Symmetry, Equivalence and Integrable Classes of Abel Equations

Abstract

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 discussed.

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arXiv:nlin/0404020v2 [nlin.SI] 4 Jul 2005
Symmetry, Equivalence and Integrable Classes
of Abel Equations
Vyacheslav M. BOYKO
Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Str., Kyiv 4, 01601 Ukraine
E-mail: boyko@imath.kiev.ua
Abstract
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
discussed.
1 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. [23,31]). 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 nontrivial symmetry (see e.g.
Lie’s book [25], the books [19, 29, 30] and review papers [19, 37]).
of the symmetry approach (and its modifications) it is possible 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 papers [2–9,12–19,26–28,30–37] may give an idea of current developments and
directions of research in the field of symmetry (algebraic) methods for investigation of
ODE.
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 [2–6,14,26–28]. 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]).
In the framework
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2 Main results
In this paper we study Abel equations having the form
˙ p(f5(y)p + f0(y)) = p3f4(y) + p2f3(y) + pf2(y) + f1(y),(1)
where p = p(y), ˙ p =
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. [1,23,31]):
dp
dy, fi, i = 0,...,5, are arbitrary smooth functions (with f1, f2,
˙ p = p3f4(y) + p2f3(y) + pf2(y) + f1(y),
˙ p(p + f0(y)) = p3f4(y) + p2f3(y) + pf2(y) + f1(y).
(2)
(3)
Equations (2), (3) along with the Riccati equation are among the “simplest” nonlinear
first-order ODE that have extensive applications. At the same time the problem of de-
scription of integrable classes of these equations stays within the focus of current research,
and was previously considered in many papers (see e.g. [5,6,10–13,28,31,32,34–36]).
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
¨ y = ˙ y4f4(y) + ˙ y3f3(y) + ˙ y2f2(y) + ˙ yf1(y),
¨ y(˙ y + f0(y)) = ˙ y4f4(y) + ˙ y3f3(y) + ˙ y2f2(y) + ˙ yf1(y),
(4)
(5)
where y = y(x), ˙ y =dy
The substitution ˙ y = p(y) reduces equations (4) and (5) respectively to the Abel
equations (2) and (3) (reduction of the order for equations (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 equations (2) and (3) respectively.
Further we will consider only the equation (5) (since equations (2)–(5) are intercon-
nected – see Remark 3). Let (5) admit a two-dimensional Lie algebra
dx, ¨ y =d2y
dx2, related to the Abel equations (2) and (3).
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 invari-
ant with respect to two-dimensional Lie algebras of the form (6) (non-equivalent realiza-
tions of the operator X2in the algebra (6) will determine canonical representatives for
equation (5)).
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It is well-known that any two-dimensional Lie algebra in the general case, by means
of choosing the basis operators X1and X2in an appropriate manner, may be reduced
to four nonequivalent cases (see e.g. [19,20,25,29]). 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,
2. [X1,X2] = 0,
3. [X1,X2] = X1,
4. [X1,X2] = X1,
5. [X1,X2] = X2,
6. [X1,X2] = X2,
rankL = 1;
rankL = 2;
rankL = 1;
rankL = 2;
rankL = 1;
rankL = 2. (7)
Further, utilising classification of two-dimensional algebras (7), we obtain that equa-
tion (5) may admit only the following realizations of two-dimensional Lie algebras (6):
1. X1= ∂x,
2. X1= ∂x,
3. X1= ∂x,
4. X1= ∂x,
5. X1= ∂x,
6. X1= ∂x,
X2= ξ(y)∂x,
X2= ξ(y)∂x+ η(y)∂y,
X2= (x + ξ(y))∂x,
X2= (x + ξ(y))∂x+ η(y)∂y,
X2= exξ(y)∂x,
X2= ex(ξ(y)∂x+ η(y)∂y),
ξ(y) ?≡ const;
ξ(y) ?≡ const or ξ(y) ≡ 0,
ξ(y) ?≡ const or ξ(y) ≡ 0;
ξ(y) ?≡ const or ξ(y) ≡ 0,
ξ(y) ?≡ 0;
η(y) ?= 0.
η(y) ?= 0;
η(y) ?= 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 [34]). 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 X2from realizations (8)).
It is straightforward to show that the most general transformations that preserve the
form of the operator X1we look as follows:
t = x + ω(y),u = g(y), (9)
where ω(y), g(y) are arbitrary smooth functions, g(y) ?≡ const.
After substitution (9) equation (5) takes the form
¨ u((1 − ω′f0)˙ u + f0g′)g′2=?f4− ω′′(1 − ω′f0) − ω′f3+ ω′2f2− ω′3f1
+?g′f3− ω′′g′f0+ g′′(1 − ω′f0) − 2ω′g′f2+ 3ω′2g′f1
+?g′2f2+ g′′g′2f0− 3ω′g′2f1
?˙ u4
?˙ u3
?˙ u2+ f1g′3˙ u, (10)
where ω′=dω
y should be expressed as functions of the variable u).
With (1 − ω′f0) ?≡ 0 equation (10) belongs again to the class of equations (5).
dy, ω′′=d2ω
dy2, g′=dg
dy, g′′=d2g
dy2(in addition in (10) all functions of the variable
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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) ?≡ 0 for the equation (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= ∂xin the algebra (6).
Remark 3. So, the transformations (9) are equivalence transformations for the class
of equations (4)–(5). Moreover, if we prolongate these transformations for ˙ u = p then
they form an equivalence transformation group for (1) and include as a subgroup in the
complete equivalence group of class (1), which are formed by the transformations
˜ y = F(y),˜ p =P1(t)p + Q1(t)
P2(t)p + Q2(t),
where F, P1, P2, Q1, Q2are arbitrary analytic functions, and P1Q2− P2Q1?= 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-dimensional Lie algebras
(6) admitted for equation (5) are reduced to the following canonical realizations:
1. X1= ∂x,
2. X1= ∂x,
3. X1= ∂x,
4. X1= ∂x,
5. X1= ∂x,
6. X1= ∂x,
X2= y∂x;
X2= ∂y;
X2= x∂x;
X2= x∂x+ y∂y;
X2= ex∂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. ¨ y = α(y)˙ y3;
2. ¨ y(˙ y + e) = d˙ y4+ c˙ y3+ b˙ y2+ a˙ y;
3. ¨ y = α(y)˙ y2;
4. y¨ y(˙ y + e) = d˙ y4+ c˙ y3+ b˙ y2+ a˙ y;
5. ¨ y(˙ y + β(y)) = α(y)˙ y3+ (1 − α(y)β(y))˙ y2− β(y)˙ y;
6. a) f0= 0 :
¨ y = dey˙ y3+ (−3dey+ c)˙ y2+ (−dey− (2c + 1) + be−y)˙ y
+ (−dey+ (c + 1) − be−y+ ae−2y);
b) f0?= 0 :
¨ y(˙ y + α(y)) = −˙ y3+ (1 − α(y))˙ y2+ α(y)˙ y,(12)
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where α(y), β(y) are arbitrary smooth functions, a, b, c, d, e are constants.
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-dimen-
sional algebras (11) up to equivalence transformations (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 algebra, 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).
3 Case of Lie’s linearization test
According to results of S. Lie [24] (see also [19–22]) second-order ODEs
¨ y = f(x,y, ˙ y)(13)
can be reduced to the form
¨ u = 0,(14)
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 equations (13) has the
form
¨ y + F3(x,y)˙ y3+ F2(x,y)˙ y2+ F1(x,y)˙ y + 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)
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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
∂w
∂y= −w2+ F2w + F3z +∂F3
∂z
∂x= z2− Fw − F1z +∂F
∂z
∂y= −zw + FF3−1
3
∂F1
∂y
+2
3
∂F2
∂x,
∂x− F1F3,
∂y+ FF2,
∂F2
∂x+2
33
∂F1
∂y
(18)
is compatible. The compatibility conditions for this system have the form
3(F3)xx− 2(F2)xy+ (F1)yy= (3F1F3− F2
3Fyy− 2(F1)xy+ (F2)xx= 3(FF3)x− 3(FF2− F2
2)x− 3(FF3)y− 3F3Fy+ F2(F1)y,
1)y+ 3F(F3)x− F1(F2)x
(19)
(subscripts x and y denote differentiations with respect to x and y, respectively).
So, following [21,22] a necessary and sufficient condition of lineariziation 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
¨ y = ˙ y3f4(y) + ˙ y2f3(y) + ˙ yf2(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).
4Conclusion
It is obvious from the above that there is an alternative way for generation of new inte-
grable 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 in-
tegrable 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
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Riccati equation that is a partial case of the equation (2) (for generation of integrable
Riccati equations an approach that is proposed in [32] may be used).
We hope that new results for classification of integrable classes of ODE may be ob-
tained also using our classification of inequivalent realizations of real low-dimensional Lie
algebras [33].
Acknowledgements
The author are grateful to Roman Popovych and Irina Yehorchenko for useful discussions
and interesting comments. the research was partially supported by National Academy of
Science of Ukraine in the form of the grant for young scientists.
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