Symmetry, equivalence and integrable classes of Abel equations
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary differential equations. The problem of linearizability of the e...
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| author | Boyko, V. Бойко, В. |
| author_facet | Boyko, V. Бойко, В. |
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"author": "V. Boyko",
"institution": "Інститут математики"
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| description | We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary differential equations. The problem of linearizability of the equations under consideration is considered. |
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Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 39–48
УДК 517.9
Symmetry, equivalence and integrable
classes of Abel equations
V.M. BOYKO
Institute of Mathematics of NAS of Ukraine, Kyiv
E-mail: boyko@imath.kiev.ua
Запропоновано пiдхiд для опису iнтегровних випадкiв рiвняння Абеля,
що базується на пiдвищеннi порядку та використаннi перетворень еквi-
валентностi для вiдповiдного звичайного диференцiального рiвняння
другого порядку. Розглянуто проблему лiнеаризацiї розглядуваного
класу рiвнянь.
We suggest an approach for description of integrable cases of the Abel
equations. It is based on increasing of the order of equations up to the
second one and using equivalence transformations for the corresponding
second-order ordinary differential equations. The problem of linearizability
of the equations under consideration is considered.
Introduction. A diversity of methods were developed to date for finding
solutions of nonlinear ordinary differential equations (ODE). Everybody
who encounters integration of a particular ODE uses, as a rule, the
accumulated databases (or reference books) of the classes of ODE and
methods for integration of them (e.g. [22, 29]). But if an ODE does not
belong to any of the described classes then it does not mean that there
is no approaches for finding solutions of this ODE in a closed form.
The symmetry approach is one of the most algorithmic approaches for
integration and lowering of the order of ODE that admit a certain nontri-
vial symmetry (see e.g. books [24,28] and review papers [19,35]). In the
framework of the symmetry approach (and its modifications) it is possi-
ble to obtain many of the known classes of integrable ODE. However,
the needs of the applications stimulate new research into development of
new methods for construction of ODE solutions in the closed form. The
works [2–9,12–19,25–27,29–35] may give an idea of current developments
and directions of research in the field of symmetry (algebraic) methods
for investigation of ODE.
40 V.M. Boyko
The problem of finding Lie symmetries for the first-order ODE is
equivalent to finding solutions for these equations, and for this reason
the direct application of the Lie method is complicated in the general
case. On of the well-known approaches in the cases when for a given ODE
it is not feasible (or not effective) to apply the Lie method directly, is
increasing of the order of the ODE under consideration (in particular, to
obtain a second-order ODE related to the respective ODE by a change
of variables). For examples of utilisation of such approach we can refer
to papers [2–6, 14, 25–27]. In such cases, if the “induced” equation of a
higher order admits a non-trivial Lie symmetry (that generated a non-
local symmetry for the initial equation), we can speak of so-called hidden
symmetries for an initial equation (for more details see [2–4]).
Main results. In this paper we study Abel equations having the
form
ṗ(f5(y)p+ f0(y)) = p3f4(y) + p2f3(y) + pf2(y) + f1(y), (1)
where p = p(y), ṗ = dp
dy , fi, i = 0, . . . , 5, are arbitrary smooth functions
(with f1, f2, f3, f4, f5 not identically vanishing simultaneously). In view
of existence of the gauge transformation of multiplication by an arbitrary
function of y, any equation (1) can be reduced to one of the following
canonical forms (respectively, Abel equations of the first and the second
kind, see e.g. [1, 22,29]):
ṗ = p3f4(y) + p2f3(y) + pf2(y) + f1(y), (2)
ṗ(p+ f0(y)) = p3f4(y) + p2f3(y) + pf2(y) + f1(y). (3)
Equations (2), (3) along with the Riccati equation are among the “simp-
lest” nonlinear first-order ODE that have extensive applications. At the
same time the problem of description of integrable classes of these equati-
ons stays within the focus of current research, and was previously consi-
dered in many papers (see e.g. [5, 6, 10–13,27,29,30,32–34,36]).
Note that the Abel equations of the first and the second kind (2), (3)
are related with each other by a local change of variables (namely, the
equation (3) can be reduced to the form (2) by means of the change of
variables p = 1/v(y) − f0). Besides, the well-known Riccati equation is
a partial case of equation (2).
Further we will consider the following second-order ODE
ÿ = ẏ4f4(y) + ẏ3f3(y) + ẏ2f2(y) + ẏf1(y), (4)
Symmetry, equivalence and integrable classes of Abel equations 41
ÿ(ẏ + f0(y)) = ẏ4f4(y) + ẏ3f3(y) + ẏ2f2(y) + ẏf1(y), (5)
where y = y(x), ẏ = dy
dx , ÿ = d2y
dx2 , related to the Abel equations (2)
and (3).
The substitution ẏ = p(y) reduces equations (4) and (5) respectively
to the Abel equations (2) and (3) (reduction of the order for equati-
ons (4) and (5)). Such reduction is induced by the Lie operator X1 = ∂x
(that corresponds to invariance of equations (4) and (5) with respect to
translations by the variable x). This is exactly the fact that explains why
we consider equations (4) and (5).
In the case when (4) or (5) are invariant with respect to another
operator (that is when (4) or (5) admit two-dimensional Lie algebras),
then equations (4) and (5) are integrable in the framework of the Lie
approach. And in this way we can obtain exact solutions of the equati-
ons (2) and (3) respectively.
Further we will consider only the equation (5) (since equations (2)–
(5) are interconnected – see Remark 3). Let (5) admit a two-dimensional
Lie algebra
L = 〈X1, X2〉, X1 = ∂x, X2 = ξ(x, y)∂x + η(x, y)∂y. (6)
We will consider a problem of description of inequivalent equations
(5) that are invariant with respect to two-dimensional Lie algebras of the
form (6) (non-equivalent realizations of the operator X2 in the algebra
(6) will determine canonical representatives for equation (5)).
It is well-known that any two-dimensional Lie algebra in the general
case, by means of choosing the basis operators X1 and X2 in an approp-
riate manner, may be reduced to four nonequivalent cases (see e.g. [19,
24, 28]). In the framework of our problem additional cases arise as we
have fixed the form of the operator X1.
So, it is quite straightforward to show that equation (5) may admit
a two-dimensional Lie algebra (6) only of one of the following types:
1. [X1, X2] = 0, rankL = 1;
2. [X1, X2] = 0, rankL = 2;
3. [X1, X2] = X1, rankL = 1;
4. [X1, X2] = X1, rankL = 2;
5. [X1, X2] = X2, rankL = 1;
6. [X1, X2] = X2, rankL = 2. (7)
42 V.M. Boyko
Further, utilising classification of two-dimensional algebras (7), we
obtain that equation (5) may admit only the following realizations of
two-dimensional Lie algebras (6):
1. X1 = ∂x, X2 = ξ(y)∂x, ξ(y) 6≡ const;
2. X1 = ∂x, X2 = ξ(y)∂x + η(y)∂y,
ξ(y) 6≡ const or ξ(y) ≡ 0, η(y) 6= 0;
3. X1 = ∂x, X2 = (x+ ξ(y))∂x, ξ(y) 6≡ const or ξ(y) ≡ 0;
4. X1 = ∂x, X2 = (x+ ξ(y))∂x + η(y)∂y,
ξ(y) 6≡ const or ξ(y) ≡ 0, η(y) 6= 0;
5. X1 = ∂x, X2 = exξ(y)∂x, ξ(y) 6≡ 0;
6. X1 = ∂x, X2 = ex(ξ(y)∂x + η(y)∂y), η(y) 6= 0. (8)
It is clear that using these realizations we can describe equations
of the form (5) that are invariant with respect to two-dimensional Lie
algebras (similarly as we have discussed in [32]). However, this way is
too cumbersome, and thus obtained types of equations (5) will be quite
complicated (functions fi, i = 0, . . . , 4 in (5) will be expressed through
coefficients of the operator X2 from realizations (8)).
It is straightforward to show that the most general transformations
that preserve the form of the operator X1 we look as follows:
t = x+ ω(y), u = g(y), (9)
where ω(y), g(y) are arbitrary smooth functions, g(y) 6≡ const.
After substitution (9) equation (5) takes the form
ü((1− ω′f0)u̇+ f0g
′)g′2 =
=
(
f4 − ω′′(1− ω′f0)− ω′f3 + ω′2f2 − ω′3f1
)
u̇4
+
(
g′f3 − ω′′g′f0 + g′′(1− ω′f0)− 2ω′g′f2 + 3ω′2g′f1
)
u̇3
+
(
g′2f2 + g′′g′2f0 − 3ω′g′2f1
)
u̇2 + f1g
′3u̇, (10)
where ω′ = dω
dy , ω
′′ = d2ω
dy2 , g′ = dg
dy , g
′′ = d2g
dy2 (in addition in (10)
all functions of the variable y should be expressed as functions of the
variable u).
With (1 − ω′f0) 6≡ 0 equation (10) belongs again to the class of
equations (5).
Symmetry, equivalence and integrable classes of Abel equations 43
Remark 1. With (1−ω′f0) ≡ 0 after the substitution (9), equation (5)
is transformed to the equation (4), that is reduced to the Abel equation
of the first kind (2).
Remark 2. It is possible to regard that (1 − ω′f0) 6≡ 0 for the equa-
tion (5) as a result of the substitution (9) (we attain that by combination
of transformations (9)).
Thus (9) are equivalence transformations for (5), and, besides, these
transformations preserve the form of the operator X1 = ∂x in the algeb-
ra (6).
Remark 3. So, the transformations (9) are equivalence transformati-
ons for the class of equations (4)–(5). Moreover, if we prolongate these
transformations for u̇ = p then they form an equivalence transformati-
on group for (1) and include as a subgroup in the complete equivalence
group of class (1), which are formed by the transformations
ỹ = F (y), p̃ =
P1(t)p+Q1(t)
P2(t)p+Q2(t)
,
where F , P1, P2, Q1, Q2 are arbitrary analytic functions, and P1Q2 −
P2Q1 6= 0.
Thus, by means of transformations (9), realizations (8) of the algebra
(6) may be reduced to the simplest canonical form. The transformations
(9) in that process will not take us out of the class of equations (5).
By means of transformations (9) the realizations (8) of two-dimen-
sional Lie algebras (6) admitted for equation (5) are reduced to the
following canonical realizations:
1. X1 = ∂x, X2 = y∂x;
2. X1 = ∂x, X2 = ∂y;
3. X1 = ∂x, X2 = x∂x;
4. X1 = ∂x, X2 = x∂x + y∂y;
5. X1 = ∂x, X2 = ex∂x;
6. X1 = ∂x, X2 = ex(∂x + ∂y). (11)
In accordance to (11) we obtain the following integrable cases for
equation (5) that are non-equivalent with respect to (9):
1. ÿ = α(y)ẏ3;
44 V.M. Boyko
2. ÿ(ẏ + e) = dẏ4 + cẏ3 + bẏ2 + aẏ;
3. ÿ = α(y)ẏ2;
4. yÿ(ẏ + e) = dẏ4 + cẏ3 + bẏ2 + aẏ;
5. ÿ(ẏ + β(y)) = α(y)ẏ3 + (1− α(y)β(y))ẏ2 − β(y)ẏ;
6. a) f0 = 0 :
ÿ = dey ẏ3 + (−3dey + c)ẏ2 + (−dey − (2c+ 1) + be−y)ẏ
+ (−dey + (c+ 1)− be−y + ae−2y);
b) f0 6= 0 :
ÿ(ẏ + α(y)) = −ẏ3 + (1− α(y))ẏ2 + α(y)ẏ, (12)
where α(y), β(y) are arbitrary smooth functions, a, b, c, d, e are cons-
tants.
The case 6a in (12) may be simplified by means of the substitution
t = x, u = ey (see (9) and (10)).
Equations (12) determine non-equivalent cases of the form (5) that
admit two-dimensional algebras (11) up to equivalence transformati-
ons (9).
Thus, summarising the above, we come to the following scheme for
integration of the Abel equation (3):
• we increase the order of equation (3), considering a second-order
equation (5);
• if a corresponding equation (5) admits a two-dimensional Lie al-
gebra, then we reduce this algebra to one of the canonical forms
(11), and thus the equation is reduced to the respective canonical
forms (12);
• we integrate the canonical form (12);
• making reverse changes of variables we obtain the solution of the
Abel equation (3).
Case of Lie’s linearization test. According to results of S. Lie [23]
(see also [19–21]) second-order ODEs
ÿ = f(x, y, ẏ) (13)
can be reduced to the form
ü = 0, (14)
Symmetry, equivalence and integrable classes of Abel equations 45
by point change of variables
t = ϕ(x, y), u = ψ(x, y), u = u(t) (15)
if equations (13) is at most cubic in the first derivative, i.e. only if equati-
ons (13) has the form
ÿ + F3(x, y)ẏ3 + F2(x, y)ẏ2 + F1(x, y)ẏ + F (x, y) = 0, (16)
where
F3(x, y) =
ϕyψyy − ψyϕyy
ϕxψy − ϕyψx
,
F2(x, y) =
ϕxψyy − ψxϕyy + 2(ϕyψxy − ψyϕxy)
ϕxψy − ϕyψx
,
F1(x, y) =
ϕyψxx − ψyϕxx + 2(ϕxψxy − ψxϕxy)
ϕxψy − ϕyψx
,
F (x, y) =
ϕxψxx − ψxϕxx
ϕxψy − ϕyψx
. (17)
For given function F3(x, y), F3(x, y), F1(x, y) and F (x, y) linearization is
possible iff the over-determined system (17) is integrable. S. Lie proved
that system (17) is integrable iff the following auxiliary system for w
and z
∂w
∂x
= zw − FF3 −
1
3
∂F1
∂y
+
2
3
∂F2
∂x
,
∂w
∂y
= −w2 + F2w + F3z +
∂F3
∂x
− F1F3,
∂z
∂x
= z2 − Fw − F1z +
∂F
∂y
+ FF2,
∂z
∂y
= −zw + FF3 −
1
3
∂F2
∂x
+
2
3
∂F1
∂y
(18)
is compatible. The compatibility conditions for this system have the form
3(F3)xx − 2(F2)xy + (F1)yy =
= (3F1F3 − F 2
2 )x − 3(FF3)y − 3F3Fy + F2(F1)y,
3Fyy − 2(F1)xy + (F2)xx =
46 V.M. Boyko
= 3(FF3)x − 3(FF2 − F 2
1 )y + 3F (F3)x − F1(F2)x (19)
(subscripts x and y denote differentiations with respect to x and y,
respectively).
So, following [20, 21] a necessary and sufficient condition of lineari-
ziation for equations of form (16) is that functions F3(x, y), F2(x, y),
F1(x, y) and F (x, y) satisfy the conditions (19).
In case f0(y) ≡ 0 equations (5) is partial case of (16), i.e. have the
following form
ÿ = ẏ3f4(y) + ẏ2f3(y) + ẏf2(y) + f1(y). (20)
This equations can be linearizable if f4(y), f3(y), f1(y) and f1(y) satisfy
the conditions (following (19))
(f2)yy = 3(f1f4)y + 3f4(f1)y − f3(f2)y,
3(f1)yy = 3(f1f3 − f2
2 )y. (21)
And point transformations (15) which linearizing (20) can be found from
system (17).
Let us note that a second-order ODE is linearizable iff it admits an
eight-dimensional Lie algebra. So, any linearizable equation (20) belongs,
up to equivalence transformations (9), belong to the set of equations (12).
Conclusion. It is obvious from the above that there is an alternative
way for generation of new integrable cases of the Abel equation based on
utilisation of the relation between the Abel equations of the first and the
second kind, and relation between the equations (4) and (5) by means
of the transformations (9). Thus, starting from some integrable Abel
equation (that is of such equation for which the solution is known) it is
possible to obtain new integrable cases of the Abel equations (solutions
of these equations will be related through transformations (9)). It would
be possible to use for this purpose even the well-known Riccati equation
that is a partial case of the equation (2) (for generation of integrable
Riccati equations an approach that is proposed in [30] may be used).
We hope that new results for classification of integrable classes of
ODE may be obtained also using our classification of inequivalent reali-
zations of real low-dimensional Lie algebras [31].
The author is grateful to Roman Popovych and Irina Yehorchenko for
useful discussions and interesting comments.
Symmetry, equivalence and integrable classes of Abel equations 47
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| resource_txt_mv | trimimathkievua/07/098ffe000af2e094cff4eacf2e6e9607.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3802020-08-13T10:42:30Z Symmetry, equivalence and integrable classes of Abel equations Класи симетрії, еквівалентності та інтегровності рівнянь Абеля Boyko, V. Бойко, В. We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary differential equations. The problem of linearizability of the equations under consideration is considered. Запропоновано підхід для опису інтегровних випадків рівняння Абеля, що базується на підвищенні порядку та використанні перетворень еквівалентності для відповідного звичайного диференціального рівняння другого порядку. Розглянуто проблему лінеаризації розглядуваного класу рівнянь. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/380 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); 39-48 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); 39-48 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); 39-48 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/380/379 Авторське право (c) 2006 В. Бойко http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Boyko, V. Бойко, В. Symmetry, equivalence and integrable classes of Abel equations |
| title | Symmetry, equivalence and integrable classes of Abel equations |
| title_alt | Класи симетрії, еквівалентності та інтегровності рівнянь Абеля |
| title_full | Symmetry, equivalence and integrable classes of Abel equations |
| title_fullStr | Symmetry, equivalence and integrable classes of Abel equations |
| title_full_unstemmed | Symmetry, equivalence and integrable classes of Abel equations |
| title_short | Symmetry, equivalence and integrable classes of Abel equations |
| title_sort | symmetry, equivalence and integrable classes of abel equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/380 |
| work_keys_str_mv | AT boykov symmetryequivalenceandintegrableclassesofabelequations AT bojkov symmetryequivalenceandintegrableclassesofabelequations AT boykov klasisimetrííekvívalentnostítaíntegrovnostírívnânʹabelâ AT bojkov klasisimetrííekvívalentnostítaíntegrovnostírívnânʹabelâ |