Differential invariants for a class of diffusion equations
We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional.The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related...
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| author | Dos Santos Cardoso-Bihlo, E. Bihlo, A. Popovych, R. Дос Сантос Кардозо-Біло, E. Біло, А. Попович, Р. |
| author_facet | Dos Santos Cardoso-Bihlo, E. Bihlo, A. Popovych, R. Дос Сантос Кардозо-Біло, E. Біло, А. Попович, Р. |
| author_institution_txt_mv | [
{
"author": "E. Dos Santos Cardoso-Bihlo",
"institution": "Memorial University of Newfoundland"
},
{
"author": "A. Bihlo",
"institution": "Memorial University of Newfoundland"
},
{
"author": "R. Popovych",
"institution": "Institute of Mathematics of NAS of Ukraine; Universitat Wien"
}
] |
| author_sort | Dos Santos Cardoso-Bihlo, E. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-13T08:52:53Z |
| description | We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional.The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration.Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation. |
| first_indexed | 2026-08-04T01:06:11Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Iíñòèòóòó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2019, ò. 16, � 1, 50�65
ÓÄÊ 517.912:512.816
Di�erential invariants for a class
of di�usion equations
E. Dos Santos Cardoso-Bihlo †, A. Bihlo †, R.O. Popovych ‡
† Memorial University of Newfoundland, Canada
E-mail: ecardosobihlo@mun.ca, abihlo@mun.ca
‡ Universit�at Wien, Austria
Institute of Mathematics of NAS of Ukraine, Kyiv
E-mail: rop@imath.kiev.ua
Çíàéäåíî ïîâíó ãðóïó åêâiâàëåíòíîñòi êëàñó (1+1)-âèìiðíèõ åâîëþöié-
íèõ ðiâíÿíü äðóãîãî ïîðÿäêó, ÿêà âèÿâèëàñÿ íåñêií÷åííîâèìiðíîþ. Ìå-
òîäîëîãiþ åêâiâàðiàíòíèõ ðóõîìèõ ðåïåðiâ çàñòîñîâàíî ó ðåãóëÿðíîìó
âèïàäêó ïðîöåäóðè íîðìàëiçàöi¨ äî ïîáóäîâè ðóõîìîãî ðåïåðà ãðóïè,
ïîâ'ÿçàíî¨ ç ãðóïîþ åêâiâàëåíòíîñòi â êîíòåêñòi ïåðåòâîðåíü åêâiâà-
ëåíòíîñòi ìiæ ðiâíÿííÿìè êëàñó. Çà äîïîìîãîþ ïîáóäîâàíîãî ðóõîìîãî
ðåïåðà îïèñàíî àëãåáðó äèôåðåíöiàëüíèõ iíâàðiàíòiâ öi¹¨ ãðóïè ÷åðåç
îòðèìàííÿ ìiíiìàëüíî¨ ãåíåðóþ÷î¨ ìíîæèíè äèôåðåíöiàëüíèõ iíâàði-
àíòiâ i ïîâíî¨ ìíîæèíè îïåðàòîðiâ iíâàðiàíòíîãî äèôåðåíöiþâàííÿ.
We �nd the complete equivalence group of a class of (1+1)-dimensional
second-order evolution equations, which is in�nite-dimensional. The equi-
variant moving frame methodology is invoked to construct, in the regular
case of the normalization procedure, a moving frame for a group related
to the equivalence group in the context of equivalence transformations
among equations of the class under consideration. Using the moving frame
constructed, we describe the algebra of di�erential invariants of the former
group by obtaining a minimum generating set of di�erential invariants and
a complete set of independent operators of invariant di�erentiation.
1. Introduction. Invariants and differential invariants of transfor-
mation groups, in particular, point symmetry groups admitted by sys-
tems of differential equations have a wide range of applications and are
therefore an intensively investigated subject. Differential invariants play
a central role in the invariant parameterization problem [1, 2, 30] and in
the problem of invariant discretization [3, 5, 7]. They are also used to
Differential invariants for a class of diffusion equations 51
construct invariant differential equations and invariant variational prob-
lems [22, 23], as well as in computer vision, integrable systems, classical
invariant theory and the calculus of variations [6, 22, 24].
Rather recently, finding differential invariants in problems related to
group classification became a research topic of interest. The idea is to
compute the differential invariants not for the point symmetry group of
a single system of differential equations but for the equivalence group
admitted by a class of such systems. The primary motivation for such
a survey is to study the equivalence of systems of differential equations.
Exploring equivalence, it is possible to explicitly determine point trans-
formations among systems from a class [28]. Such a mapping between
two systems of differential equations is especially helpful if wide sets
of exact solutions are known for one of the systems involved. These
solutions then can be mapped to solutions of the equivalent system. An-
other case of particular interest is the mapping between nonlinear and
linear elements of a class of systems of differential equations [19]. For
the solution of the equivalence problem, finding differential invariants
for the equivalence group is a main ingredient. There are a number of
papers where some low-order differential invariants of the equivalence
groups of various physically relevant classes of systems of differential
equations were computed using the Lie infinitesimal method; see, e.g.,
[11, 12, 13, 14, 15, 17, 32, 33, 34, 35] and references therein.
In the present paper we will be concerned with differential invariants
for a group1 related to the equivalence group of the class of diffusion
equations
ut = uxx + f(u, ux) (1)
in the context of equivalence transformations among equations of this
class. This subject was originally considered in [32], using the infinitesi-
mal method and restricting the order of differential invariants up to two.
We revisit the construction of differential invariants for the class (1)
from the very beginning, analyzing differential invariants of which group
should be found. Then, we apply the method of equivariant moving
frames in the formulation originally proposed and formulated by Fels
and Olver [9, 10], which was later generalized to infinite-dimensional
Lie (pseudo)groups in [6, 25, 26], and this is the setting that is needed
1In fact, this object and the “equivalence group” of the class (1) are Lie pseu-
dogroups of locally defined point transformations. We use the term “group” for
brevity since this does not lead to any confusion.
52 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
to study differential invariants for the class (1). The advantage of mo-
ving frames is that they allow for a canonical process of invariantization,
which associates to each object, such as functions, differential functions,
differential forms and total differentiation operators, its invariant coun-
terpart. For the problem of finding differential invariants of a Lie trans-
formation (pseudo)group, this property is especially convenient. The
invariantization of the jet-space coordinate functions yields the so-called
normalized differential invariants. The invariantized coordinate func-
tions whose transformed counterparts were involved in the construction
of the corresponding moving frame via the normalization procedure are
equal to the respective constants chosen in the course of normalization.
This is why these objects are called phantom normalized differential in-
variants. The non-phantom normalized differential invariants constitute
a complete set of functionally independent differential invariants. As
a further asset, the method of moving frames also permits to study the
algebra of differential invariants by deriving relations, called syzygies,
between invariant derivatives of non-phantom normalized differential in-
variants. Finding syzygies can aid in the establishment of a minimum
generating set of differential invariants. See e.g. [6, 8, 22, 25, 26] for
more details and an extensive discussion on the computation of differen-
tial invariants for both finite-dimensional Lie symmetry groups and for
infinite-dimensional Lie (pseudo)groups using moving frames.
The further organization of this paper is as follows. In Section 2
we compute the equivalence group and the equivalence algebra of the
class (1). Section 3 is devoted to the selection of a group to be consid-
ered and a preliminary analysis of equivariant moving frames associated
with this group. The structure of the algebra of differential invariants
is determined in the main Section 4. This includes a description of a
minimum generating set of differential invariants and a complete set of
independent operators of invariant differentiation, which serve to ex-
haustively describe the set of differential invariants. Moreover, for each
k ∈ N0 we explicitly present a functional basis of differential invariants
of order not greater than k.
2. The equivalence group. The auxiliary system for the class (1),
which is satisfied by the arbitrary element f , is ft = fx = fut = futt =
futx = fuxx = 0. By definition [27, 28, 29, 31], the (usual) equiva-
lence group G∼ of the class (1) consists of the point transformations in
the space with coordinates (t, x, u, ut, ux, utt, utx, uxx, f) that have the
following properties:
Differential invariants for a class of diffusion equations 53
• they are projectable to the space with the coordinates (t, x, u),
• their components for derivatives of u are found by prolongation
using the chain rule, and
• they map every equation from the class (1) to an equation from
the same class.
To begin finding the group G∼, we fix an arbitrary equation of the
class (1), ut = uxx + f(u, ux), and aim to find point transformations in
the space with coordinates (t, x, u),
t̃ = T (t, x, u), x̃ = X(t, x, u), ũ = U(t, x, u), (2)
that transform the fixed equation to an equation of the same class,
ũt̃ = ũx̃x̃ + f̃(ũ, ũx̃). (3)
A preliminary simplification is obtained from noting that the class (1)
is a subclass of the class of second-order (1+1)-dimensional semi-linear
evolution equations. Any point transformation between two equations
from the latter class satisfies the constraints Tx = Tu = Xu = 0, i.e.,
t̃ = T (t), x̃ = X(t, x), and TtXxUu 6= 0. See [16, 18, 21] for further
details. After taking into account the above constraints, the required
transformed derivatives read
ũt̃ =
1
Tt
(
DtU −
Xt
Xx
DxU
)
, ũx̃ =
1
Xx
DxU,
ũx̃x̃ =
(
1
Xx
Dx
)2
U,
where Dt and Dx are the usual total derivative operators with respect
to t and x, respectively. Substituting these expressions and ut = uxx+f
into Eq. (3), we split the resulting equation with respect to uxx yielding
Tt = X 2
x . The remaining equation is
f =
Tt
Uu
f̃ − Ut +
Xt
Xx
(Ux + Uuux) + Uxx + 2Uxuux + Uuuu
2
x. (4)
The differential consequences of Eq. (4) that are obtained by separate
differentiations with respect to t and x can be split with respect to
derivatives of f̃ since they are regarded as independent for equivalence
transformations. This yields the equations
Ttt = Xxt = Xtt = Ut = Ux = 0.
54 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
The equation (4) itself gives the f -component of equivalence transfor-
mations.
The arbitrary element f in fact depends only on u and ux. The space
with coordinates (t, x, u, ux, f) is preserved by all elements of G∼. This
is why we can assume this space as the underlying space for G∼ and
present merely the transformation components for its coordinates.
As a result, we have proved the following theorem.
Theorem 1. The equivalence group G∼ of the class (1) is constituted
by the transformations
t̃ = C2
1 t+ C0, x̃ = C1x+ C1C2t+ C3, ũ = ϕ(u),
ũx̃ = C−1
1 ϕ′ux, f̃ = C−2
1
(
ϕ′f − C2ϕ
′ux − ϕ′′u2
x
)
,
(5)
where C0, C1, C2, C3 ∈ R, ϕ is an arbitrary smooth function of u and
C1ϕ
′ 6= 0.
The infinitesimal generators of one-parameter subgroups of G∼, which
constitute the equivalence algebra g∼ of the class (1), can be derived
from (5) by differentiation, cf. the proof of Corollary 11 in [20] or the
proof of Corollary 6 in [4]. These generators coincide with those de-
termined in [32]. As we will later need them for the description of the
algebra of differential invariants of a group related to G∼ in the context
of the G∼-equivalence among equations of the class (1), we present them
here. The general element of g∼ is
Q = τ∂t + ξ∂x + φ∂u + η∂ux + θ∂f ,
where the components are of the form
τ = 2c1t+ c0, ξ = c1x+ c2t+ c3, φ = φ(u),
η = (φ′ − c1)ux, θ = (φ′ − 2c1)f − c2ux − φ′′u2
x,
in which c0, c1, c2 and c3 are arbitrary real constants, and φ is an arbi-
trary smooth function of u. In other words, the equivalence algebra g∼
of the class (1) is spanned by the vector fields
∂t, 2t∂t + x∂x − ux∂ux − 2f∂f , t∂x − ux∂f ,
φ∂u + φ′ux∂ux + (φ′f − φ′′u2
x)∂f ,
where φ runs through the set of smooth functions of u.
Differential invariants for a class of diffusion equations 55
3. Preliminary analysis of moving frames. Let us first clar-
ify the space of independent and dependent variables to be used and
the group to be considered. While formally the arbitrary element f
is a smooth function on the second-order jet space with coordinates
(t, x, u, ut, ux, utt, utx, uxx), practically it explicitly depends only on u
and ux. This is why subsequently we will only consider the projection of
the equivalence transformations to the space with coordinates (u, ux, f).
As a shorthand, we denote v := ux and ṽ := ũx̃ = V (u, v) := C−1
1 ϕ′(u)v.
In other words, we will in fact study differential invariants of the pro-
jection G1 of G∼ to the space with coordinates (u, v, f), where u and v
are the independent variables and f is the dependent variable. The in-
finitesimal counterpart of G1 is the projection g1 of g∼ to the space with
coordinates (u, v, f).
In order to describe the algebra of differential invariants of the group
G1, we now construct a moving frame for this group. Since it is infinite-
dimensional, we have to use the machinery developed for Lie pseudo-
groups, see [6, 25] for an extensive description of this subject.
The first step in the construction of the moving frame is the com-
putation of the lifted horizontal coframe, the dual of which yields the
implicit total differentiation operators Dũ and Dṽ. For the equivalence
transformations (5), the lifted horizontal coframe is
dhũ = (DuU) du+ (DvU) dv = ϕ′ du,
dhṽ = (DuV ) du+ (DvV ) dv =
ϕ′′
C1
v du+
ϕ′
C1
dv.
Computing the dual, we derive that
Dũ =
1
ϕ′
Du −
ϕ′′
(ϕ′)2
vDv, Dṽ =
C1
ϕ′
Dv (6)
are the required implicit differentiation operators. Acting with them on
the transformation component for f , we find that
f̃ij =
∂i+j f̃
∂ũi∂ṽj
= D i
ũD j
ṽ F,
where i, j ∈ N0 := N ∪ {0} and
f̃00 = f̃ = F :=
1
C2
1
(ϕ′f − C2ϕ
′v − ϕ′′v2)
56 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
is the f -component of equivalence transformations. In particular, the
derivatives up to order 2 are exhausted by
f̃10 =
1
C2
1ϕ
′
(
ϕ′fu + ϕ′′(f − vfv)− ϕ′′′v2 + 2
(ϕ′′)2
ϕ′
v2
)
,
f̃01 =
1
C1ϕ′
(ϕ′fv − C2ϕ
′ − 2ϕ′′v) ,
f̃20 =
1
C2
1ϕ
′
(
fuu −
ϕ′′
ϕ′
(fu−2vfuv) +
(
ϕ′′
ϕ′
)2
v2fvv
+
(
ϕ′′
ϕ′
)′
(f−vfv)− (ϕ′)2
(
1
ϕ′
(
1
ϕ′
)′′)′
v2
)
,
f̃11 =
1
C1ϕ′2
(
ϕ′fuv − ϕ′′vfvv − 2ϕ′′′v + 4
ϕ′′2
ϕ′
v
)
,
f̃02 =
1
ϕ′2
(ϕ′fvv − 2ϕ′′).
There are a relative invariant and a relative conditional invariant
which play a significant role in the following consideration. By taking
the difference f̃00 − ṽf̃01 we exclude the inessential constant C2, which
only arises in f̃00 and f̃01,
f̃00 − ṽf̃01 =
1
C2
1
(
ϕ′(f − vfv) + ϕ′′v2
)
.
Combining further 2(f̃00 − ṽf̃01) + ṽ2f̃02 to exclude ϕ′′, we obtain
W̃ =
1
C2
1
W, where
W = 2f − 2vfv + v2fvv,
W̃ = 2f̃ − 2ṽf̃ṽ + ṽ2f̃ṽṽ,
i.e., W is a relative invariant of G1. In other words, the condition W = 0
is preserved by any equivalence transformation in the class (1). Analo-
gously, the combination 2f̃10 − vf̃11 gives
S̃ =
1
C2
1
S +
1
C2
1
ϕ′′
ϕ′
W, where
S = 2fu − vfuv,
S̃ = 2f̃ũ − ṽf̃ũṽ.
(7)
This means that S is a relative invariant of G1 if the condition W = 0 is
satisfied. Values of the differential functions W and S determine which
normalization conditions should be chosen.
Differential invariants for a class of diffusion equations 57
We next find appropriate normalization conditions, which form the
basis for the construction of an equivariant moving frame. As ϕ arises
only in U , we can set U to any value including zero. The value of V
can be set to any constant excluding zero, and all these possibilities are
equivalent. We find it convenient to put V = 1 and express ϕ′ = C1/v.
The constraint W = 0 singles out the singular case for the moving
frame construction, which has to be investigated separately. Within
this singular case, there is the ultra-singular subcase associated with the
constraint S = 0. Indeed, under the constraint W = 0 the equation (7)
can be solved for C1 if and only if S 6= 0.
4. Differential invariants for the regular case. In this paper, we
only consider the regular case for moving frames of G1, where W 6= 0.
In this case, the following normalization conditions can be used to de-
termine a complete moving frame
ũ = 0, ṽ = 1, f̃ = 1, f̃01 = 0, f̃02 = 0,
f̃i0 = −v
2ϕ(i+2)
C 2
1 (ϕ′)i
+
1
C 2
1
i∑
i′=0
(
i
i′
)
1
(ϕ′)i′
(
ϕ′′
(ϕ′)2
)i−i′
fi′,i−i′
+ · · · = 0, i ∈ N.
(8)
In the expression for f̃i0, we presented only the summands with the
highest-order derivatives of ϕ and f , which are ϕ(i+2) and fi′,i−i′ , i
′ =
0, . . . , i, respectively. We solve the first five equations with respect to C1,
C2, ϕ, ϕ′ and ϕ′′ and substitute the obtained expressions into the other
equations. For each fixed i ∈ N, we solve the modified equation f̃i0 = 0
in view of the similar equations with lower values of i and thus find an
expression for ϕ(i+2), the explicit form of which is essential for further
consideration only for i = 3. This yields the following complete moving
frame:
C1 =
W
2v
, C2 = fv − vfvv,
ϕ = 0, ϕ′ =
W
2v2
, ϕ′′ =
W
4v2
fvv,
ϕ′′′ =
W
4v4
(
2fu + (f − vfv + v2fvv)fvv
)
,
ϕ(i+2) =
W
2v1
i∑
i′=0
(
i
i′
)(
v2
W
)i−i′
fi′,i−i′ + · · · , i = 2, 3, . . . .
(9)
58 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
In the expression for ϕ(i+2), we presented only the summands with the
highest-order derivatives fi′,i−i′ , i
′ = 0, . . . , i. The invariantization Iij =
ι(fij) of the derivatives fij of f̃ that are not involved in the normalization
conditions (8) gives rise to a complete set of functionally independent
differential invariants of G1. The lowest-order non-phantom normalized
differential invariant is I11, and it reads
I11 = −2v2 4fu − 2vfuv + (2f − 2vfv + v2fvv)fvv
(2f − 2vfv + v2fvv)2
.
This differential invariant is of second order. For each tuple (i, j) with
i + j > 3 and j 6= 0, the maximal orders of derivatives of f and ϕ
appearing in the expression for f̃ij are i+ j and i+ 2, respectively. This
is why the maximal order of derivatives of f in the expression for f̃ij
cannot be lowered in the course of the invariantization, i.e., the order
of the normalized differential invariant Iij is i+ j. Therefore, there are
precisely 1
2k(k+1)−2 functionally independent differential G1-invariants
of order not greater than k > 2. They are given by the functions I11
and Iij with 3 6 i+ j 6 k and j 6= 0.
Apart from finding the complete set of functionally independent dif-
ferential invariants of G1 for each fixed order by successively invarianti-
zing all the derivatives fij , the moving frame (9) can be used to deter-
mine the operators of invariant differentiation. They are found upon
invariantizing the operators of total differentiation (6) and read
Di
u =
2v2
2f − 2vfv + v2fvv
(
Du −
1
2
vfvvDv
)
, Di
v = vDv. (10)
We now aim to investigate the structure of the algebra of differential
invariants of G1. The starting point for this investigation is the uni-
versal recurrence relation, which relates the differentiated invariantized
differential functions or differential forms with the invariantization of
the respective differentiated objects. This universal recurrence relation
reads [25]
dι(Ω) = ι
(
dΩ +Q(∞)(Ω)
)
. (11)
The first step in our study is the evaluation of (11) for the independent
variables u and v and the derivatives fij , i, j ∈ N0,
dhι(u) = ω1 + ι(φ), dhι(v) = ω2 + ι(η),
Differential invariants for a class of diffusion equations 59
dhI
ij = dhι(fij) = ι(fi+1,jdu+ fi,j+1dv + θij)
= Ii+1,jω1 + Ii,j+1ω2 + ι(θij),
where ω1 = ι(du), ω2 = ι(dv), and
θij = D i
uD j
v (θ − φf10 − ηf01) + φfi+1,j + ηfi,j+1
= (j − 2)c1fij − (j − 1)
i∑
i′=0
(
i
i′
)
φ(i′+1)fi−i′,j
−
i∑
i′=1
(
i
i′
)(
φ(i′)fi−i′+1,j + vφ(i′+1)fi−i′,j+1
)
− c2δ0i(δ0jv + δ1j)− φ(i+2)(δ0jv
2 + 2δ1jv + 2δ2j)
is the fij-component of the infinite prolongation of the vector field φ∂u+
η∂v + θ∂f . Here δij is the Kronecker delta. The respective recurrence
relations then split into two kinds, the first being the so-called phantom
recurrence relations. For a well-defined moving frame cross-section, they
can be uniquely solved for the invariantized Maurer–Cartan forms, which
arise due to the presence of the correction term ι
(
Q(∞)(Ω)
)
in (11).
Then, plugging these invariantized Maurer–Cartan forms into the second
kind of recurrence relations, the non-phantom ones, gives a complete
description of the relation between the normalized and differentiated
differential invariants, see [6, 25] for more details. For the chosen cross-
section (8), the phantom recurrence relations read
0 = dhι(u) = ω1 + ι(φ) = ω1 + φ̂,
0 = dhι(v) = ω2 + ι(η) = ω2 + φ̂′ − ĉ1,
0 = dhI
00 = ι(θ) = φ̂′ − 2ĉ1 − ĉ2 − φ̂′′,
0 = dhI
01 = I11ω1 + ι(θ01) = I11ω1 − ĉ2 − 2φ̂′′,
0 = dhI
02 = I12ω1 + I03ω2 + ι(θ02) = I12ω1 + I03ω2 − 2φ̂′′,
0 = dhI
i0 = Ii1ω2 + ι(θi0)
= Ii1ω2 + φ̂(i+1) − φ̂(i+2) −
i−1∑
i′=1
(
i
i′
)
Ii−i
′,1φ̂(i′+1), i ∈ N,
where the forms ĉ1, ĉ2 and φ̂(i), i ∈ N0, are the invariantizations of the
parameters c1, c2 and φ(i) of the infinitely prolonged general element of
the algebra g1, respectively, ĉ1 = ι(c1), ĉ2 = ι(c2) and φ̂(i) = ι(φ(i)).
60 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
More rigorously, here the parameters c1, c2 and φ(i), i ∈ N0, are in-
terpreted as the coordinate functions on the infinite prolongation of g1.
Recall that under the prolongation we consider u and v to be the indepen-
dent variables and f to be the dependent variable. In other words, these
coefficients are first-order differential forms in the jet space J∞(u, v| f).
Hence their invariantizations are also forms, which are called invari-
antized Maurer–Cartan forms.
The above system can be solved to yield the following invariantized
Maurer–Cartan forms
ĉ1 =
(
1
2
I12 − I11
)
ω1 +
(
1
2
I03 − 1
)
ω2,
ĉ2 = (I11 − I12)ω1 − I03ω2,
φ̂ = −ω1, φ̂′ =
(
1
2
I12 − I11
)
ω1 +
(
1
2
I03 − 2
)
ω2,
φ̂′′ =
1
2
I12ω1 +
1
2
I03ω2,
φ̂(i+2) = φ̂(i+1) −
i−1∑
i′=1
(
i
i′
)
Ii−i
′,1φ̂(i′+1) + Ii1ω2, i ∈ N.
(12)
The explicit expression for the invariantized form φ̂(i+2), i ∈ N, as a
combination of ω1 and ω2 with coefficients being polynomials of normal-
ized differential invariants is obtained by expanding the above expression
when successively going over the values of i. In particular,
φ̂′′′ =
1
2
I12ω1 +
(
I11 +
1
2
I03
)
ω2,
φ̂(4) =
(
1
2
I12 − I11I12
)
ω1 +
(
I11 +
1
2
I03 + I21 − I11I03
)
ω2.
For i > 3, the greatest value of i′ + j′ for the normalized differential
invariants Ii
′j′ that are involved in φ̂(i+2) is i+ 1, and Ii1ω2 is the only
summand with this value.
The non-phantom recurrence relations are
dhI
11 = I21ω1 + I12ω2 + ι(θ11)
=
(
I21 + 2(I11)2 − I11I12 − I12
)
ω1
+
(
I12 − I11I03 + I11 − I03
)
ω2,
Differential invariants for a class of diffusion equations 61
dhI
ij = Ii+1,jω1 + Ii,j+1ω2 + ι(θij), i+ j > 3, j 6= 0,
with
ι(θij) = (j − 2)Iij ĉ1 − (j − 1)
i∑
i′=0
(
i
i′
)
Ii−i
′,j φ̂(i′+1)
−
i∑
i′=1
(
i
i′
)(
Ii−i
′+1,j φ̂(i′) + Ii−i
′,j+1φ̂(i′+1)
)
− δ0i(δ0j + δ1j)ĉ2 − (δ0j + 2δ1j + 2δ2j)φ̂
(i+2).
The first non-phantom recurrence relation splits into
Di
uI
11 = I21 + 2(I11)2 − I11I12 − I12,
Di
vI
11 = I12 − I11I03 + I11 − I03.
Therefore, the normalized differential invariants I12 and I21 are ex-
pressed in terms of invariant derivatives of I11 and I03,
I12 = Di
vI
11 + I11I03 − I11 + I03,
I21 = Di
uI
11 − 2(I11)2
+ (I11 + 1)(Di
vI
11 + I11I03 − I11 + I03).
(13)
In view of the above discussion on the invariantize forms φ̂(i′), i ∈ N, the
expression for ι(θij) with i + j > 3 and j 6= 0 implies that the greatest
value of i′ + j′ for Ii
′j′ involved in ι(θij) is i + j. Hence splitting the
recurrence relation with dhI
ij leads to expressions for Ii+1,j and Ii,j+1
in terms of invariant derivatives of Ii
′j′ with i′+ j′ 6 i+ j. For example,
from the non-phantom recurrence relation
dhI
03 = I13ω1 + I04ω2 + ι(θ03)
=
(
I13 + I11I03 − I12I03
2
)
ω1 +
(
I04 + I03 − (I03)2
2
)
ω2
we derive
Di
uI
03 = I13 + I11I03 − 1
2
I12I03,
Di
vI
03 = I04 − 1
2
(I03)2 + I03.
62 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
This implies by induction, where the expressions (13) for I12 and I21 give
the base case, that any non-phantom normalized differential invariant
can be expressed in terms of invariant derivatives of I11 and I03.
To find a minimum generating set of differential invariants for the
projected group G1, we should additionally check whether I03 can be
expressed in terms of invariant derivatives of I11. We use (11) to compute
the commutator between the operators of invariant differentiation. This
is done upon evaluating (11) for the basis horizontal forms du and dv,
dhι(du) = ι(φ′du) = ι(φ′) ∧ ι(du)
=
(
2− 1
2
I03
)
ω1 ∧ ω2 = −Y 1
12 ω
1 ∧ ω2,
dhι(dv) = ι
(
φ′′vdu+ (φ′ − c1)dv
)
= ι(φ′′v) ∧ ι(du) = −1
2
I03ω1 ∧ ω2 = −Y 2
12 ω
1 ∧ ω2.
The commutation relation then evaluates as
[Di
u,D
i
v] = Y 1
12Di
u + Y 2
12Di
v =
(
1
2
I03 − 2
)
Di
u +
1
2
I03Di
v,
see [25] for details of the technique applied. Evaluating [Di
u,D
i
v]I
11, we
can derive the following expression for I03:
I03 :=
2v3fvvv
2f − 2vfv + v2fvv
= 2
2Di
uI
11 + [Di
u,D
i
v]I
11
Di
uI
11 + Di
vI
11
.
As a result, we have proved the following theorem.
Theorem 2. The algebra of differential invariants of the group G1,
which is the projection of the equivalence group G∼ of the class of diffu-
sion equations (1) to the space with coordinates (u, v, f), is generated by
the single differential invariant
I11 = −2v2 4fu − 2vfuv + (2f − 2vfv + v2fvv)fvv
(2f − 2vfv + v2fvv)2
along with the two operators of invariant differentiation
Di
u =
2v2
2f − 2vfv + v2fvv
(
Du −
1
2
vfvvDv
)
, Di
v = vDv.
All other differential invariants are functions of I11 and invariant deriva-
tives thereof.
Differential invariants for a class of diffusion equations 63
Corollary 1. A functional basis of differential invariants of order not
greater than k ∈ N0 in terms of invariant derivatives of non-phantom
normalized differential invariants is exhausted by(
Di
u
)i(
Di
v
)j
I11, i+ j 6 k − 2,
(
Di
v
)j′
I03, j′ 6 k − 3.
Acknowledgements. This research was undertaken, in part, thanks
to funding from the Canada Research Chairs program, the NSERC Dis-
covery Grant program and the InnovateNL LeverageR&D program. AB
is a recipient of an APART Fellowship of the Austrian Academy of Sci-
ences. The research of ROP and EMDSCB was supported by the Aus-
trian Science Fund (FWF), projects P25064 and P29177.
[1] Bihlo A., Dos Santos Cardoso-Bihlo E., Popovych R.O., Invariant parameteri-
zation of geostrophic eddies in the ocean, arXiv:1908.06345, 21 pp.
[2] Bihlo A., Dos Santos Cardoso-Bihlo E.M., Popovych R.O., Invariant param-
eterization and turbulence modeling on the beta-plane, Phys. D 269 (2014),
48–62, arXiv:1112.1917.
[3] Bihlo A., Popovych R.O., Invariant discretization schemes for the shallow-water
equations, SIAM J. Sci. Comput. 34 (2012), B810–B839, arXiv:1201.0498.
[4] Bihlo A., Popovych R.O., Group classification of linear evolution equations,
J. Math. Anal. Appl. 448 (2017), 982–1005, arXiv:1605.09251.
[5] Bihlo A., Valiquette F., Symmetry-preserving numerical schemes, in Symmetries
and Integrability of Difference Equations, CRM Ser. Math. Phys., Springer,
Cham, 2017, 261–324.
[6] Cheh J., Olver P.J., Pohjanpelto J., Algorithms for differential invariants of
symmetry groups of differential equations, Found. Comput. Math. 8 (2008),
501–532.
[7] Dorodnitsyn V., Applications of Lie groups to difference equations, Differential
and Integral Equations and their Applications, Vol. 8, Chapman & Hall/CRC,
Boca Raton, FL, 2011.
[8] Dos Santos Cardoso-Bihlo E.M., Differential invariants for the Korteweg–de
Vries equation, in Proceedings of 6th International Workshop “Group Analysis
of Differential Equations & Integrable Systems” (June 17–21, 2012, Protaras,
Cyprus), University of Cyprus, Nicosia, 2013, 71–79.
[9] Fels M., Olver P.J., Moving coframes: I. A practical algorithm, Acta Appl.
Math. 51 (1998), 161–213.
[10] Fels M., Olver P.J., Moving coframes. II. Regularization and theoretical foun-
dations, Acta Appl. Math. 55 (1999), 127–208.
[11] Gandarias M.L., Torrisi M., Tracina R., On some differential invariants for a
family of diffusion equations, J. Phys. A: Math. Theor. 40 (2007), 8803–8813.
64 E. Dos Santos Cardoso-Bihlo, A. Bihlo, R.O. Popovych
[12] Ibragimov N.H., Invariants of hyperbolic equations: Solution of the Laplace
problem, J. Appl. Mech. Tech. Phys. 45 (2004), 158–166.
[13] Ibragimov N.H., Meleshko S.V., Thailert E., Invariants of linear parabolic differ-
ential equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008), 277–284.
[14] Ibragimov N.H., Sophocleous C., Differential invariants of the one-dimensional
quasi-linear second-order evolution equation, Commun. Nonlinear Sci. Numer.
Simul. 12 (2007), 1133–1145.
[15] Ibragimov N.H., Torrisi M., Valenti A., Differential invariants of nonlinear equa-
tions vtt = f(x, vx)vxx + g(x, vx), Commun. Nonlinear Sci. Numer. Simul. 9
(2004), 69–80.
[16] Ivanova N.M., Popovych R.O., Sophocleous C., Group analysis of variable
coefficient diffusion-convection equations. I. Enhanced group classification,
Lobachevskii J. Math. 31 (2010), 100–122.
[17] Johnpillai I.K., Mahomed F.M., Singular invariant equation for the (1+1)
Fokker–Planck equation, J. Phys. A: Math. Gen. 34 (2001), 11033–11051.
[18] Kingston J.G., Sophocleous C., On form-preserving point transformations of
partial differential equations, J. Phys. A: Math. Gen. 31 (1998), 1597–1619.
[19] Kumei S., Bluman G.W., When nonlinear differential equations are equivalent
to linear differential equations, SIAM J. Appl. Math. 42 (1982), 1157–1173.
[20] 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.
[21] Magadeev B.A., Group classification of nonlinear evolution equations, St. Pe-
tersburg Math. J. 5 (1994), 345–359.
[22] Olver P.J., Generating differential invariants, J. Math. Anal. Appl. 333 (2007),
450–471.
[23] Olver P.J., Differential invariant algebra, Comtemp. Math. 549 (2011), 95–121.
[24] Olver P.J., Modern developments in the theory, applications of moving frames,
London Math. Soc. Impact150 Stories 1 (2015), 14–50.
[25] Olver P.J., Pohjanpelto J., Moving frames for Lie pseudo-groups, Canadian J.
Math. 60 (2008), 1336–1386.
[26] Olver P.J., Pohjanpelto J., Differential invariant algebras of Lie pseudo-groups,
Adv. Math. 222 (2009), 1746–1792.
[27] 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.
[28] Ovsiannikov L.V., Group analysis of differential equations, Acad. Press, New
York, 1982.
[29] Popovych R.O., Classification of admissible transformations of differential equa-
tions, Collection of Works of Institute of Mathematics, Kyiv 3 (2006), no. 2,
239–254.
[30] Popovych R.O., Bihlo A., Symmetry preserving parameterization schemes,
J. Math. Phys. 53 (2012), 073102, 36 pp., arXiv:1010.3010.
Di�erential invariants for a class of di�usion equations 65
[31] 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.
[32] Torrisi M., Tracina R., Second-order differential invariants of a family of diffu-
sion equations, J. Phys. A: Math. Gen. 38 (2005), 7519–7526.
[33] Torrisi M., Tracina R., Valenti A., On the linearization of semilinear wave equa-
tions, Nonlin. Dyn. 36 (2004), 97–106.
[34] Tracina R., Invariants of a family of nonlinear wave equations, Commun. Non-
linear Sci. Numer. Simul. 9 (2004), 127–133.
[35] Tsaousi C., Sophocleous C., Tracina R., Invariants of two- and three-dimen-
sional hyperbolic equations, J. Math. Anal. Appl. 349 (2009), 516–525.
|
| id | oai:trim.imath.kiev.ua:article-367 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:06:11Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/59/8eec58aaf57f706d721ae01332be6259.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3672020-08-13T08:52:53Z Differential invariants for a class of diffusion equations Диференціальні інваріанти класу рівнянь дифузії Dos Santos Cardoso-Bihlo, E. Bihlo, A. Popovych, R. Дос Сантос Кардозо-Біло, E. Біло, А. Попович, Р. We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional.The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration.Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation. Знайдено повну групу еквівалентності класу (1+1)-вимірних еволюційних рівнянь другого порядку, яка виявилася нескінченновимірною.Методологію еквіваріантних рухомих реперів застосовано у регулярному випадку процедури нормалізації до побудови рухомого репера групи, пов'язаної з групою еквівалентності в контексті перетворень еквівалентності між рівняннями класу.За допомогою побудованого рухомого репера описано алгебру диференціальних інваріантів цієї групи через отримання мінімальної генеруючої множини диференціальних інваріантів і повної множини операторів інваріантного диференціювання. Інститут математики НАН України 2019-09-03 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/367 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 1 (2019): Symmetry and Integrability of Equations of Mathematical Physics; 50-65 Сборник Трудов Института математики НАН Украины; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 50-65 Збірник Праць Інституту математики НАН України; Том 16 № 1 (2019): Симетрія та інтегровність рівнянь математичної фізики; 50-65 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/367/362 Авторське право (c) 2019 E. Дос Сантос Кардозо-Біло, А. Біло, Р. Попович |
| spellingShingle | Dos Santos Cardoso-Bihlo, E. Bihlo, A. Popovych, R. Дос Сантос Кардозо-Біло, E. Біло, А. Попович, Р. Differential invariants for a class of diffusion equations |
| title | Differential invariants for a class of diffusion equations |
| title_alt | Диференціальні інваріанти класу рівнянь дифузії |
| title_full | Differential invariants for a class of diffusion equations |
| title_fullStr | Differential invariants for a class of diffusion equations |
| title_full_unstemmed | Differential invariants for a class of diffusion equations |
| title_short | Differential invariants for a class of diffusion equations |
| title_sort | differential invariants for a class of diffusion equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/367 |
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