Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients
We describe the equivalence groupoid of the class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients.This class is shown to reduce by a family of equivalence transformations to a subclass with a four-dimensional usual equivalence group.Classified are admissible transfor...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2019
|
| Online Access: | https://trim.imath.kiev.ua/index.php/trim/article/view/371 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Transactions of Institute of Mathematics of NAS of Ukraine |
| Download file: |  |
Institution
Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552867802382336 |
|---|---|
| author | Opanasenko, S. Опанасенко, С. |
| author_facet | Opanasenko, S. Опанасенко, С. |
| author_institution_txt_mv | [
{
"author": "S. Opanasenko",
"institution": "Institute of Mathematics of NAS of Ukraine; Memorial University of Newfoundland"
}
] |
| author_sort | Opanasenko, S. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | We describe the equivalence groupoid of the class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients.This class is shown to reduce by a family of equivalence transformations to a subclass with a four-dimensional usual equivalence group.Classified are admissible transformations of this subclass and singled out its subclasses admitting maximal nontrivial conditional equivalence groups.All of them turn out to have dimension higher than four.In particular, few new examples of nontrivial cases of normalization in the generalized sense of classes of differential equationsappeared this way. Neither of classes discussed possesses a unique effective generalized equivalence group. |
| first_indexed | 2026-08-04T01:06:16Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 131�154
ÓÄÊ 517.912:512.816
Equivalence groupoid of a class
of general Burgers�Korteweg�de Vries
equations with space-dependent
coe�cients
S. Opanasenko
Memorial University of Newfoundland, Canada
E-mail: sopanasenko@mun.ca
Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine
Îïèñàíî ãðóïî¨ä åêâiâàëåíòíîñòi êëàñó çàãàëüíèõ ðiâíÿíü Áþðãåðñà�
Êîðòåâåãà�äå Ôðiçà ç ïðîñòîðîâèìè êîåôiöi¹íòàìè. Ïîêàçàíî, ùî öåé
êëàñ çâîäèòüñÿ ñiì'¹þ ïåðåòâîðåíü åêâiâàëåíòíîñòi äî ñâîãî ïiäêëàñó
ç ÷îòèðèâèìiðíîþ çâè÷àéíîþ ãðóïîþ åêâiâàëåíòíîñòi. Ïðîêëàñèôiêî-
âàíî äîïóñòèìi ïåðåòâîðåííÿ öüîãî ïiäêëàñó òà âèîêðåìëåíi ïiäêëà-
ñè, ùî äîïóñêàþòü ìàêñèìàëüíi íåòðèâiàëüíi óìîâíi ãðóïè åêâiâàëåíò-
íîñòi. Âèÿâëÿ¹òüñÿ, ùî âñi âîíè ìàþòü ðîçìiðíiñòü áiëüøó çà ÷îòèðè.
Çîêðåìà, çíàéäåíî äåêiëüêà íîâèõ êëàñiâ äèôåðåíöiàëüíèõ ðiâíÿíü,
íîðìàëiçîâàíèõ â óçàãàëüíåíîìó ñåíñi. Æîäåí ç íèõ íå äîïóñê๠¹äèíó
åôåêòèâíó óçàãàëüíåíó ãðóïó åêâiâàëåíòíîñòi.
We describe the equivalence groupoid of the class of general Burgers�
Korteweg�de Vries equations with space-dependent coe�cients. This class
is shown to reduce by a family of equivalence transformations to a subclass
with a four-dimensional usual equivalence group. Classi�ed are admissible
transformations of this subclass and singled out its subclasses admitting
maximal nontrivial conditional equivalence groups. All of them turn out to
have dimension higher than four. In particular, few new examples of nontri-
vial cases of normalization in the generalized sense of classes of di�erential
equations appeared this way. Neither of classes discussed possesses a unique
e�ective generalized equivalence group.
1. Introduction. A number of evolution equations that are impor-
tant in mathematical physics are of the general form
ut + C(t, x)uux =
r∑
k=0
Ak(t, x)uk +B(t, x). (1)
132 S. Opanasenko
In particular, this includes Burgers, Korteweg–de Vries (KdV), Kuramo-
to–Sivashinsky, Kawahara, and generalized Burgers–KdV equations.
Here and in the following the integer parameter r is fixed, and r > 2.
We require the condition CAr 6= 0 guaranteeing that equations from the
class (1) are nonlinear and of genuine order r. Throughout the paper we
use the standard index derivative notation ut = ∂u/∂t, uk = ∂ku/∂xk.
The class (1) and its various subclasses were subject to studying from
the symmetry analysis point of view, see [6] for an extensive list of refer-
ences. Recently, the class (1) became a source of examples of nontrivial
equivalence groups [6]. In fact, the first examples of classes with gener-
alized and extended generalized equivalence groups are of the form (1)
(with some additional restrictions). Moreover, detailed studying thereof
allowed the authors to introduce the concept of an effective generalized
equivalence group of a class of differential equations. Furthermore, the
structure of this class is so flexible, that a “reasonable” singled out sub-
class thereof is likely to possess normalization properties in some sense.
Nonetheless, it is not the case for a subclass F̄ of equations with the
arbitrary elements being time-independent,
ut + C(x)uux =
r∑
k=0
Ak(x)uk +B(x), where ArC 6= 0. (2)
The aim of this paper is to thoroughly study admissible transformations
of the class F̄ . In a nutshell, the results of this paper comprise the
following four facts. Any equation in F̄ is mapped by an equivalence
transformation of F̄ to an equation in the subclass F of reduced general
Burgers–Korteweg–de Vries equations with space-dependent coefficients,
singled out by conditions C = 1 and A1 = 0. The subclass F is not nor-
malized in any sense, and its usual equivalence group is four-dimensional.
Classified are admissible transformations of the class F and singled out
are its subclasses admitting maximal nontrivial conditional equivalence
subgroups of the equivalence group of F ,
F̂I,1 : ut + uux =
(
α+ 2
a01
b1 + a01|x+ β|α
)
u
+ (x+ β)
(
b2|x+ β|2α + b1|x+ β|α − b21(α+ 1)
a2
01
)
+
r∑
j=2
aj(x+ β)j |x+ β|αuj with αara01 6= 0,
Equivalence groupoid of a class of stationary BKdV equations 133
F̂I,01 : ut + uux =
r∑
j=2
aj(x+ β)j |x+ β|αuj + a00u
+ (x+ β)
(
b2|x+ β|2α − α+ 1
(α+ 2)2
a2
00
)
with (α+ 2)ar 6= 0,
F̂I,00 : ut + uux =
r∑
j=2
aj(x+ β)j−2uj + b0(x+ β)
+ b2(x+ β)−5 with ar 6= 0,
F̂II,0 : ut + uux =
r∑
j=2
aj(x+ β)juj + a00u+ b0 with ar 6= 0,
F̂II,1 : ut + uux =
r∑
j=2
aj(x+ β)juj + (a01 ln |x+ β|+ a00)u
+(x+ β)
(
−a
2
01
4
ln2 |x+ β|+
(
a2
01
4
− a00a01
2
)
ln |x+ β|+ b0
)
with ara01 6= 0,
FIII : ut + uux =
r∑
j=2
aje
αxuj + (a01e
αx + a00)u+ b2e
2αx
− a00a01
α
eαx − a2
00 + a00
2α
with αar 6= 0,
FIV,1 : ut + uux =
r∑
j=2
ajuj + a0u+ b1x+ b0
with αar
r−1∑
j=2
|aj | 6= 0,
Fr>2
IV,0 : ut + uux = arur + a0u+
r − 1
(r − 2)2
a2
0x+ b0
with αar 6= 0, r > 2,
Fr=2
IV,0 : ut + uux = a2u2 + b1x+ b0 with αar 6= 0.
All these subclasses but F̂II,0 are normalized in the generalized sense.
The class F̂II,0 is normalized in the usual sense.
134 S. Opanasenko
The main result of the paper is described in the following theorem.
Theorem 1. The usual equivalence group of the class F of reduced
general Burgers–Korteweg–de Vries equations with space-dependent co-
efficients is four-dimensional. The list of maximal nontrivial conditional
equivalence subgroups is exhausted by the generalized equivalence groups
of the normalized subclasses F̂I,1, F̂I,01, F̂I,00, F̂II,1, FIII, FIV,1, Fr>2
IV,0,
Fr=2
IV,0 and the usual equivalence group of the normalized subclass F̂II,0.
The equivalence groupoid of the class F is generated by its usual equiv-
alence group and the equivalence groups of the above subclasses.
For all classes normalized in the generalized sense, we can take their
effective generalized equivalence subgroups as maximal conditional equiv-
alence groups. Denote by F0 the complement to the union of the above
subclasses in the class F . It is a normalized class in the usual sense, and
its equivalence group coincides with that of F .
Corollary 2. The class F is a union of the normalized (in either the
generalized or the usual sense) classes F̂I,1, F̂I,01, F̂I,00, F̂II,1, F̂II,0, FIII,
FIV,1, Fr>2
IV,0, Fr=2
IV,0 and F0.
The structure of this paper is as follows. Firstly, we remind in Sec-
tion 2 theoretical foundations related to equivalence within classes of
differential equations. Following [6] in Section 3 we recall the structure
of the equivalence groupoids of the superclass of general Burgers–Korte-
weg–de Vries equations, its subclass of equations with time-independent
coefficients and gauging of these classes to the corresponding subclasses
of reduced equations. In Section 4 we give the complete classification of
admissible transformations of the class F of reduced general Burgers–
KdV equations with space-dependent coefficients. In [6] there were found
subclasses of the class F possessing admissible transformations that are
not generated by the equivalence transformations of F . But the ques-
tion of a structure of equivalence groupoids of these subgroups was not
addressed there. Here we fill this gap by comprehensive description of all
these subclasses and their equivalence groups (for subclasses normalized
in the generalized sense we present either the entire generalized equiv-
alence group, or its effective generalized equivalence group or both of
them). By partitioning if necessary these subclasses we achieve a nor-
malization of “subsubclasses” in either usual or generalized sense. Thus
we present the superclass F as a union of normalized classes of differential
Equivalence groupoid of a class of stationary BKdV equations 135
equations described in Theorem 1. For the two normalized subclasses to
be able to have a closed form of group transformations we apply a non-
standard approach, the technical crux of which is as follows. First we
gauge the class under consideration by a family of equivalence transfor-
mations thereof to a nice normalized subclass. Then every equivalence
transformation in the class under consideration would be a composition
of the gauging mapping, an equivalence transformation within the nice
subclass and the inverse of a (not the same as before because we con-
sider not symmetry but equivalence transformations of the superclass)
gauging mapping. This procedure may explain an appearance of gener-
alized equivalence groups for most of the considered subclasses. In fact,
the determining systems of ODEs are exactly solvable for all but the
two equivalence groups and this procedure is only lurking in the back-
ground, but we could use it almost everywhere. In this case, even if
a nice underlying subclass is normalized in the usual sense, we compose
its equivalence transformations with transformations from the families
parameterized by arbitrary elements of the superclass, and thus para-
meterize the equivalence transformations thereof by arbitrary elements,
making them generalized.
2. Equivalence of classes of differential equations. We recall
the essential notions for the present paper only. See [6, 8, 9] for more
details. Let Lθ denote a system of differential equations of the form
L
(
x, u(r), θ
(
x, u(r)
))
= 0,
where x = (x1, . . . , xn) is the n independent variables, u = (u1, . . . , um)
is the m dependent variables, and L is a tuple of differential functions
in u. We use the standard short-hand notation u(r) to denote the tuple
of derivatives of u with respect to x up to order r, which also includes u
as the derivatives of order zero. The system Lθ is parameterized by the
tuple of functions θ = (θ1(x, u(r)), . . . , θk(x, u(r))), called the arbitrary
elements running through the solution set S of an auxiliary system of
differential relations in θ. Thus, the class of (systems of) differential
equations L|S is the parameterized family of systems Lθ, such that θ lies
in S.
Equivalence of classes of differential equations is based on studying
how equations from a given class are mapped to each other. The notion
of admissible transformations, which constitute the equivalence groupoid
of the class L|S , formalizes this study. An admissible transformation is
a triple (θ, θ̃, ϕ), where θ, θ̃ ∈ S are arbitrary-element tuples associated
136 S. Opanasenko
with equations Lθ and Lθ̃ from the class LS that are similar, and ϕ is a
point transformation in the space of (x, u) that maps Lθ to Lθ̃.
A related notion of relevance in the group classification of differen-
tial equations is that of equivalence transformations. Usual equivalence
transformations are point transformations in the joint space of indepen-
dent variables, derivatives of u up to order r and arbitrary elements that
are projectable to the space of (x, u(r′)) for each r′ = 0, . . . , r, with re-
spect the contact structure of the rth order jet space coordinatized by
the r-jets (x, u(r)) and map every system from the class L|S to a system
from the same class. The Lie (pseudo)group constituted by the equiva-
lence transformations of L|S is called the usual equivalence group of this
class and denoted by G∼.
Each equivalence transformation T ∈ G∼ generates a family of admis-
sible transformations parameterized by θ,
G∼ 3 T →
{
(θ, T θ, π∗T ) | θ ∈ S
}
⊂ G∼,
and therefore the usual equivalence group G∼ gives rise to a subgroupoid
of the equivalence groupoid G∼. The function π is the projection of the
space of (x, u(r), θ) to the space of equation variables only, π(x, u(r), θ) =
(x, u). The pushforward π∗T of T by π is then just the restriction of T
to the space of (x, u).
The projectability property for equivalence transformations can be
neglected. Then these equivalence transformations constitute a Lie pse-
udogroup Ḡ∼ called the generalized equivalence group of the class. See
the first discussion of this notion in [3, 4] and the further development
in [8, 9]. When the generalized equivalence group coincides with the
usual one the situation is considered to be trivial. Similarly to usual
equivalence transformations, each element of Ḡ∼ generates a family of
admissible transformations parameterized by θ,
Ḡ∼ 3 T →
{
(θ′, T θ′, π∗(T |θ=θ′(x,u))) | θ′ ∈ S
}
⊂ G∼,
and thus the generalized equivalence group Ḡ∼ also generates a sub-
groupoid H̄ of the equivalence groupoid G∼.
Definition 3. Any minimal subgroup of Ḡ∼ that generates the same
subgroupoid of G∼ as the entire group Ḡ∼ does is called an effective
generalized equivalence group of the class L|S .
If the entire group Ḡ∼ is effective itself, then its uniqueness is ev-
ident. At the same time, there exist classes of differential equations,
Equivalence groupoid of a class of stationary BKdV equations 137
where effective generalized equivalence groups are proper subgroups of
the corresponding generalized equivalence groups that are even not nor-
mal. Hence each of these effective generalized equivalence groups is not
unique since it differs from some of subgroups non-identically similar to
it, and all of these subgroups are also effective generalized equivalence
groups of the same class.
The class of differential equations L|S is normalized in the usual (resp.
generalized) sense if the subgroupoid induced by its usual (resp. generali-
zed) equivalence group coincides with the entire equivalence groupoid G∼
of L|S . The normalization of L|S in the usual sense is equivalent to the
following conditions. The transformational part ϕ of each admissib-
le transformation (θ′, θ′′, ϕ) ∈ G∼ does not depend on the fixed initial
value θ′ of the arbitrary-element tuple θ and, therefore, is appropriate
for any initial value of θ.
The normalization properties of the class L|S are usually established
via computing its equivalence groupoid G∼, which is realized using the
direct method. Here one fixes two arbitrary systems from the class,
Lθ : L(x, u(r), θ(x, u(r))) = 0 and Lθ̃ : L(x̃, ũ(r), θ̃(x̃, ũ(r))) = 0, and aims
to find the (nondegenerate) point transformations, ϕ: x̃i = Xi(x, u),
ũa = Ua(x, u), i = 1, . . . , n, a = 1, . . . ,m, connecting them. For this,
one changes the variables in the system Lθ̃ by expressing the derivatives
ũ(r) in terms of u(r) and derivatives of the functions Xi and Ua as well as
by substituting Xi and Ua for x̃i and ũa, respectively. The requirement
that the resulting transformed system has to be satisfied identically for
solutions of Lθ leads to the system of determining equations for the
components of the transformation ϕ.
Imposing additional constraints on arbitrary elements of the class,
we may single out its subclass whose equivalence group is not contained
in the equivalence group of the entire class. Let L|S′ be the subclass of
the class L|S , which is constrained by the additional system of equations
S ′(x, u(r), θ(q′)) = 0 and inequalities Σ′(x, u(r), θ(q′)) 6= 0 with respect to
the arbitrary elements θ = θ(x, u(r)). Here S ′ ⊂ S is the set of solutions
of the united system S = 0, Σ 6= 0, S ′ = 0, Σ′ 6= 0. We assume that the
united system is compatible for the subclass L|S′ to be nonempty.
Definition 4. The equivalence group G∼(L|S′) of the subclass L|S′ is
called a conditional equivalence group of the entire class L|S under the
conditions S ′ = 0, Σ′ 6= 0. The conditional equivalence group is called
nontrivial if it is not a subgroup of G∼(L|S).
138 S. Opanasenko
Conditional equivalence groups may be trivial not with respect to the
equivalence group of the entire class but with respect to other conditional
equivalence groups. Indeed, if S ′ ⊂ S ′′ and G∼(L|S′) ⊂ G∼(L|S′′) then
the subclass L|S′ is not interesting from the conditional symmetry point
of view. Therefore, the set of additional conditions on the arbitrary
elements can be reduced substantially.
Definition 5. The conditional equivalence group G∼L|S′
of the class L|S
under the additional conditions S ′ = 0, Σ′ 6= 0 is called maximal if for
any subclass L|S′′ of the class L|S containing the subclass L|S′ we have
G∼L|S′
6⊂ G∼L|S′′ .
3. Preliminary analysis of equivalence groupoid. We start
studying admissible transformations of the class F by presenting the
equivalence groupoid of its superclass (1) and then descend therefrom to
the class under study.
Proposition 6. The class (1) is normalized in the usual sense. Its usual
equivalence group G∼(1) consists of the transformations in the joint space
of (t, x, u, θ) whose (t, x, u)-components are of the form
t̃ = T (t), x̃ = X(t, x), ũ = U1(t)u+ U0(t, x),
where T = T (t), X = X(t, x), U1 = U1(t) and U0 = U0(t, x) are
arbitrary smooth functions of their arguments such that TtXxU
1 6= 0.
Following [6] we can gauge the arbitrary elements C = 1 and A1 = 0
by a family of equivalence transformations of the class (1) and obtain
the class of reduced general Burgers–KdV equations
ut + uux =
r∑
j=2
Aj(t, x)uj +A0(t, x)u+B(t, x). (3)
As before, the arbitrary elements run through the set of smooth functions
of (t, x) with ArC 6= 0.
Theorem 7. The class of reduced (1+1)-dimensional general rth order
Burgers–KdV equations (3) is normalized in the usual sense. Its usual
equivalence group G∼ consists of the transformations of the form
t̃ = T (t), x̃ = X1(t)x+X0(t), ũ =
X1
Tt
u+
X1
t
Tt
x+
X0
t
Tt
, (4)
Equivalence groupoid of a class of stationary BKdV equations 139
Ãj =
(X1)j
Tt
Aj , Ã0 =
1
Tt
(
A0 + 2
X1
t
X1
− Ttt
Tt
)
, (5)
B̃ =
X1
(Tt)2
B +
1
Tt
(
X1
t
Tt
)
t
x+
1
Tt
(
X0
t
Tt
)
t
−
(
X1
t
Tt
x+
X0
t
Tt
)
Ã0, (6)
where j = 2, . . . , r, and T = T (t), X1 = X1(t) and X0 = X0(t) are
arbitrary smooth functions of their arguments with TtX
1 6= 0.
The subclass F̄ of general Burgers–KdV equations with space-depen-
dent coefficients is singled out from the class (1) by the constraints
Akt = 0, k = 0, . . . , r, Bt = 0 and Ct = 0. Therefore, its usual equiv-
alence group G∼F̄ is a subgroup of G∼(1) that consists of transformations
preserving the above constraints.
Proposition 8. The usual equivalence group G∼F̄ of the class F̄ of
general Burgers–Korteweg–de Vries equations with space-dependent co-
efficients consists of the transformations in the joint space of (t, x, u, θ)
whose (t, x, u)-components are of the form
t̃ = c1t+ c2, x̃ = X(x), ũ = c′3u+ U0(x),
where c1, c2 and c′3 are arbitrary constants and X = X(x) and U0 =
U0(x) are arbitrary smooth functions of x such that c1Xxc
′
3 6= 0.
The existence of classifying conditions [6]
Tt
(Xx)r
XtÃ
r
x̃ +
(
Tt
(Xx)r
)
t
Ãr = 0,
TtU
1
Xx
XtC̃x̃ +
(
TtU
1
Xx
)
t
C̃ = 0,
for admissible transformations of the class F̄ implies that it is definitely
not normalized in any sense. At the same time, we can gauge the arbi-
trary elements C and A1 again by means of equivalence transformations
of the class F̄ and produce the class F of reduced general Burgers–KdV
equations with space-dependent coefficients,
ut + uux =
r∑
j=2
Aj(x)uj +A0(x)u+B(x).
Proposition 9. The usual equivalence group G∼F of the class F is four-
dimensional and consists of transformations of the form
t̃ = c1t+ c2, x̃ = c3x+ c4, ũ =
c3
c1
u,
140 S. Opanasenko
Ãj =
(c3)j
c1
Aj , Ã0 =
1
c1
A0, B̃ =
c3
(c1)2
B,
where j = 2, . . . , r, and c’s are arbitrary constants with c1c3 6= 0.
Nor the class F neither its superclass F̄ are normalized in any sense.
Thus, the problem of describing the equivalence groupoid G∼F of the
class F should be considered as the classification of admissible trans-
formations up to G∼F -equivalence, see [9, Sections 2.6 and 3.4]. The
class F is a subclass of the class (3), whence G∼F is a subgroupoid of the
equivalence groupoid of the class (3), and the results of Theorem 7 are
valid here, although they should be further specified. This is achieved by
differentiating the relations (5)–(6), solved with respect to the source ar-
bitrary elements, with respect to t. This gives the classifying conditions
for admissible transformations,(
X1
t x+X0
t
)
Ãjx̃ +
(
Ttt
Tt
− jX
1
t
X1
)
Ãj = 0, (7)
(
X1
t x+X0
t
)
Ã0
x̃ +
Ttt
Tt
Ã0 =
1
Tt
(
2
X1
t
X1
− Ttt
Tt
)
t
, (8)
(
X1
t x+X0
t
)
B̃x̃ +
(
2
Ttt
Tt
− X1
t
X1
)
B̃ = − Tt
X1
(
X1
t x+X0
t
)2
Ã0
x̃
− X1
T 2
t
(
Tt
X1
t x+X0
t
X1
)
t
Ã0 +
X1
T 2
t
(
Tt
X1
(
X1
t x+X0
t
Tt
)
t
)
t
, (9)
where the initial space variable x should be substituted, after expanding
all derivatives, by its expression via x̃, x = (x̃ − X0)/X1. Note that
admissible transformations with Ttt = X0
t = X1
t = 0 are generated by
the usual equivalence group G∼F .
4. Nontrivial conditional equivalence subgroups. In [6] with
a help of the method of furcate splitting, cf. [5, 7] the classifying condi-
tions (7)–(9) for admissible transformations of the class F were solved,
but the obtained admissible transformations were presented superficially.
More precisely, they were parameterized by solutions of some ODEs.
Here we study the question in more depth and present explicit forms
of group parameters of the nontrivial conditional equivalence groups.
Besides, following [6] for simplicity we consider only subclasses of the
classes FI and FII, defined below, admitting proper subgroups of maxi-
mal conditional equivalence groups. In fact, these subgroups are the quo-
tients thereof by the space-translations. Note that given in Theorem 1
Equivalence groupoid of a class of stationary BKdV equations 141
are the subclasses admitting maximal nontrivial conditional equivalence
subgroups.
I. The class FI of equations
ut + uux =
r∑
j=2
ajx
j |x|αuj +
(
a00 + a01|x|α
)
u
+ x
(
b0 + b1|x|α + b2|x|2α
)
with αar 6= 0 naturally partitions into two G∼FI
-invariant subclasses FI,0
and FI,1 singled out by the conditions a01 = 0 and a01 6= 0, respectively,
since the arbitrary element a01 is easily shown to be transformed by the
rule ã01 = c4a01 under admissible transformations of the class, c4 6= 0.
The class FI,1 admits additional admissible transformations if and only if
a00 = (α+2)b1/a01 and b0 = −b21(1+α)/a2
01, so we reduce the arbitrary-
elements tuple of the class by a00 and b0 and denote the subclass obtained
again by FI,1.
Proposition 10. The class FI,1 is normalized in the generalized sense.
Its generalized equivalence group consists of the point transformations in
the relevant space, which are of the form
t̃ = T̄ , x̃ = X̄1x, ũ =
X̄1
T̄t
u− X̄1
t
T̄t
x,
α̃ = α, ãj = c̄4aj , ã01 = c̄4a01, b̃2 = c̄24b2, b̃1 = c̄5,
where T̄ is a smooth function of t and the arbitrary elements θ,
T̄ (t, θ) =
1
c̄5
ln
∣∣∣∣c̄5(c1 e−b1αt/a01 − 1
−b1α/a01
+ c2
)
+ 1
∣∣∣∣ ,
θ = (α, aj , a01, b2, b1),
taking the form at the singular points
T̄ (t, θ) = c̄1
e−b1αt/a01 − 1
−b1α/a01
+ c̄2 if c̄5 = 0 and b1 6= 0,
T̄ (t, θ) =
1
c̄5
ln
∣∣c̄5(c̄1t+ c̄2)
∣∣ if c̄5 6= 0 and b1 = 0,
T̄ (t, θ) = c̄1t+ c̄2 if (c̄5, b1) = (0, 0),
142 S. Opanasenko
c̄’s are arbitrary functions of θ with c̄1c̄4
∂(ã2,...,ãr,ã01,b̃1,b̃2)
∂(a2,...,ar,a01,b1,b2) 6= 0 as well
as X̄1(t, θ) = (c̄4T̄t)
−1/α if α is odd or rational in the reduced form with
an odd numerator and X̄1(t) = ε|c̄4T̄t|−1/α with ε = ±1 and c̄4T̄t > 0
otherwise.
Remark 11. The function T is a solution of an ODE smoothly depen-
ding on parameters, so it is a smooth function of these parameters and
initial conditions [1, Corollary 6, p. 97] (α, b1 and a01 are the parameters
of the equation in this case, c’s are the initial conditions). This argu-
mentation is valid for the group parameters in the equivalence groups
below, where appropriate, as well. In fact, in these cases it follows from
the transformation for Ã0 (the equation (5)) that the function T satisfies
the equation
γ = δ
1
Tt
+
(
1
Tt
)
t
= 0
for some constants γ and δ, having the general solution
T (t) =
1
γ
ln
∣∣∣∣γ (c1 eδt − 1
δ
+ c2
)
+ 1
∣∣∣∣ .
The continuity of this function is evident and at the singular points the
function takes the form
T (t) = c1
eδt − 1
δ
+ c2 if γ = 0 and δ 6= 0,
T (t) =
1
γ
ln |γ(c1t+ c2) + 1| if γ 6= 0 and δ = 0,
T (t) = c1t+ c2 if (γ, δ) = (0, 0).
The transformations in Proposition 10 indeed form a group, which is
straightforward to show. Therefore the equivalence group of the class FI,1
is a local Lie group of transformations (all equivalence group here and
below in the paper are finite-dimensional so we do not need to talk about
Lie pseudogroups). If the function T is of the form 1
γ ln |γ(c1t+ c2) + 1|
and γc2 = −1, then T (t) degenerates into an affine function. To avoid
this, in all such situations thereafter we implicitly assume otherwise.
The notation ∂(·,...,·)
∂(·,...,·) stands for the determinant of the corresponding
Jacobian matrix. Thereafter, we will not call attention to these facts.
Equivalence groupoid of a class of stationary BKdV equations 143
Since the arbitrary element α is invariant under admissible trans-
formations, it is convenient to consider the two subclasses FI,00 and FI,01
of FI,0 singled out by conditions α = −2 and α 6= −2, respectively. To
achieve an extension of a number of admissible transformations in the
latter class we need to consider its subclass (denoted again FI,01) singled
out by the conditions b1 = 0 and b0 = −(α+ 1)a2
00/(α+ 2)2.
Proposition 12. The class FI,01 is normalized in the generalized sense.
Its generalized equivalence group consists of the point transformations of
the form
t̃ = T̄ (t), x̃ = X̄1(t)x, ũ =
X̄1
T̄t
u− X̄1
t
T̄t
x,
α̃ = α, ãj = c̄4aj , ã00 = c̄5, b̃2 = c̄24b2,
where T̄ is a smooth function of t and the arbitrary elements θ,
T̄ (t, θ) =
1
c̄5
ln
∣∣∣∣c̄5(c̄1 ea00αt/(α+2) − 1
a00α/(α+ 2)
+ c̄2
)
+ 1
∣∣∣∣ .
The function T̄ takes at the singular points the following forms
T̄ (t, θ) = c̄1
ea00αt/(α+2) − 1
a00α/(α+ 2)
+ c̄2 if c̄5 = 0 and a00 6= 0,
T̄ (t, θ) =
1
c̄5
ln |c̄5(c̄1t+ c̄2) + 1| if c̄5 6= 0 and a00 = 0,
T̄ (t, θ) = c̄1t+ c̄2 if (c̄5, a00) = (0, 0).
Here c̄’s are arbitrary functions of θ with c̄1c̄4
∂(ã2,...,ãr,ã00,b̃2)
∂(a2,...ar,a00,b2) 6= 0 as well
as X̄1(t, θ) = (c̄4T̄t)
−1/α if α is odd or rational in the reduced form with
an odd numerator and X̄1(t, θ) = ε|c̄4T̄t|−1/α with ε = ±1 and c̄4T̄t > 0
otherwise.
A description of the equivalence group of the class FI,00 is more com-
plicated and we present its equivalence groupoid first. In accordance
with our standard approach we consider its subclass singled out by the
conditions a00 = b1 = 0.
Proposition 13. A point transformation connects the two equations in
the class FI,00 if and only if its components are of the form
t̃ = T (t), x̃ = X1(t)x, ũ =
X1
Tt
u− X1
t
Tt
x,
144 S. Opanasenko
where (X1(t))2 = c4Tt and the smooth function T of t satisfies the equa-
tion (
Ttt
Tt
)
t
− 1
2
(
Ttt
Tt
)2
= 2b̃0T
2
t − 2b0.
Here c4 is an arbitrary constant and c4Tt > 0.
The last equation is an autonomous ordinary differential equation
on T which can be integrated in quadratures with standard techniques,
but proceeding this way one can write an explicit form of the general
solution only for specific values of parameters. On the other hand,
for any equation in FI,00 there is an equivalent one to it in the sub-
class Fb0=0
I,00 singled out by the condition b0 = 0. The corresponding point
transformation is t̃ = T (t), x̃ =
√
Ttx, ũ = u/
√
Tt − Tttx/(2
√
(Tt)3),
where a smooth function T of t is a solution of the equation (Ttt/Tt)t −
1
2 (Ttt/Tt)
2
+2b0 = 0, for which the general solution can be found explic-
itly, although a particular solution will suffice for our purposes. Thus, if
b0 = b2 > 0, then T (t) = e2bt is a particular solution; if b0 = −b2 < 0,
then T (t) = tan(bt) is a particular solution, b > 0 in both cases.
Proposition 14. The class Fb0=0
I,00 is normalized in the usual sense. Its
usual equivalence group is constituted by the point transformations of the
form
t̃ = T (t), x̃ = X1(t)x, ũ =
X1
Tt
u− X1
t
Tt
x,
ã2 = c4a2, b̃2 = c24b2,
where X1(t) = ε
√
c4Tt with ε = ±1, T = (c1t + c2)/(c3t + c0) and c’s
are arbitrary constants, with δ = c1c0 − c2c3 6= 0 and c0, c1, c2 and c3
being defined up to a nonzero constant, and c4δ > 0.
On the other hand, any admissible transformation of the class FI,00
can be represented as a composition of an admissible transformation with
a source equation in FI,00 and a target equation in Fb0=0
I,00 , an admissi-
ble transformation generated by an equivalence transformation in Fb0=0
I,00
and an admissible transformation back. In this way we avoid implicit
quadrature expressions arising in a previous approach. Note that the
parameter-function T is defined as a solution of a third-order ODE pa-
rameterized by b0 and b̃0 and thus should be parameterized by three
Equivalence groupoid of a class of stationary BKdV equations 145
constants to agree with the Picard–Lindelöf theorem. This is indeed the
case.
Proposition 15. The class FI,00 is normalized in the generalized sense.
Its effective generalized equivalence group is constituted by the point
transformations of the form
t̃ = P 2(T (P 1(t))), x̃ =
√
P 2
t̄ P
1
t X
1(t̂)x,
ũ =
1
P 2
t̄
(
X1
Tt̂P
1
t
u−
(
X1P 1
tt
2Tt̂(P
1
t )3/2
+
X1
t̂
√
P 1
t
Tt̂
+
P 2
t̄t̄X
1
√
P 1
t
2P 2
t̄
)
x
)
,
ãj = c4aj , b̃2 = c24b2,
b̃0 =
1
(P 2
t̄ )2
(
1
(P 1
t )2
(
b0 −
(
P 1
tt
2P 1
t
)2
+
1
2
(
P 1
tt
P 1
t
)
t
)
−
(
P 2
t̄t̄
2P 2
t̄
)2
+
1
2
(
P 2
t̄t̄
P 2
t̄
)
t̄
)
,
where t̂ = P 1(t), t̄ = T (t̂), t̃ = P 2(t̄), X1(t̂) = ε(c4Tt̂)
1/2, T = (c1t̂ +
c2)/(c3t̂+ c0), with δ = c1c0 − c2c3 6= 0, c’s are arbitrary constants,
P 1(t) =
t if b0 = 0,
tan(
√
−b0t) if b0 < 0,
e2
√
b0t if b0 > 0;
P 2(t̄) runs through the set of smooth functions
{
t̄, 1
c5
ln |t̄|, 1
2c5
arctan t̄
}
,
with c4δ > 0, ci, i = 0, 1, 2, 3, are defined up to a nonzero constant, and
P 2
t̄ > 0 and ε = ±1.
The arbitrary element b̃0 of the target equation takes the value of c25
if P 2(y) = 1
c5
ln |y|, of −c25 if P 2(y) = 1
2c5
arctan y and of 0 otherwise.
The functions P 2(T (P 1(t))) give a three-parameter family of solutions
to the nonlinear third-order equation on T above parameterized by b0
and b̃0.
Remark 16. The point transformations in Proposition 15 form a group
by construction, and thus constitute an effective generalized equivalence
group of the class FI,00. To obtain the entire generalized equivalence
group thereof one allows c’s to vary through the set of arbitrary smooth
functions of the arbitrary elements of the class.
146 S. Opanasenko
II. A class FII of differential equations of the form
ut + uux =
r∑
j=2
ajx
juj + (a01 ln |x|+ a00)u
+ x
(
−a
2
01
4
ln2 |x|+
(
a2
01
4
− a00a01
2
)
ln |x|+ b0
)
is partitioned into two subclasses FII,0 and FII,1 that are singled out by
conditions a01 = 0 and a01 6= 0, respectively, and invariant under the
admissible transformations of the class FII.
Proposition 17. The class FII,0 is normalized in the usual sense. Its
equivalence group is constituted by the point transformations of the form
t̃ = c1t+ c2, x̃ = c4e
c3tx, ũ =
c4e
c3t
c1
(u+ c3x),
ãj =
aj
c1
, ã00 =
a00 + 2c3
c1
, b̃0 =
b0 − c23
(c1)2
,
where c’s are arbitrary constants with c1c4 6= 0.
The class FII,0 is the only owner of a conditional group normalized
in the usual sense.
Proposition 18. The class FII,1 is normalized in the generalized sense.
Its generalized equivalence group Ḡ∼II,1 is constituted by the point trans-
formations of the form
t̃ = c̄1t+ c̄2, x̃ = X̄1x, ũ =
X̄1
c̄1
(
u+
c̄4a01
2
ea01t/2x
)
,
ãj =
aj
c̄1
, ã01 =
a01
c̄1
, ã00 =
1
c̄1
(a00 − a01c̄3),
b̃0 =
1
4c̄21
(
4b0 − a2
01(c̄23 + c̄3) + 2a00a01c̄3
)
,
where X̄1 := exp
(
c̄3 + c̄4 exp(a01t
2 )
)
, and c̄’s are smooth functions of the
arbitrary elements a00, a01, aj and b0 with c̄1
∂(ã2,...,ãr,ã01,ã00,b̃0)
∂(a2,...,ar,a01,a00,b0) 6= 0.
To extract an effective generalized equivalence group from the genera-
lized equivalence group, we set c̄2 := c2/a01, c̄3 := −c3/a01 and get rid
of the dependence of other c̄’s on the arbitrary elements.
Equivalence groupoid of a class of stationary BKdV equations 147
Proposition 19. An effective generalized equivalence group Ĝ∼II,1 of the
class FII,1 is constituted by the point transformations of the form
t̃ = c1t+
c2
a01
, x̃ = X1(t)x, ũ =
X1(t)
c1
(
u+
c4a01
2
ea01t/2x
)
,
ãj =
aj
c1
, ã01 =
a01
c1
, ã00 =
1
c1
(a00 + c3),
b̃0 =
1
4c21
(
4b0 + (a01 − 2a00)c3 − c23
)
,
where X1(t) := exp
(
− c3
a01
+ c4 exp(a01t
2 )
)
and c’s are arbitrary constants
with c1 6= 0.
The effective generalized equivalence group Ĝ∼II,1 is not a normal sub-
group of Ḡ∼II,1, which is readily seen after writing the time-transformation
out. Therefore, it is not unique as an effective generalized equivalence
group as conjugate subgroups in Ḡ∼II,1 are also effective generalized equiv-
alence groups. Thus, the existence of a class of differential equations with
unique nontrivial (proper) effective generalized equivalence group is still
a question.
III. A class of differential equations of the form
ut + uux =
r∑
j=2
aje
αxuj + (a01e
αx + a00)u+ b2e
2αx
+ b1e
αx + b0 with αar 6= 0
admits additional admissible transformations if and only if
b0 = −a
2
00 + a00
2α
and b1 = −a00a01
α
.
Proposition 20. The class FIII is normalized in the generalized sense.
Its generalized equivalence group is constituted by the point transforma-
tions of the form
t̃ = T̄ , x̃ = c̄5x−
c̄5
α
ln |c̄4T̄t|, ũ =
c̄5
T̄t
(
u− T̄tt
αT̄t
)
,
α̃ =
α
c̄5
, ãj = c̄4c̄
j
5aj , ã01 = c̄4a01, ã00 = c̄3, b̃2 = c̄24c̄5b2,
148 S. Opanasenko
where the function T of t and the arbitrary elements θ is defined by
T̄ (t, θ) =
1
c̄3
ln
∣∣∣∣c̄3(c̄1 ea00t − 1
a00
+ c̄2
)
+ 1
∣∣∣∣ ,
and takes the following values at the singular points
T̄ (t, θ) = c̄1
ea00t − 1
a00
+ c̄2 if c̄3 = 0 and a00 6= 0,
T̄ (t, θ) =
1
c̄3
ln |c̄3(c̄1t+ c̄2)| if a00 = 0 and c̄3 6= 0,
T̄ (t, θ) = c̄1t+ c̄2 if (a00, c̄3) = (0, 0),
c̄’s are smooth functions of θ with c̄1c̄4c̄5
∂(ᾱ,ã2,...,ãr,ã00,ã01,b̃2)
∂(α,a2,...,ar,a00,a01,b2) 6= 0.
To find an effective generalized equivalence group of the class FIII we
resort to the following heuristic speculation. The arbitrary element ã00
may take any real value. Thus, it sufficient to parameterize ã00 to be
a00 + c3, c3 ∈ R. We preserve the number of initial conditions parame-
terizing T and guaranteeing the necessary domain for values of ã00. To
satisfy another condition of an effective generalized equivalence group we
drop any dependence of remaining c̄’s on the arbitrary elements. In fact,
we chose a correct parameterization for them already in the theorem.
Proposition 21. An effective generalized equivalence group Ĝ∼III of the
class FIII is constituted by the point transformations of the form
t̃ = T, x̃ = c5x−
c5
α
ln |c4Tt|, ũ =
c5
Tt
(
u− Ttt
αTt
)
, α̃ =
α
c5
,
ãj = c4c
j
5aj , ã01 = c4a01, ã00 = a00 + c3, b̃2 = c24c5b2,
where the function T is equal to
T (t) =
1
a00 + c3
ln
∣∣∣∣(a00 + c3)
(
c1
ea00t − 1
a00
+ c2
)
+ 1
∣∣∣∣ ,
and takes the following values at the singular points
T (t) = c1
ea00t − 1
a00
+ c2 if c3 = −a00 6= 0,
T (t) =
1
c3
ln |c3(c1t+ c2)| if a00 = 0 and c3 6= 0,
Equivalence groupoid of a class of stationary BKdV equations 149
T (t) = c1t+ c2 if (a00, c3) = (0, 0),
and c’s are arbitrary constants with c1c4c5 6= 0.
Guided by the same logic as for the class FII,1, we can show nonuni-
queness of effective generalized equivalence groups for FIII as well.
IV. Finally we discuss the last subclass FIV of F admitting additional
admissible transformations. It consists of equations
ut + uux =
r∑
j=2
ajuj + a0u+ b1x+ b0.
Since the arbitrary elements aj are scaled under the action of the equiv-
alence group of the class, it is reasonable to single out two subclasses of
the class under question: FIV,0 with aj = 0 for all j = 2, . . . , r − 1, and
complementary to it the subclass FIV,1 with at least one aj nonzero.
Proposition 22. The class FIV,1 is normalized in the generalized sense.
Its generalized equivalence group is constituted by the point transforma-
tions of the form
t̃ = T̄ 1t+ T̄ 0, x̃ = X̄1x+ X̄0, ũ =
X̄1
T̄ 1
u+
X̄0
t
T̄ 1
,
ãj =
(X̄1)r
T̄ 1
aj , a0 =
a0
T̄ 1
, b̃1 =
b1
(T̄ 1)2
,
b̃0 =
1
(T̄ 1)2
(
X̄1b0 + c̄3
)
,
where
X̄0(t, θ) =
c̄1e
λ1t + c̄2e
λ2t + c̄3 if λ1 6= 0, D > 0,
c̄1t+ c̄2e
λ2t + c̄3 if λ1 = 0, D > 0,
c̄1e
b1t/2 + c̄2te
b1t/2 + c̄3 if b1 6= 0, D = 0,
c̄1t
2 + c̄2t+ c̄3 if b1 = 0, D = 0,
eb1t/2
(
c̄1 sin(
√
−Dt)
+c̄2 cos(
√
−Dt)
)
+ c̄3 if D < 0,
where D = b21 + 4a0 and λ1,2 = (b1 ±
√
D)/2 with |λ1| < |λ2|, X̄1,
T̄ 0, T̄ 1 and c̄’s run through the set smooth functions of the arbitrary
elements θ = (aj , a0, b1, b0) with X̄1T̄ 1 ∂(ã2,...,ãr,ã0,b̃1,b̃0)
∂(a2,...,ar,a0,b1,b0) 6= 0.
150 S. Opanasenko
Here the function X0(t) is a solution of the ordinary differential equa-
tion X0
ttt − b1X
0
tt − a0X
0
t = 0 and thus it smoothly depends on the
parameters b1, a0 and all the initial conditions.
The equivalence groupoid of the class FIV,0 depends essentially on the
order r of equations therein. So we consider both the cases separately.
First assume that r > 2 and denote the class of such equations Fr>2
IV,0.
This class admits additional admissible transformations if and only if
b1 = a2
0(r − 1)/(r − 2)2, so we reduce a tuple of the arbitrary elements
thereof by the element b1.
Proposition 23. The class Fr>2
IV,0 is normalized in the generalized sense.
Its generalized equivalence group is constituted by the point transforma-
tions of the form
t̃ = T̄ , x̃ = X̄1x+ X̄0, ũ =
X̄1
T̄t
u+
X̄1
t
T̄t
x+
X̄0
t
T̄t
,
ãr =
(X̄1)r
T̄t
ar, ã0 = c̄3, b̃0 = c̄5,
where the pair of smooth functions
(
T̄ , X̄0
)
of t and the arbitrary ele-
ments θ equal to
(c̄1t+ c̄2, c̄7t
2 + c̄6t+ c̄5) if a0 = 0 and c̄3 = 0,(
1
c̄3
ln |c̄3(c̄1t+ c̄2)|, c̄5r
2
c̄23(r − 1)
+
c̄6t+ c̄7
|t+ c̄2/(c̄1c̄3)|1/r
− c̄23b0X̄
1
2
(
t+
c̄2
c̄1c̄3
)2)
if a0 = 0 and c̄3 6= 0,
(r − 2
c̄3r
ln
∣∣∣∣ 1
c̄1
e
a0rt
r−2 +
c̄2
c̄1
∣∣∣∣ , c̄5(r − 2)2
c̄23(r − 1)
+
(
c̄6e
a0rt
r−2 + c̄7
)
∣∣∣c̄2/c̄1 + e
a0rt
r−2
∣∣∣1/r
+
(r − 2)2b0
(r − 1)a2
0
X̄1
)
if a0c̄3 6= 0,(
c̄1e
a0rt
r−2 + c̄2,
c̄5c̄
2
1
2
e
2a0rt
r−2 + c̄6e
a0rt
r−2 + c̄7 −
(r − 2)2b0
(r − 1)a2
0
X̄1
)
if a0 6= 0 and c̄3 = 0,
Equivalence groupoid of a class of stationary BKdV equations 151
where c̄’s are arbitrary smooth functions of θ with c̄4T̄t
∂(ãr,ã0,b̃0)
∂(ar,a0,b0) 6= 0 as
well as X̄1(t, θ) = ε(c̄4T̄t)
1/r with ε = ±1 and c̄4T̄t > 0 if r is even and
ε = 1 otherwise.
The function X̄0 in the second pair in the second set gives a general
solution of the linear inhomogeneous equation on X0(t),
c̄5 = b0
X1
Tt2
+
1
Tt
(
X0
t
Tt
)
t
− c̄3
X0
t
Tt
− c̄23(r − 1)
(r − 2)2
X0,
parameterized by the function T in the set and the corresponding X1(t).
Any particular solution of this equation seems impossible to be found
with standard techniques. Here instead, we used a method used for the
class FI,00 with gauging the arbitrary elements a0 and b0 to 0 first and
composing equivalence transformations thereafter.
Due to the above condition on the arbitrary elements b1 and a0, the
class Fr=2
IV,0 admits additional admissible transformations if and only if
a0 = 0. Abusing notations we denote the subclass singled out by this
condition again by Fr=2
IV,0.
Proposition 24. The point transformation of the form
t̃ = T (t), x̃ = X1(t)x+X0(t), ũ =
X1
Tt
u+
X1
t
Tt
x+
X0
t
Tt
connects the source and target equations in the class Fr=2
IV,0 if and only
if (X1)2/Tt = const 6= 0, the parameter function T runs through the
solution set of the system(
Ttt
Tt
)
t
− 1
2
(
Ttt
Tt
)2
= 2b̃1T
2
t − 2b1,
and the parameter function X0 of t satisfies the equation
1
Tt
(
X0
t
Tt
)
t
− b̃1X0 = b̃0 − b0
X1
Tt2
.
The last equation is linear inhomogeneous with respect to X0(T ) for a
given T (t), while the differential equation on T is integrated in quadra-
tures as an autonomous equation on ln |Tt| with standard techniques.
Nonetheless, using the similar trick as was used for the class FI,00, one
152 S. Opanasenko
can do better. More precisely, we gauge the arbitrary elements b0 and b1
to zeros by the point transformation of the form
t̃ = T (t), x̃ =
√
Ttx+X0(t), ũ =
u√
Tt
+
Tttx
2(Tt)3/2
+
X0
t
Tt
,
where(
T,X0
)
=
(
e2
√
b1t, 4b0(2
√
b1)3/2e
√
b1t
)
if b1 > 0;(
T,X0
)
=
(
tan(
√
−b1t),
−b0(−b1)3/4
cos
√
−b1t
)
if b1 < 0,
obtaining the subclass Fr=2
IV,00 of Fr=2
IV,0. Thereafter we present the equiv-
alence groupoid of Fr=2
IV,0 by composing an equivalence transformation
within the subclass Fr=2
IV,00 with point transformations mapping equa-
tions in the superclass to equations in the subclass and vice versa.
Proposition 25. The class Fr=2
IV,00 is normalized in the usual sense. Its
usual equivalence group is constituted by point transformations of the
form
t̃ = T (t), x̃ = X1(t)x+X0, ũ =
X1
Tt
u+
X1
t
Tt
x, ã2 = c4a2,
where X1(t) = ε(c4δ)
1/2/(c3t + c0), T = (c1t + c2)/(c3t + c0), X0 and
c’s are arbitrary constants with δ = c1c0 − c2c3 6= 0, c0, c1, c2 and c3
being defined up to a nonzero constant, c4δ > 0 and ε = ±1.
The point transformation TT̃ ,X̃0 which maps an equation in Fr=2
IV,00 to
an equation in Fr=2
IV,0 is of the same form as above,
t = T̃ (t̃), x =
√
|T̃t̃|x̃+ X̃0(t̃), u =
ũ√
|T̃t̃|
+
T̃t̃t̃x̃
2|T̃t̃|3/2
+
X̃0
t̃
T̃t̃
,
where T̃ (T (t)) = t and X̃0(t̃) = −(X0/Tt)(T̃ (t̃)), that is,
(
T̃ , X̃0
)
=
(
ln |t̃|
2
√
b̃1
,
b̃0
b̃1
)
if b̃1 > 0;
(
T̃ , X̃0
)
=
(
arctan(t̃)√
−b̃1
,
b̃0
b̃1
)
if b̃1 < 0.
Equivalence groupoid of a class of stationary BKdV equations 153
Proposition 26. The class Fr=2
IV,0 is normalized in the generalized sense.
Its effective generalized equivalence group is constituted by point trans-
formations of the form
t̃ = P 2(T (P 1(t))),
x̃ =
√
P 2
t̄ P
1
t X
1(t̂)x+
√
P 2
t̄
(
X1R1 +X0
)
+R2,
ũ =
X1u√
P 2
t̄ P
1
t Tt̂
+
X1P 1
tt
2Tt̂
√
P 2
t̄ (P 1
t )3
+
X1
t̂
√
P 1
t√
P 2
t̄ Tt̂
+
P 2
t̄t̄X
1
2(P 2
t̄ )2
√
P 1
t
x
+
X1R1
t
Tt̂
√
P 2
t̄ P
1
t
+
X1
t̂
R1
Tt̂
+
P 2
t̄t̄
2(P 2
t̄ )3/2
(
X1R1 +X0
)
+
R2
t̄
P 2
t̄
,
ã2 = c4a2, b̃1 =
1
(P 2
t̄ )2
(
1
(P 1
t )2
(
b1 +
1
2
(
P 1
tt
P 1
t
)
t
− 1
4
(
P 1
tt
P 1
t
)2
)
+
1
2
(
P 2
t̄t̄
P 2
t̄
)
t̄
− 1
4
(
P 2
t̄t̄
P 2
t̄
)2
)
,
b̃0 =
1
(P 2
t̄ )3/2
(
b0
(P 1
t )3/2
+
1
P 1
t
(
R1
t
P 1
t
)
t
)
+
1
P 2
t̄
(
R2
t̄
P 2
t̄
)
t̄
− b̃1R2,
where t̂ = P 1(t), t̄ = T (t̂), t̃ = P 2(t̄), X1(t̂) = ε(c4Tt̂)
1/2, T = (c1t̂ +
c2)/(c3t̂+ c0) with δ = c1c0 − c2c3 6= 0;
(P 1(t), R1(t)) =
(t, −b0t2/2) if b1 = 0,
(tan(
√
−b1t), −b0(−b1)3/4/ cos(
√
−b1t)) if b1 < 0,
(e2
√
b1t, 4b0(2
√
b1)−3/2e
√
b1t) if b1 > 0;
X0 and c’s are arbitrary constants and the pair of smooth functions
(P 2(t̄), R2(t̄)) runs through the set{(
t̄,
c6t̄
2
2
)
,
(
ln |t̄|
2c5
,
c6
c25
)
,
(
arctan t̄
c5
, −c6
c25
)}
,
with ci, i = 0, . . . , 3, being defined up to a nonzero constant, c4δ > 0,
P 2
t̄ > 0 and ε = ±1.
In the notation of Proposition 26, the point transformation TP 2,R2
maps an equation in Fr=2
IV,00 to an equation in Fr=2
IV,0 with arbitrary-
element tuples (b0, b1) equal to (c6, 0), (c6, c
2
5) and (c6,−c25), respectively.
154 S. Opanasenko
Acknowledgments. The author is grateful to Roman O. Popovych
for useful discussions and interesting comments, and acknowledges the
support of the Natural Sciences and Engineering Research Council of
Canada.
[1] Arnol’d V.I., Ordinary differential equations, Springer-Verlag, Berlin, 1992.
[2] Kurujyibwami C., Basarab-Horwath P., Popovych R.O., Algebraic method for
group classification of (1+1)-dimensional linear Schrödinger equations, Acta
Appl. Math. 157 (2018), 171–203, arXiv:1607.04118.
[3] Meleshko S.V., Group classification of equations of two-dimensional gas mo-
tions, J. Appl. Math. Mech. 58 (1994), 629–635.
[4] Meleshko S.V., Generalization of the equivalence transformations, J. Nonlinear
Math. Phys. 3 (1996), 170–174.
[5] Nikitin A.G., Popovych R.O., Group classification of nonlinear Schrödinger
equations, Ukrainian Math. J. 53 (2001), 1255–1265, arXiv:math-ph/0301009.
[6] Opanasenko S., Bihlo A., Popovych R.O., Group analysis of general Burgers–
Korteweg–de Vries equations, J. Math. Phys. 58 (2017), 081511, 37 pp.,
arXiv:1703.06932.
[7] Opanasenko S., Boyko V., Popovych R.O., Enhanced group classifica-
tion of reaction-diffusion equations with gradient-dependent diffusion,
arXiv:1804.08776.
[8] Popovych R.O., Classification of admissible transformations of differential equa-
tions, in Collection of Works of Institute of Mathematics, Kyiv 3 (2006), no. 2,
239–254.
[9] Popovych R.O., Kunzinger M., Eshraghi H., Admissible transformations and
normalized classes of nonlinear Schrödinger equations, Acta Appl. Math. 109
(2010), 315–359, arXiv:math-ph/0611061.
[10] Vaneeva O.O., Johnpillai A.G., Popovych R.O., Sophocleous C., Enhanced
group analysis and conservation laws of variable coefficient reaction-diffusion
equations with power nonlinearities, J. Math. Anal. Appl. 330 (2007), 1363–
1386, arXiv:math-ph/0605081.
|
| id | oai:trim.imath.kiev.ua:article-371 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:16Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a5/7abc4120d7a794fb0620d7282b0ce6a5.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3712020-08-13T08:52:53Z Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients Групоїд еквівалентності класу загальних рівнянь Бюргерса-Кортевега-де Фріза з просторовими коефіцієнтами Opanasenko, S. Опанасенко, С. We describe the equivalence groupoid of the class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients.This class is shown to reduce by a family of equivalence transformations to a subclass with a four-dimensional usual equivalence group.Classified are admissible transformations of this subclass and singled out its subclasses admitting maximal nontrivial conditional equivalence groups.All of them turn out to have dimension higher than four.In particular, few new examples of nontrivial cases of normalization in the generalized sense of classes of differential equationsappeared this way. Neither of classes discussed possesses a unique effective generalized equivalence group. Описано групоїд еквівалентності класу загальних рівнянь Бюргерса-Кортевега-де Фріза з просторовими коефіцієнтами. Показано, що цей клас зводиться сім'єю перетворень еквівалентності до свого підкласу з чотиривимірною звичайною групою еквівалентності. Прокласифіковано допустимі перетворення цього підкласу та виокремлені підкласи, що допускають максимальні нетривіальні умовні групи еквівалентності. Виявляється, що всі вони мають розмірність більшу за чотири. Зокрема, знайдено декілька нових класів диференціальних рівнянь, нормалізованих в узагальненому сенсі. Жоден з них не допускає єдину ефективну узагальнену групу еквівалентності. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/371 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 131-154 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 131-154 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 131-154 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/371/368 Авторське право (c) 2019 С. Опанасенко |
| spellingShingle | Opanasenko, S. Опанасенко, С. Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title | Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title_alt | Групоїд еквівалентності класу загальних рівнянь Бюргерса-Кортевега-де Фріза з просторовими коефіцієнтами |
| title_full | Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title_fullStr | Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title_full_unstemmed | Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title_short | Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients |
| title_sort | equivalence groupoid of a class of general burgers-korteweg-de vries equations with space-dependent coefficients |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/371 |
| work_keys_str_mv | AT opanasenkos equivalencegroupoidofaclassofgeneralburgerskortewegdevriesequationswithspacedependentcoefficients AT opanasenkos equivalencegroupoidofaclassofgeneralburgerskortewegdevriesequationswithspacedependentcoefficients AT opanasenkos grupoídekvívalentnostíklasuzagalʹnihrívnânʹbûrgersakortevegadefrízazprostorovimikoefícíêntami AT opanasenkos grupoídekvívalentnostíklasuzagalʹnihrívnânʹbûrgersakortevegadefrízazprostorovimikoefícíêntami |