Acta Polytechnica
DOI:10.14311/AP.2020.60.0428
Acta Polytechnica 60(5):428–434, 2020 © Czech Technical University in Prague, 2020
available online at https://ojs.cvut.cz/ojs/index.php/ap
PROPERTIES OF A DIFFERENTIAL SEQUENCE BASED UPON
THE KUMMER-SCHWARZ EQUATION
Adhir Maharaja, ∗, Kostis Andriopoulosb, Peter Leacha, b
a Durban University of Technology, Steve Biko Campus, Department of Mathematics, Durban, 4000, Republic of
South Africa
b University of KwaZulu-Natal, School of Mathematical Sciences, Private Bag X54001, Durban,4000, Republic of
South Africa
∗ corresponding author: adhirm@dut.ac.za
Abstract. In this paper, we determine a recursion operator for the Kummer-Schwarz equation, which
leads to a sequence with unacceptable singularity properties. A different sequence is devised based upon
the relationship between the Kummer-Schwarz equation and the first-order Riccati equation for which
a particular generator has been found to give interesting and excellent properties. We examine the
elements of this sequence in terms of the usual properties to be investigated – symmetries, singularity
properties, integrability, alternate sequence – and provide an explanation of the curious relationship
between the results of the singularity analysis and a consideration of the solution of each element
obtained by quadratures.
Keywords: Lie symmetries, singularity analysis, differential sequence.
1. Introduction
In a prescient paper of 1977 Olver [1], the idea of a re-
cursion operator for an evolution of partial differential
equations, but with definite indication of extension
to wider classes of partial differential equations was
introduced1. Given an evolution differential equation,
ut = F(t,x,u,ux, . . . ,unx), (1)
R[u] is a recursion operator for (1) if it satisfies the
equation [1] [p 1213, (8)]
[L[F] −Dt,R[u]]LB v ∈ (ut −F) , (2)
where L[F] is the linearised operator
L[F] =
∂F
∂unx
Dnx + . . . +
∂F
∂ux
Dx +
∂F
∂u
, (3)
Dt and Dx denote the total differentiation with respect
to t and x, respectively, and v is a solution of the
linearised equation, ie Lv = 0.
Over the last thirty years, recursion operators have
found a wide application in the study of nonlinear
evolution partial differential equations with partic-
ular reference to their integrability in terms of the
possession of an infinite number of conservation laws.
More recently, Euler et al [4] and Petersson et al [5]
examined the linearisability of nonlinear hierarchies of
evolution equations in (1 + 1) dimensions with a par-
ticular reference to the generalised x-hodograph trans-
formation. As a natural development from this work,
Euler et al [6] initiated a parallel study of recursion
1In this respect Olver made use of a number of results due
to various authors [2, 3]. The beauty of the subsequent work is
in his synthesis.
operators applied to ordinary differential equations.
This was subsequently amplified by Andriopoulos et al
[7] in their detailed study of the Riccati Differential Se-
quence2. A further development was the introduction
of the concept of an alternate sequence [8].
In this paper, we construct a first-order recursion
operator for a sequence of nonlinear ordinary differen-
tial equations based upon the Schwarzian differential
invariant in its expression as the Kummer-Schwarz
equation and investigate the properties of the higher-
order differential equations so generated. As our pri-
mary interest is in differential equations integrable
in the sense of Poincaré (iewith solutions which are
analytically away from isolated polelike singularities)
we are disappointed with the results. However, we are
able to develop a sequence based upon a different gen-
erator and find that the elements of that sequence have
a combination of very interesting properties, which
help to illustrate a facet of this type of analysis not
displayed in the literature. In addition to the exami-
nation of the singularity properties of this sequence,
we also demonstrate its complete symmetry group
and provide a differential generator for an alternate
representation of the elements of the sequence.
2. The first-order recursion
operator for the
Kummer-Schwarz equation
We write the Kummer-Schwarz equation in the form
u′′′
2u′
−
3u′′2
4u′2
= 0, (4)
2The change in terminology is justified in that paper.
428
https://doi.org/10.14311/AP.2020.60.0428
https://ojs.cvut.cz/ojs/index.php/ap
vol. 60 no. 5/2020 Properties of a differential sequence based upon the Kummer-Schwarz. . .
where the prime denotes the differentiation with re-
spect to the independent variable, x, to emphasise its
connection with the Schwarzian derivative, (u′−1/2)′′.
We construct a first-order recursion operator for
(4), R = D + A, where we write D as the operator
of the total differentiation with respect to the single
independent variable, x, and A is a function to be
determined. The linearised operator for (4) is
L[u] =
1
2u′
D3 −
3u′′
2u′2
D2 −
(
u′′′
2u′2
−
3u′′2
2u′3
)
D. (5)
As we are dealing with an ordinary differential equa-
tion, (2) is somewhat simpler. We calculate the Lie
Bracket of L and R and require its action upon v
to give zero when the (linearised) equation for v is
satisfied. This gives a large equation with terms in
v′′, v′ and v. The coefficient of each of these terms is
required separately to be zero. That of v′′ gives
6A′
u′
−
6u′′2
u′3
+
6u′′′
u′2
= 0 (6)
which has the solution
A = −
u′′
u′
(7)
up to an arbitrary constant of integration, which we
consider to be zero. When the right side (7) is substi-
tuted into the remaining terms, the result is identically
zero.
We apply the recursion operator
R = D −
u′′
u′
(8)
to the Kummer-Schwarz equation, (4), to initiate the
construction of the elements of the sequence. The first
equation we obtain is
u(4)
u′
− 5
u′′u(3)
u′2
+ 9
u′′3
2u′3
= 0. (9)
When we seek the leading-order behaviour of (9) by set-
ting u = αχp, where χ = x−x0 and x0 is the location
of the putative singularity, we find that p = −1(bis), 1.
This is not acceptable for a scalar equation!
We recall that the Kummer-Schwarz equation is
closely related to the equation
v′′ + v′2 = 0 (10)
which is the autonomous version of the potential form
of Burgers equation for which the first-order recur-
rence relation is D + v′. This recurrence relation
becomes that of the Kummer-Schwarz equation ob-
tained above, under the transformation connecting
the two equations.
Equation (10) is an elementary Riccati equation in
the dependent variable v′. In terms of y = v′ it is
y′ + y2 = 0. (11)
The connection of this form of the Riccati equation to
the Kummer-Schwarz equation is well known. It is a
simple calculation to show that the recursion operator
for (11) is D + 2y. However, it has been shown [6, 7]
that the operator, D + y, generates a sequence of
differential equations based upon the Riccati equation,
which has very satisfying properties in terms of their
singularity characteristics and algebras. There is a
price to pay. This operator is not a recursion operator
– hence the descriptor ‘generator of sequence’ [7] to
emphasise the distinction – and so one cannot expect
to obtain the properties normally associated with
the recursion operator for an hierarchy of evolution
equations. However, there is much to be said for the
attractive properties which we do find.
In terms of the dependent variable of the Kummer-
Schwarz equation, the generator of the sequence, D+y,
becomes D −u′′/2u′ and it is this operator which we
employ henceforth.
3. The early elements of the new
sequence
We now apply the generator
R = D −
u′′
2u′
(12)
to the Kummer-Schwarz equation, (4), to initiate the
construction of the elements of the sequence and then
apply it to each new element in turn. The process
may be continued for as long as the memory of one’s
symbolic manipulator can contain the results. We
simply list the first few members to give an indication
of the structure of these equations. We include the
first element, the original Kummer-Schwarz equation,
for the sake of completeness in terms of the notation
used. The KS should need no explanation. The initial
‘1’ indicates that the sequence is generated by a first-
order operator. The two subsequent digits indicate
the place of the particular equation as an element of
the sequence. We obtain the following sequence,
KS101: =
u′′′
u′
−
3u′′2
2u′2
= 0
KS102: =
u′′′′
u′
−
9u′′u
′′′
2u′2
+
15u′′3
4u′3
= 0
KS103: =
u′′′′′
u′
−
6u′′u′′′′
u′2
−
9u′′′2
2u′2
+
45u′′2u′′′
2u′3
−
105u′′4
8u′4
= 0
KS104: =
u′′′′′′
u′
−
15u′′u′′′′′
2u′2
−
15u′′′u′′′′
u′2
+
75u′′2u′′′′
2u′3
+
225u′′u′′′2
4u′3
−
525u′′3u′′′
4u′4
+
945u′′5
16u′5
= 0
...
429
A. Maharaj, K. Andriopoulos, P. Leach Acta Polytechnica
KS10n:=
(
D −
u′′
2u′
)n (
u′′′
u′
−
3u′′2
2u′2
)
= 0.
4. Singularity analysis
We perform the singularity analysis in the usual fash-
ion by firstly making the substitution
u = αχp, (13)
where χ = x−x0 and x0 is the location of the putative
singularity, to determine the possible exponents of the
leading-order term and the corresponding coefficients.
The results for the first few elements of the sequence
are given in Table 1.
There are three points to note. Firstly, the elements
of the sequence are of degree zero in u and so the coef-
ficient of the leading-order term is arbitrary. Secondly,
if one removes the fractions by multiplying by the
highest power of u′ in the denominator, the possibility
of p = 0 has multiple occurrences. For the sequence as
we have written it, these possibilities are spurious. As
further analysis is more convenient to perform when
the denominators are removed, we ignore these values.
Thirdly, there is always the possibility that p = 1.
Such a value is without the ambit of the singularity
analysis. It is amusing to note that KS101 is invariant
under the transformation u −→ 1/u. This property
does not persist for higher elements of the sequence.
The second step in the singularity analysis is to de-
termine at what powers of χ the additional constants
of integration occur. The performance of this com-
putation is greatly facilitated by the removal of the
fractions involving the derivative of the dependent
variable. In Table 2, we list the resonances for the
various permissible values of the exponents, p.
5. Complete Symmetry Group
The element KS101 possesses six Lie point symmetries
with the algebra sl(2,R)⊕sl(2,R). All other elements
of the differential sequence possess just four Lie point
symmetries, namely Γ1 = ∂x, Γ2 = x∂x, Γ3 = ∂u and
Γ4 = u∂u with the algebra 2A1 ⊕ 2A1. Neither alge-
bra is sufficient to specify the corresponding equation
completely. It is necessary to have a recourse to non-
local symmetries for the complete specification. The
determination of the nonlocal symmetries is facilitated
by the fact that all elements of the sequence may be
linearised by means of the same transformation, which
linearises the Kummer-Schwarz equation. We recall
that (4) can be written as u′1/2(u′−1/2)′′ = 0. Conse-
quently the linearising transformation is u′ = 1/w2.
The linear equation corresponding to the nth element
of the sequence is3
w(n+1) = 0 (14)
for which a representation of its complete symmetry
group [9, 10] is
Σi = xi∂w, i = 0, n, and Σn+1 = w∂w. (15)
When we reverse the transformation, the symme-
tries in (15) become
Σ̄i =
{∫
xiu′3/2dx
}
∂u, i = 0, n, and Σ̄n+1 = u∂u.
(16)
To these n + 2 symmetries, we add the symmetry
behind the reduction of order to (14), namely ∂u. It is
a simple matter to demonstrate that this indeed is a
representation of the complete symmetry group. The
algebra is A2 ⊕s (n + 1)A1. The (n + 1)-dimensional
abelian subalgebra is comprised of the symmetries Σ̄i,
i = 0, n.
6. The alternate sequence
In the Introduction, we mention the work of Euler
and Leach [8] in which they demonstrated that the
elements of a given sequence of ordinary differential
equations of increasing order could be written in terms
of equations of a lower order with a nonhomogeneous
term of increasing complexity as one rose through
the sequence. This is also the case with the present
sequence. We illustrate the procedure with the first
few elements of the sequence. We recall that the
principle of the construction of the alternate sequence
is based upon the relationship between one member
of the sequence and the next member through the
generator of sequences. The initial step is to solve the
equation (
D −
u′′
2u′
)
q1 = 0. (17)
Thereafter, one proceeds in a sequential manner by
solving (
D −
u′′
2u′
)
qn = qn−1, n = 1, . . . , (18)
where we adopt the convention that q0 = 0. The
solution of (17) is
q1 = C0u′1/2, (19)
where u′−1/2 is an integrating factor for KS1014. The
representation of the second member of the sequence
3In a usual manner of dealing with differential equations
one forgets about the factor 1/w corresponding to the u′1/2.
When it comes to the generation of differential sequences, there
is no place for such slackness. Were one to wish to consider
the differential sequence corresponding to that of the Kummer-
Schwarz sequence, the base equation would have to be w′′/w =
0. As we see below, the removal of denominators entails an
adjustment of the generator.
4This property was firstly noted in the case of recursion
operators. It appears to be somewhat robust!
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vol. 60 no. 5/2020 Properties of a differential sequence based upon the Kummer-Schwarz. . .
Element identifier Possible Exponents
KS101 -1, 1
KS102 -3, -1, 1
KS103 -5, -3, -1, 1
KS104 -7, -5, -3, -1, 1
KS105 -9, -7, -5, -3, -1, 1
KS106 -11, -9, -7, -5, -3, -1, 1
KS107 -13, -11, -9, -7, -5, -3, -1, 1
KS108 -15, -13, -11, -9, -7, -5, -3, -1, 1
...
KS10n -2n+1, -2n+3, -2n+5,... -7, -5, -3, -1, 1
Table 1. Kummer-Schwarz Sequence: possible exponents of the leading-order term for the first eight elements
is then
KS102 :
u′′′
u′
−
3u′′2
2u′2
= C0u′1/2. (20)
It is an easy matter to demonstrate that in general
KS10n :
u′′′
u′
−
3u′′2
2u′2
=
(
n−1∑
i=0
1
i!
Cn−1−ix
i
)
u′1/2.
(21)
7. Conclusion
In the article, we have reported the results of our
analysis of the elements of this sequence. It remains
to interpret the results of the singularity analysis in
the light of our knowledge of a route to obtain the
solution of each element of the sequence in terms of a
quadrature. The nth element of the sequence has n
possible values for the exponent of the leading-order
term, which are acceptable, ie negative integers. As
was noted in Table 1, for the first eight elements
of the sequence, the precise values are −(2i − 1),
i = 1, 2, . . . , n. In Table 2, the resonances for each
of the negative exponents of the leading-order term
are given. Several features are to be noted. Firstly,
there is the possibility of repeated resonances, which
indicates the introduction of a logarithmic term into
the Laurent expansion for the solution. The num-
ber of exponents of the leading-order term for which
this happens increases as one proceeds down the se-
quence. Secondly, the highest exponent leads to a
Right Painlevé Series, albeit with the intrusion of the
unhappy logarithm, and the lowest exponent to what
would be a Left Painlevé Series were it not for a single
positive resonance equal in value to the negative of
the exponent. For the exponents between the highest
and the lowest, one sees a selection of both negative
and positive resonances (apart from the generic −1),
which indicates a full Laurent expansion, again with
the possibility of logarithmic terms occurring. As
should be by now well known, the Right Painlevé
Series is a Laurent expansion in the neighbourhood of
the singularity convergent on a punctured disc and the
Left Painlevé Series has the nature of an asymptotic
expansion in that it exists on the exterior of a disc
centred on the singularity. The full series is defined
over an annulus centred on the similarity and the ex-
istence of multiple instances of these series indicates a
succession of annuli, the bounding circles of which are
defined by successive singularities. Thirdly, the value
of the highest resonance shows some variability. For
patterns of resonances, in which there is no repeated
resonance or the repeated resonance is the highest
resonance, the highest resonance is the negative of
the exponent of the leading-order term. For the other
patterns of the resonances – always starting from the
more positive exponents – this value is exceeded to
the extent that it is necessary to provide a full number
of constants of integration.
Evidently, some explanation of these features is
required! Fortunately, we are in a position to describe
the solution of each element of the sequence due to
the property of the linearisation noted in §5. To
provide the explanation, we make use of the third
element of the sequence, which has the double merits
of featuring in almost every property (the repetition
of logarithms is not one of them) mentioned in the
previous paragraph.
The third element of the sequence, KS103,
u′′′′′
u′
−
6u′′u′′′′
u′2
−
9u′′′2
2u′2
+
45u′′2u′′′
2u′3
−
105u′′4
8u′4
= 0,
(22)
takes the linear form
w(4) = 0 (23)
under the transformation u′ −→ w−2. The solution
of (23) is
w(x) = P3(x), (24)
431
A. Maharaj, K. Andriopoulos, P. Leach Acta Polytechnica
Element identifier Exponents Resonances
KS101 -1 -1, 0, 1
KS102 -1 -1, 0, 1 (bis)
-3 -2, -1, 0, 3
KS103 -1 -1, 0, 1 (bis), 2
-3 -2, -1, 0, 1, 3
-5 -3, -2, -1, 0, 5
KS104 -1 -1, 0, 1 (bis), 2, 3
-3 -2, -1, 0, 1, 2, 3
-5 -3, -2, -1, 0, 1, 5
-7 -4, -3, 2, -1, 0, 7
KS105 -1 -1, 0, 1 (bis), 2, 3, 4
-3 -2, -1, 0, 1, 2, 3 (bis)
-5 -3, -2, -1, 0, 1, 2, 5
-7 -4, -3, -2, -1, 0, 1, 7
-9 -5, -4, -3, -2, -1, 0, 9
KS106 -1 -1, 0, 1 (bis), 2, 3, 4, 5
-3 -2, -1, 0, 1, 2, 3 (bis), 4
-5 -3, -2, -1, 0, 1, 2, 3, 5
-7 -4, -3, -2, -1, 0, 1, 2, 7
-9 -5, -4, -3, -2, -1, 0, 1, 9
-11 -6, -5, -4, -3, -2, -1, 0, 11
KS107 -1 -1, 0, 1 (bis), 2, 3, 4, 5, 6
-3 -2, -1, 0, 1, 2, 3 (bis), 4, 5,
-5 -3, -2, -1, 0, 1, 2, 3, 4, 5
-7 -4, -3, -2, -1, 0, 1, 2, 3, 7
-9 -5, -4, -3, -2, -1, 0, 1, 2, 9
-11 -6, -5, -4, -3, -2, -1, 0, 1, 11
- 13 -7, -6, -5, -4, -3, -2, -1, 0, 13
KS108 -1 -1, 0, 1 (bis), 2, 3, 4, 5, 6, 7
-3 -2, -1, 0, 1, 2, 3 (bis), 4, 5, 6
-5 -3, -2, -1, 0, 1, 2, 3, 4, 5 (bis)
-7 -4, -3, -2, -1, 0, 1, 2, 3, 4, 7
-9 -5, -4, -3, -2, -1, 0, 1, 2, 3, 9
-11 -6, -5, -4, -3, -2, -1, 0, 1, 2, 11
- 13 -7, -6, -5, -4, -3, -2, -1, 0, 1, 13
-15 -8, -7, -6, -5, -4, -3, -2, -1, 0, 15
Table 2. Kummer-Schwarz Sequence: Resonances for the permissible values of the exponent of the leading-order
term
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vol. 60 no. 5/2020 Properties of a differential sequence based upon the Kummer-Schwarz. . .
where P3(x) is a polynomial of degree three in x.
Consequently, the solution of (22) can be written in
terms of the quadrature,
u(x) =
∫
dx
P3(x)2
+ M. (25)
The evaluation of the quadrature in (25) is, in princi-
ple, a simple matter due to the Fundamental Theorem
of Algebra and the use of partial fractions. However,
the interpretation of the results of the singularity anal-
ysis obtained above requires that the quadrature is
to be approached with a certain degree of delicacy
to illustrate the different possibilities. We commence
with the case p = −1 and write the solution in terms
of a polynomial about the singularity at x0 in terms
of the variable χ = x−x0. We have
P3(x) = Kχ(χ−a)(χ− b), (26)
where we have written the factors in the sense that
0 < |a| ≤ |b| to empathise that we are dealing with a
simple pole at x0. When we substitute (26) into (25),
we obtain
u(x) =
∫
dx
(Kχ(χ−a)(χ− b))2
+ M. (27)
We may write the integrand of (27) as
1
K2χ2a2b2
(
1 −
χ
a
)−2 (
1 −
χ
b
)−2
. (28)
After applying the binomial expansion and simpli-
fying, we may write (28) as
1
K2a2b2
[
1
χ2
− 2
(
1
a
+
1
b
)
1
χ
+
(
3
a2
+
3
b2
+
4
ab
)
+2
(
2
a3
+
2
b3
−
3
ab2
−
3
a2b
)
χ...
]
. (29)
When we substitute (29) into (27) and perform the
quadrature, we obtain
u(x) = M −
1
K2a2b2
[
1
χ
+ 2
(
1
a
+
1
b
)
log χ
−
(
3
a2
+
3
b2
−
4
ab
)
χ
−
(
2
a3
+
2
b3
−
3
ab2
−
3
a2b
)
χ2...
]
. (30)
The occurrence of the unfortunate logarithm and the
requisite number of constants of integration (five in
this case, K,a,b,x0 and M) is obvious. The quadra-
ture evaluated above is valid to the next singularity
at χ = a.
For the case p = −5, we may write the integrand
of (27) as
1
k2χ6
(
1 −
a
K
)−2 (
1 −
b
K
)−2
. (31)
After the application of the binomial expansion and
simplification of (31), we obtain
1
k2χ6
[
1 −
2(a + b)
χ
+
3(a2 + b2) + 4ab
χ2
−
4(a3 + b3) + 6a2b2 + 6ab2
χ3
...
]
. (32)
When we substitute (32) into (27) and perform the
quadrature, we have
u(x) =
1
K2
[
−
1
5χ5
+
(a + b)
4χ6
−
3(a2 + b2) + 4ab
7χ7
+
2(a3 + b3) + 3a2b + 3ab2
4χ8
...
]
+ M, (33)
where the requisite number of constants of integration
is also obvious.
Finally, we consider the case p = −3 where the
full series is defined over an annulus centred on the
singularity and we write the factors in the sense that
|a| < |χ| < |b|. We now write integrand of (27) as
1
K2b2χ4
(
1 −
a
χ
)−2 (
1 −
χ
b
)−2
. (34)
By applying the above method to (34) we obtain
u(x) = M +
1
b2K2
[(
−
4
b3
...
)
log χ−
1
χ
(
3
b2
+
8a
b3
...
)
−
1
2χ2
(
−
2
b
−
6a
b2
−
12a2
b3
...
)
−
1
3χ3
(
1 +
4a
b
+
9a2
b2
+
16a3
b3
...
)
−
1
4χ4
(
−2a−
6a2
b
−
12a3
b2
...
)
−
1
5χ5
(
3a2 +
8a3
b
...
)]
. (35)
We observe that (35) contains an unfortunate loga-
rithmic term, which is not indicated in the resonances
for p = −3. This is possible since we have a complete
Laurent series, which must necessarily be convergent
in an annulus centred on the singularity.The Kummer-
Schwarz equation and the members of the sequence
generated from it by the generator (12) can be easily
integrated. What we have done here is to explore its
properties from a wider perspective so that a greater
appreciation of its properties can be gained. The ex-
ploration of different aspects of such sequences adds
to our understanding of them.
Acknowledgements
PGLL would like to thank the National Research foun-
dation of the Republic of South Africa, University of
KwaZulu-Natal and Durban University of Technology for
their continued support. AM would like to thank the Dur-
ban University of Technology for their continued support.
433
A. Maharaj, K. Andriopoulos, P. Leach Acta Polytechnica
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http://dx.doi.org/10.1063/1.523393
http://dx.doi.org/10.1063/1.1665772
http://dx.doi.org/10.1070/RM1975v030n05ANEH001522
http://dx.doi.org/10.1111/1467-9590.t01-1-00236
http://dx.doi.org/10.111/j.0022-2526.2004.02511.x
http://dx.doi.org/10.2991/jnmp.2007.14.2.11
http://dx.doi.org/10.1007/s11232-009-0038-y
http://dx.doi.org/10.1006/jmaa.2001.7570
Acta Polytechnica 60(5):428–434, 2020
1 Introduction
2 The first-order recursion operator for the Kummer-Schwarz equation
3 The early elements of the new sequence
4 Singularity analysis
5 Complete Symmetry Group
6 The alternate sequence
7 Conclusion
Acknowledgements
References