) =
∏
θ∈S Gθ.
The class L|S is called strongly normalized in generalized sense if it
is normalized in generalized sense and ∀θ0 ∈S: G∼gen|θ=θ
0
(x,u) =
∏
θ∈Sθ0
Gθ,
where Sθ0 = {θ′ ∈ S | G∼gen|θ=θ
′
(x,u) = G∼gen|θ=θ
0
(x,u)}.
Definition 7. The class L|S is called semi-normalized if ∀(θ, θ̃, ϕ) ∈
T(L|S) ∃ϕ̃∈Gθ, ∃Φ∈G∼ : ϕ = ϕ̃ ◦ Φ|(x,u), i.e.
T(L|S) = {(θ,Φθ, ϕ̃ ◦ Φ|(x,u)) | θ∈S, ϕ̃∈Gθ, Φ∈G∼}.
(T(L|S) = {(θ0,Φθ0, ϕ̃ ◦ Φ|θ=θ0
(x,u)) | θ
0 ∈ S, ϕ̃ ∈ Gθ, Φ ∈ G∼gen} if L|S is
semi-normalized in generalized sense.)
Roughly speaking, the class L|S is normalized if any admissible trans-
formation in this class belongs to the equivalence group G∼ and is
strongly normalized if additionally G∼|(x,u) is generated by elements
from Gθ, θ ∈ S. The set of admissible transformations of a semi-nor-
malized class is generated by the transformations from the equivalence
group of the whole class and the transformations from the Lie symmetry
groups of equations of this class.
Intersection of normalized subclasses of the class L|S with the same
equivalence group G∼0 is a normalized subclass possessing G∼0 as a sub-
group of the equivalence group, which generates the whole corresponding
set of admissible transformations. Indeed, let L|S′ and L|S′′ be normali-
zed subclasses of the class L|S and G∼(L|S′) = G∼(L|S′′) = G∼0 . If
Φ ∈ G∼0 then (θ,Φθ,Φ|(x,u)) ∈ T(L|S′∩S′′) for any θ ∈ S ′ ∩ S ′′, i.e.
Φ ∈ G∼(L|S′∩S′′). In view of normalization of L|S′ or L|S′′, for any
(θ, θ̃, ϕ) ∈ T(L|S′∩S′′) there exist Φ ∈ G∼0 such that θ̃ = Φθ and ϕ =
Φ|(x,u). Therefore, L|S′∩S′′ is a normalized subclass. The proof in case of
normalization in generalized sense is analogous.
9. Examples of normalized classes. There exist a number of obvi-
ous examples of normalized classes. Thus, it is intuitively understandable
that the extreme cases of classes formed by either a single system of di-
fferential equations or all systems having a fixed number of independent
variables, unknown functions and differential equations with or without
restriction of order are normalized. Let us demonstrate it within the
framework of the above formal approach.
Classification of admissible transformations 249
Consider a system L(x, u(p)) = 0 of l differential equations for m
unknown functions u of n independent variables x, which admits the
maximal point symmetry group G. We assume that the tuple θ consists
of a single arbitrary element denoted also as θ and L depends on θ
constantly. The auxiliary system S for the arbitrary element θ is possible
to be chosen in different ways. Here we discuss two possibilities.
The first one is to constrain θ with a single (algebraic or differential)
equation, for example, θ = 0. Hence, S is a one-element set consisting of
the function identically vanishing on J (p), T(L|S) = { (0, 0, ϕ) | ϕ ∈ G }
and G∼ = { (x̃, ũ) = ϕ(x, u), θ̃ = F (x, u(p), θ)θ | ϕ ∈ G, F (·, ·, 0) 6=
0 }, i.e. in view of definition 1 the class L|S is normalized. It possesses
the nonempty trivial equivalence group G∼triv = { (x̃, ũ) = (x, u), θ̃ =
F (x, u(p), θ)θ | F (·, ·, 0) 6= 0} which should be neglected, and G∼/G∼triv =
{ (x̃, ũ) = ϕ(x, u), θ̃ = θ | ϕ ∈ G }.
The second possibility is to demand no constraints on θ, so S is
the whole set of p-th order differential functions of (x, u), T(L|S) =
{ (θ, θ̃, ϕ) | θ, θ̃ ∈ S, ϕ ∈ G} and G∼ = { (x̃, ũ(p)) = prp ϕ(x, u(p)), θ̃ =
F (x, u(p), θ) | ϕ ∈ G, ∂F/∂θ 6= 0 }. Therefore, L|S is normalized. This
class gives an example of classes without one-to-one correspondence
between arbitrary elements and systems of differential equations.
The class of all systems of l differential equations for m unknown
functions of n independent variables, which have order no greater than p,
(here l, m, n and p are fixed integers) can be included within the frame-
work of the formal approach after putting the left part of equations them-
selves as arbitrary elements and taking the empty auxiliary system S, i.e.
k = l, L ≡ θ and S is the whole set of l-tuples of functionally independent
p-th order differential functions of (x, u). Then T(L|S) = { (θ, θ̃, ϕ) |
θ ∈ S, θ̃ = F (x, u(p),prp ϕ) ◦ θ, |∂ϕ/∂(x, u)| 6= 0, ∂F/∂θ|θ=0 6= 0 } and
G∼ = {Φ = (ϕ(x, u), F (x, u(p), θ)) | |∂ϕ/∂(x, u)| 6= 0, ∂F/∂θ|θ=0 6= 0 }
that obviously shows normalization of this class.
Normalization property has been proved in some ways for a number
of different classes of differential equations being important for applicati-
on. For example, generalized Burgers equations [8], eikonal equations of
space dimensions 1, 2 and 3 [3], quasi-linear one-dimensional evoluti-
ons equations [2, 25], different multi-dimensional quasi-linear parabolic
equations [23], (1 + 1)-dimensional generalized nonlinear wave equati-
ons [14], different kinds of (1 + 1)-dimensional nonlinear Schrödingher
equations [5, 6, 19, 21, 22, 26], multi-dimensional generalized nonlinear
Schrödingher equations [11].
250 R.O. Popovych
10. Normalized classes and group classification problems.
The notion of normalized classes was implicitly used in solving the group
classification problems for many classes of system of differential equati-
ons. The most known classical group classification problems such as the
Lie’s classifications of second-order ordinary differential equations [13]
and of second-order two-dimensional linear partial differential equati-
ons [12] were solved with essential usage of strong normalization of the
above classes. Similar classification technique implicitly based on the
properties of normalized classes was recently applied in solving group
classification problems by a number of authors (see e.g. [2,3,5,14,21,25,
26]).
Proposition 4. Let the class L|S be normalized and Gi, i = 1, 2, be local
groups of point transformations in the space of (x, u), for which Si = {θ∈
S |Gp
θ = Gi} 6= ∅. Then S1 ∼ S2 mod G∼ iff G1 ∼ G2 mod G∼.
Proposition 5. Two systems from a semi-normalized class are trans-
formed each to other by a point transformation iff they are equivalent
with respect to the equivalence group of this class.
Proposition 6. Any normalized class of systems of differential equati-
ons is semi-normalized.
Proposition 7. Let the class L|S be normalized and a subset S ′ of S
determine a subclass L|S′ which is invariant under action of G∼(L|S).
Then the subclass L|S′ is normalized (in the same sense). G∼(L|S) is a
subgroup of G∼(L|S′), which generates T(L|S′) and, if L|S is normali-
zed in usual sense, coincides with G∼(L|S′) up to gauge equivalence
transformations in L|S′.
Proof. G∼(L|S′) ⊃ G∼(L|S), since for any Φ ∈ G∼(L|S) and for any
θ ∈ S ′ we have Φθ ∈ S ′, i.e. (θ,Φθ,Φ|(x,u)) ∈ T(L|S′) that implies Φ ∈
G∼(L|S′). Since T(L|S′) ⊂ T(L|S), for any (θ, θ̃, ϕ) ∈ T(L|S′) there exists
Φ ∈ G∼(L|S) such that θ̃ = Φθ and ϕ = Φ|(x,u), i.e. the subclass L|S′
is normalized. The above part of the proof is simply extended to the
generalized case.
Any Ψ ∈ G∼(L|S′) and any θ ∈ S ′ give the admissible transformation
(θ,Ψθ,Ψ|(x,u)) ∈ T(L|S′). Therefore, there exists Φ ∈ G∼(L|S) such that
Ψ|(x,u) = Φ|(x,u) and Ψθ = Φθ.
Note that under the above supposition the subclass L|S\S′ has similar
properties.
Classification of admissible transformations 251
Given the class L|S and a local (connected) group G of point transfor-
mations of (x, u) such that G = Gp
θ for some θ∈S, consider the subsets
of S
SG = { θ∈S | Gp
θ ⊃ G}, S̄G = { θ∈S | Gp
θ ⊃ G mod G∼},
S ′G = { θ∈S | Gp
θ = G}, S̄ ′G = { θ∈S | Gp
θ = G mod G∼}.
Corollary 1. Let the class L|S be normalized. Then L|S̄G and L|S̄′G are
normalized subclasses of L|S . G∼(L|S) is a subgroup of G∼(L|S̄G) and
G∼(L|S̄′G) and generates T(L|S̄G) and T(L|S̄′G).
Proposition 8. The subclass L|S0
is invariant with respect to G∼(L|S),
where S0 = S ′G∩, G∩ = G∩p(L|S).
Proof. Let us fix any Φ ∈ G∼(L|S) and any θ ∈ S0. It is necessary
to show that Φθ ∈ S0. G
p
Φθ = AdΦG
p
θ = AdΦG
∩, where AdΦ is the
action of Φ on transformation groups: G 3 ψ → ϕ−1 ◦ ψ ◦ ϕ ∈ AdΦG,
ϕ := Ψ|(x,u). Since Φθ ∈ L|S , Gp
Φθ ⊃ G∩. If Gp
Φθ = AdΦG
∩ 6= G∩
then G∩ 6= AdΦ−1G∩ ⊂ G∩. But AdΦ−1G∩ = Gp
Φ−1θ, Φ−1θ ∈ L|S and,
therefore, AdΦ−1G∩ ⊃ G∩ that implies a contradiction. That is why,
Gp
Φθ = G∩, i.e. Φθ ∈ S0.
Proposition 9. L|S′G is normalized in usual sense if L|S is normalized
in usual sense. T(L|S′G) is generated by the group G∼(L|S′G) ∩G∼(L|S)
the projection of which in (x, u) is the normalizer of G in G∼(L|S)|(x,u).
Proof. Let us fix arbitrary (θ, θ̃, ϕ) ∈ T(L|S′G). Since T(L|S′G) ⊂ T(L|S),
there exists Φ ∈ G∼(L|S) such that θ̃ = Φθ and ϕ = Φ|(x,u), θ, θ̃ ∈ S ′G,
hence G = Gp
θ̃
= ϕ−1 ◦Gp
θ ◦ ϕ = ϕ−1 ◦G ◦ ϕ, i.e. ϕ = Φ|(x,u) belongs to
the normalizer of G in G∼(L|S)|(x,u).
Consider any Φ ∈ G∼(L|S) such that ϕ = Φ|(x,u) belongs to the
normalizer ofG inG∼(L|S)|(x,u). Then (θ,Φθ, ϕ) ∈ T(L|S′G) for arbitrary
θ ∈ S ′G since Φθ ∈ S ′G. Indeed, Φθ ∈ S and G = Gp
Φθ = ϕ−1 ◦Gp
θ ◦ ϕ =
ϕ−1 ◦G ◦ ϕ = G. Therefore, Φ ∈ G∼(L|S′G).
Proposition 10. G∩p(L|SG) = G. G∼(L|SG) ⊂ G∼(L|S′G). If L|S is
normalized in usual sense, projections of these groups in (x, u) coincide.
Proof. The first statement trivially follows from the definition of L|SG .
Then in view of proposition 8 L|S′G is invariant with respect toG∼(L|SG),
i.e. G∼(L|SG) ⊂ G∼(L|S′G). Proposition 9 implies the latter statement.
In particular, G∼(L|SG) ∩G∼(L|S) = G∼(L|S′G) ∩G∼(L|S).
252 R.O. Popovych
Note 2. In general, the class L|SG is not normalized.
In view of the above propositions, the group classification problem
in any normalized class of differential equations is reduced to subgroup
analysis of the corresponding equivalence group. The property of strong
normalization allows us to hope that essential part of subgroups will
be Lie symmetry groups of systems from the class under consideration.
Moreover, under classification a hierarchy of normalized classes corres-
ponding to symmetry extension cases are naturally obtained.
11. Normalized subclasses and admissible transformations.
Investigation of normalization of the class L|S or its subclasses is necessa-
ry for description of T(L|S) and can be included as a step in stu-
dying T(L|S). The problem of classification of admissible transformati-
ons can be assumed solved, for example, in the following cases.
In view of the definition of normalized classes, the set of admissible
transformations is known if the class proves to be normalized and its
equivalence group are calculated. Then
T(L|S) = { (θ,Φθ,Φ|(x,u)) | θ ∈ S,Φ ∈ G∼}.
Suppose that the class L|S is presented as a union of disjoint normali-
zed subclasses, and there are no admissible transformations between
systems from different subclasses. That is, S =
⋃
γ∈Γ Sγ , L|Sγ is normali-
zed for any γ ∈ Γ, Sγ ∩ Sγ′ = ∅ and T(θ, θ′) = ∅, where θ ∈ Sγ ,
θ′ ∈ Sγ′, γ 6= γ′. Then obviously G∼(L|Sγ ) ⊃ G∼(L|S) for any γ ∈ Γ and
T(L|S) is the union of the simply constructed sets T(L|Sγ ) of admissible
transformations in the subclasses:
T(L|S) =
⋃
γ∈Γ
{ (θ,Φθ,Φ|(x,u)) | θ ∈ Sγ ,Φ ∈ G∼(L|Sγ ) }.
The class of nonlinear Schrödinger equations with potentials and
general modular nonlinearities has the set of admissible transformations
of the above structure for all space dimensions [11,18,19].
A more nontrivial situation is when normalized subclasses intersect
each other. Let S ′,S ′′ ⊂ S, S ′∩S ′′ 6= ∅, the subclasses L|S′ and L|S′′ are
normalized, S ′ = G∼(L|S′) S ′ ∩ S ′′ and S ′′ = G∼(L|S′′) S ′ ∩ S ′′. Then
any admissible transformation (θ′, θ′′, ϕ) with θ′∈ S ′ and θ′′∈ S ′′, can be
presented in the form (θ′,Φ2(Φ1θ′), (Φ1◦Φ2)|(x,u)), where Φ1 ∈ G∼(L|S′),
Φ2 ∈ G∼(L|S′′) and Φ1θ′ ∈ S ′ ∩ S ′′.
Classification of admissible transformations 253
A set of admissible transformations of such structure arises under
investigation of a class of variable coefficient diffusion–reaction equa-
tions [24].
12. Conclusion. Consideration in this paper is quite informal. The
aim was to give a description of major tools of modern group analysis
and to present a new treatment of group classification problems. Most of
adduced definitions and statements are flexible and can be made rigorous
after fixing a class of system of differential equations under investigation.
The author is grateful to Prof. M. Kunzinger for fruitful discussion.
The research of the author was supported by the Austrian Science Fund
(FWF), Lise Meitner project M923-N13.
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|
| id | oai:trim.imath.kiev.ua:article-468 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:49Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
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| resource_txt_mv | trimimathkievua/2f/5f488b14606490e77b1de8a8e7dc082f.pdf |
| spelling | oai:trim.imath.kiev.ua:article-4682020-08-13T10:42:30Z Classification of admissible transformations of differential equations Класифікація допустимих перетворень диференціальних рівнянь Popovych, R. Попович, Р. The framework of group classification is modified and extended to classification of admissible transformations in classes of differential equations. For this purpose, existing notions of group analysis are revised. Recently introduced notions (conditional equivalence group, normalized class of differential equations) are described and their properties are investigated. Концепцію групової класифікації модифіковано і поширено до класифікації допустимих перетворень у класах диференціальних рівнянь.З цією метою переглянуто існуючі поняття групового аналізу. Описано недавно введені поняття (умовна група еквівалентності, нормалізований клас диференціальних рівнянь) та досліджено їх властивості. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/468 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); 239-254 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); 239-254 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); 239-254 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/468/464 Авторське право (c) 2020 Р. Попович http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Popovych, R. Попович, Р. Classification of admissible transformations of differential equations |
| title | Classification of admissible transformations of differential equations |
| title_alt | Класифікація допустимих перетворень диференціальних рівнянь |
| title_full | Classification of admissible transformations of differential equations |
| title_fullStr | Classification of admissible transformations of differential equations |
| title_full_unstemmed | Classification of admissible transformations of differential equations |
| title_short | Classification of admissible transformations of differential equations |
| title_sort | classification of admissible transformations of differential equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/468 |
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