Some General Solutions to the Painlevé – PII Equation (New Version 11/07/2014)

Some General Solutions to the

Painlevé – PII Equation

Solomon M. Antoniou

SKEMSYS Scientific Knowledge Engineering

and Management Systems

Corinthos 20100, Greece [email protected]

Revised Version: 11/07/2014

Abstract

We initiate a solution procedure to the Painlevé PII equation. The method used is

the Riccati equation method with variable expansion coefficients. The solutions

are expressed in terms of the Airy functions.

Keywords: Painlevé equations, Painlevé PII equation,the extended Riccati

equation method, nonlinear equations, exact solutions, Airy functions.

2

1. Introduction.

The Painlevé equations (numbered as PI-PVI) is a special class of second order

nonlinear ODEs which have no movable critical points (branch points or essential

singularities). These equations were discovered under a number of assumptions at

the end of nineteenth century/beginning of twentieth century by Painlevé and

Gambier. They also appear in many physical applications. Some reviews and

further developments the reader can consult, are the classical book by E. L. Ince

(Ince [10], Chapter XIV) and the articles by A. S. Fokas and M. J. Ablowitz

(Fokas and Ablowitz [9]) and P. A. Clarkson (Clarkson [7]). Closely related to

Painlevé equations is the so-called ARS conjecture (after Ablowitz, Ramani and

Segur [1], [2] and [11]) which is an integrability test for ordinary differential

equations. The ARS conjecture was extended to PDEs by J. Weiss, M. Tabor and

G. Carnevale (Weiss, Tabor and Carnevale [13]) who introduced the so-called

singular manifold method (for a review and further examples see Ramani,

Grammaticos and Bountis [12]). This method (also named as Painlevé truncation

method) serves also as a solution method (Weiss [14], [15] and [16]) in the sense

that it can determine the Lax pairs and the Bäcklund transformations.

No explicit solutions of the Painlevé equations have been found so far. In some

cases only rational solutions are available (Airlaut [4] and Wenjum and Yezhou

[17]). Some relations have also been established between solutions (Fokas and

Ablowitz [9], Clarkson [7]).

In this paper we introduce a solution method and find some closed-form solutions

to the Painlevé PII equation using the extended Riccati equation method with

variable expansion coefficients. The solutions are expressed in terms of the Airy

functions. The paper is organized as follows: In Section 2 we describe the

extended Riccati equation method with variable expansion coefficients. This

method was introduced in Antoniou ([5] and [6]) and applied successfully in two

cases of nonlinear PDEs, the Burgers and KdV equations. In Section 3 we solve

3

the Painleve PII equation wzw2dz

wd 32

2⋅+= . We find two families of solutions,

expressed in terms of the Airy functions. In Section 4 we consider the

−′ )G/G( expansion method with variable expansion coefficients. We obtain two

third order differential equations which, when solved, can in principle determine

the unknown function G.

2. The Method.

We suppose that a nonlinear ordinary differential equation

0),u,u,u,x(F xxx =L (2.1)

with unknown function )x(u admits a solution expressed in the form

∑∑==

+=n

1kkk

n

0k

kk

Y

bYa)x(u (2.2)

where all the expansion coefficients depend on the variable x,

)x(aa kk ≡ , )x(bb kk ≡ for every n,,2,1,0k L=

The function )x(YY ≡ satisfies Riccati’s equation

2YBA)x(Y ⋅+=′ (2.3)

where the coefficients A and B depend on the variable x as well.

In solving the nonlinear ODE (2.1), we consider the expansion (2.2) and then we

balance the nonlinear term with the highest derivative term of the function )x(u

which determines n (the number of the expansion terms). Equating the

coefficients of the different powers of the function )x(Y to zero, we can

determine the various expansion coefficients )x(ak , )x(bk and the functions

)x(A , )x(B . We finally solve Riccati's equation and then find the solutions of the

equation considered.

4

3. The Painleve PII equation and its solutions.

We consider the Painleve PII equation

wzw2dz

wd 32

2⋅+= (3.1)

where )z(ww ≡ is the unknown function and z the independent variable,

considered complex in general.

We shall use the extended Riccati equation method in solving equation (3.1). In

this case we consider the expansion

∑∑==

+=n

1kkk

n

0k

kk

Y

bYa)z(w (3.2)

and balance the second order derivative term with the second order nonlinear term

of (3.1). We then find 1n = and thus

Y

bYaa)z(w 1

10 ++= (3.3)

where all the coefficients 0a , 1a and 1b depend on z, and Y satisfies Riccati’s

equation

)z(Y)z(B)z(A)z(Y 2+=′ (3.4)

The prime will always denote derivative with respect to the variable z. From

equation (3.3) we obtain, taking into account 2BYAY +=′

2

2112

110Y

)BYA(b

Y

b)BYA(aYaa)z(w

+−′

+++′+′=′ (3.5)

)BYA(YB2YBA(a)BYA(a2Yaa)z(w 221

2110 ++′+′++′+′′+′′=′′

2

21

21

211

Y

)BYA(BYb2)YBA(b)BYA(b2

Y

b ++′+′++′−

′′+

3

221

Y

)BYA(b2 ++ (3.6)

Therefore equation (3.1), under the substitutions (3.3) and (3.6), becomes

5

)BYA(YB2YBA(a)BYA(a2Yaa 221

2110 ++′+′++′+′′+′′

2

21

21

211

Y

)BYA(BYb2)YBA(b)BYA(b2

Y

b ++′+′++′−

′′+

++⋅+

++=++Y

bYaaz

Y

bYaa2

Y

)BYA(b2 110

31

103

221 (3.7)

Upon expanding and equating the coefficients of Y to zero, we obtain from the

above equation a system of seven ordinary differential equations from which we

can determine the various expansion coefficients. We obtain

coefficient of 3Y :

0a2Ba2 31

21 =− (3.8)

coefficient of 2Y :

0Ba2Baaa6 11210 =′+′+− (3.9)

coefficient of Y:

0aaa6ba6azBAa2 11201

2111 =′′+−−− (3.10)

coefficient of 0Y :

0Aa2BbAaBb2baa12a2aza 11111103000 =′+′−′+′−−−−′′ (3.11)

coefficient of 1Y − :

0ABb2ba6ba6bzb 12111

2011 =+−−−′′ (3.12)


0Abba6Ab2 12101 =′−−′− (3.13)


0b2Ab2 31

21 =− (3.14)

We are to solve the system of equations (3.8)-(3.14), supplemented by Riccati's

equation 2BYAY +=′ .

6

From equations (3.8) and (3.14), ignoring the trivial solutions, we obtain

Ba1 ±= and Ab1 ±= (3.15)

respectively. We examine four cases separately. Two of them, leading to

inconsistent results are examined in Appendix B.

3.I. Case I. The First Solution.

We first consider the case

Ba1 = and Ab1 = (3.16)

We then obtain from (3.9) and (3.13)

BB

21

a0′

= and AA

21

a0′

−= (3.17)

respectively. Equating the two different expressions of 0a , we obtain 0BB

AA =

′+

′

and by integration, we find that

pBA = (3.18)

where p is a constant to be determined.

From equation (3.10) we get

0)zp4(B

B

2

3

B

B 2

=+−

′−

′′ (3.19)


0)zp4(A

A

2

3

A

A 2

=+−

′−

′′ (3.20)


0BB

)zp8(BB

21

BB 3

=

′+−

′−

″

′ (3.21)

We first solve equation (3.19). Under the substitution

BB

F′

= (3.22)

equation (3.19) transforms into

7

)zp4(F21

F 2 ++=′ (3.23)

which is a Riccati differential equation. Under the standard substitution

uu

2F′

−= (3.24)

Riccati's equation (3.23) becomes

0)z(u2

zp4)z(u =⋅++′′ (3.25)

The previous equation can be transformed into the Airy equation. In fact, under

the substitution µ+λ= yz , equation (3.25) takes on the form

0)y(u2

)p4(y

2dy

)y(ud 23

2

2=

+µλ+λ+ (3.26)

The choices 23 −=λ and p4−=µ transforms (3.26) into

0)y(uydy

)y(ud2

2=− (3.27)

which is the Airy differential equation with general equation

)y(BiC)y(AiC)y(u 21 += (3.28)

where )y(Ai and )y(Bi are Airy's functions of the first and second kind

respectively (see for example Abramowitz and Stegun [3]). Going back to the

original variable (taking into account y)2(p4z 3/1−=+ ) we obtain the general

solution of equation (3.25)

−++

−+=

3/123/11)2(

p4zBiC

)2(

p4zAiC)z(u (3.29)

We now have to determine the function Y which satisfies Riccati's equation

2BYAY +=′ . Under the substitution

vv

B1

Y′

⋅−= (3.30)

Riccati's equation 2BYAY +=′ becomes

8

0v)AB(vBB

v =+′

′−′′ (3.31)

Since uu

2BB ′

−=′

and pBA = , equation (3.31) becomes 0uvp2vu2vu =+′′+′′

which can be written as

0v)uup2()vu( =′′−+′′⋅ (3.32)

The substitution

vu ⋅=Ψ (3.33)

transforms (3.32) into

0uu

p2 =Ψ

′′−+Ψ ′′ (3.34)

Since z21

p2uu −−=

′′ (because of (3.25)), equation (3.34) takes on the form

0z21

p4 =Ψ

++Ψ ′′ (3.35)

The above equation can be transformed into the Airy's equation (the same way

(3.25) was transformed into (3.27)) and admits the general solution

−++

−+=Ψ

3/123/11)2(

p8zBiC

~

)2(

p8zAiC

~ (3.36)

Integrating uu

2BB ′

−=′

we obtain

)z(u

KB

2= (3.37)

where K is a constant. Since uu

vv ′

−ΨΨ′

=′

(this equation comes from u/v Ψ=

by differentiation) we obtain from (3.30), using also (3.37)

′−

ΨΨ′

−=u

u

K

uY

2 (3.38)

where Ψ and u are given by (3.36) and (3.29) respectively.

9

So far we have not taken into account equations (3.20) and (3.21). It is obvious

that not every solution of (3.19) satisfies both (3.20) and (3.21). We thus have to

find the range of values the parameters and the constants should attain in order to

have compatible equations. The coefficients A and B of Riccati's equations are

connected through the relation 0BB

AA =

′+

′ and satisfy equations (3.19) and (3.20)

respectively. We thus have to examine the compatibility condition between (3.19)

and (3.20) first, taking into account 0BB

AA =

′+

′. We state and prove the following

Lemma. If 0BB

AA =

′+

′ then

2

B

B2

B

B

A

A

′=

′′+

′′.

Proof. We let AA

H′

= . Since 0FH =+ , we also have FH ′−=′ and then

22

BB

FAA

HAA

′+′−=

′+′=

′′ and

2

BB

FBB

′+′=

′′. Adding the last two

equations, we obtain 2

BB

2BB

AA

′=

′′+

′′ and the Lemma is proved. ■

Adding now equations (3.19) and (3.20) and taking into account the previous

Lemma, we obtain the equation 0)zp4(2BB 2

=+−

′− from which we obtain

further, in view of uu

2BB ′

−=′

, that

02z

p2uu 2

=

++

′ (3.39)

The above equation is the compatibility condition between (3.19) and (3.20) and

should be satisfied for every z.

We finally consider equation (3.21). This equation takes the form

F)p8z(F21

F 3 +=−′′ (3.40)

10

where F is defined in (3.22). This equation should be combined with (3.23).

From (3.23) multiplying by F we obtain F)zp4(FFF21 3 ++′−=− and because

of that, equation (3.40) takes on the form Fp4FFF =′−′′ . Differentiating (3.23)

we get 1FFF =′−′′ . We thus obtain the equation 1Fp4 = . Since uu

2F′

−= , we

derive the compatibility condition p8

1uu −=

′, i.e.

0uup8 =+′ (3.41)

This last equation should hold for every z.

Equations (3.39) and (3.41) should also be compatible each other. Equations

(3.39) and (3.41) are considered in Appendix A. In that Appendix, expanding the

function )z(u given by (3.29) we find the conditions between the parameters and

the various constants in order equations (3.39) and (3.41) should be true for every

z. According to the results of Appendix A, the compatibility equations (3.39) and

(3.41) lead to the same conditions

0);C,C(X 21 =ω and 0);C,C(Z 21 =ω (3.42)

where );C,C(X 21 ω and );C,C(Z 21 ω are defined by

)(BiC)(AiC);C,C(X 2121 ω+ω=ω (3.43)

)(iBC)(iAC);C,C(Z 2121 ω′+ω′=ω (3.44)

p)3i1(2 3/2 −=ω (3.45)

and the prime denotes the usual derivative

ω==ω′z

)z(Aidzd

)(iA and ω==ω′z

)z(Bidzd

)(iB

The two equations (3.42) hold simultaneously, in view of (3.43) and (3.44), if

)(Ai)(Bi

C

C

2

1

ωω−= and

)(iA)(iB

C

C

2

1

ω′ω′

−= (3.46)

11

Equating the two different expressions of the ratio 21 C/C , we arrive at the

condition

0)(Bi)(iA)(iB)(Ai =ω⋅ω′−ω′⋅ω (3.47)

The above condition determines the constant p.

Equation (3.29) has to be written, in view of (3.46), into the form

−+ω−

−+ω=

3/13/1 )2(

p4zAi)(Bi

)2(

p4zBi)(AiC)z(u (3.48)

where we have set )(Ai

CC 2

ω= .

Conclusion. The solution of the Painleve PII equation wzw2dz

wd 32

2⋅+= is

given by Y

bYaa)z(w 1

10 ++= where BB

21

a0′

= , Ba1 = , Ab1 = with pBA = ,

)z(u

K)z(B

2= and

′−

ΨΨ′

−=u

u

K

uY

2, where Ψ and u are given by (3.36) and

(3.48) respectively. Therefore )z(w is given by

1

uu

p)z(w−

ΨΨ′

−′

+ΨΨ′

−= (3.49)

where p is determined by (3.47) and Ψ , u are given by (3.36) and (3.48)

respectively.

3.II. Case II. The Second Solution.

We next consider the case

Ba1 −= and Ab1 −= (3.50)


BB

21

a0′

−= and AA

21

a0′

= (3.51)

respectively. Equating the two different expressions of 0a , we find that

pAB = (3.52)

12

where p is a constant.


0)zp4(B

B

2

3

B

B 2

=+−

′−

′′ (3.53)


0)zp4(A

A

2

3

A

A 2

=+−

′−

′′ (3.54)


0BB

)zp8(BB

21

BB 3

=

′+−

′−

″

′ (3.55)

Using the same reasoning as in Case I, we obtain the following

Conclusion. The solution of the Painleve PII equation wz)z(w2dz

wd 32

2⋅+= is

given by Y

bYaa)z(w 1

10 ++= where BB

21

a0′

−= , Ba1 −= , Ab1 −= with

pBA = , )z(u

K)z(B

2= and

′−

ΨΨ′

−=uu

Ku

Y2

, where Ψ and u are given by

(3.36) and (3.48) respectively. Therefore )z(w is given by

1

uu

p)z(w−

ΨΨ′

−′

+ΨΨ′

= (3.56)

where p is determined by (3.47) and Ψ , u are given by (3.36) and (3.48)

respectively.

Note. Considering the equation

α+⋅+= wz)z(w2dz

wd 32

2

we find that the solution does not depend on the constant α , a feature for which

we cannot provide an explanation.

13

4. The −

′GG expansion method with variable expansion

coefficients.

We consider that equation (3.1) admits a solution of the form

′+=

)z(G)z(G

)z(a)z(a)z(w 10 (4.1)

Upon substituting (4.1) into (3.3) we obtain an equation, which when arranged in

powers of G, can take the form

]Gaa6Ga2GaGazGa[)x(G

1)a2aza( 2

0111113000 ′−′′′+′′′+′−′′′+−−′′

])G(aa6GGa3)G(a2[)x(G

1 20

211

212

′−′′′−′′−+

0])G(a2)G(a2[)x(G

1 31

3313

=′+′−+ (4.2)

Equating to zero all the coefficients of )x(G in the above equation, we obtain the

following system of ordinary differential equations

0a2aza 3000 =−−′′ (4.3)

0Gaa6Ga2GaGazGa 2011111 =′−′′′+′′′+′−′′′ (4.4)

0)G(aa6GGa3)G(a2 20

211

21 =′−′′′−′′− (4.5)

0)G(a2)G(a2 31

331 =′+′− (4.6)

Equation (4.3) is essentially equation (3.3) and thus admits all the solutions found

in Section 3. From equation (4.6), ignoring the trivial solution, we obtain that

1a1 ±= (4.7)

From equation (4.4), taking into account the above values of 1a , we derive the

equation

za6GG 2

0 +=′′′′

(4.8)

14

From equation (4.5), we obtain

0a2GG −=

′′′

for 1a1 = and 0a2GG =

′′′

for 1a1 −= (4.9)

We thus derive the following equations, dividing (4.8) by (4.9):

Solution I. For 1a1 = , the function )x(G is determined as solution of the

equation 0

20

a2

za6

GG +−=

′′′′′

where 0a is any solution of equation (4.3).

Solution II. For 1a1 −= , the function )x(G is determined as solution of the

equation 0

20

a2

za6

GG +=

′′′′′

where 0a is any solution of equation (4.3).

In both the above equations, 0a is given by (3.47) and (3.54) (simply substitute

)z(w by 0a ). Because of the complexity of the expressions for 0a , the integration

of the differential equations might require supercomputing facilities with symbolic

capabilities. A remarkable feature of the −′ )G/G( expansion with variable

expansion coefficients is that a repeated use of the method in determining 0a

leads to a proliferation of solutions, a rather unique feature of the Riccati –

−′ )G/G( expansion.

Appendix A. In this Appendix we find the conditions the various parameters

and the constants should satisfy so as equations (3.39) and (3.41) should be true

for every value of the parameter z.

We first consider equation (3.39) which can be written as a couple of equations

0u2

zp2iu =++′ (A.1)

and

0u2

zp2iu =+−′ (A.2)

where )z(u is given by (3.29).

15

Upon expanding )z(u in power series, we find that (A.1) can be written as

);C,C(Z)3i1(4

2);C,C(Xp2i 21

3/2

21 ω−+ω

z)};C,C(Z)i1(p24);C,C(X)p162i({p8

121

6/121

2/3 ω+⋅+ω−+

)3i()p2[(8);C,C(X)p2128ip322i({p128

1 6/121

32/32/3

++ω++−+

221

3/22/5 z)};C,C(Z]2p)13i(4 ω−+

+ω−++ );C,C(X)pi2640i23p2048({p3072

121

32/92/5

)i3(p2512)13i(2p128[ 46/13/22/5 +⋅−−+

321

6/1 z)};C,C(Z]p)3i(212 ω+⋅−

)z(O 4+ (A.3)

where we have introduced the notation

)(BiC)(AiC);C,C(X 2121 ω+ω=ω (A.4)

)(iBC)(iAC);C,C(Z 2121 ω′+ω′=ω (A.5)

p)3i1(2 3/2 −=ω (A.6)

and the prime denotes the usual derivative

ω==ω′z

)z(Aidzd

)(iA and ω==ω′z

)z(Bidzd

)(iB

Similarly expanding )z(u in power series, we find that (A.2) can be written as

);C,C(Z)3i1(4

2);C,C(Xp2i 21

3/2

21 ω−+ω−

z)};C,C(Z)i1(p24);C,C(X)p162i({p8

121

6/121

2/3 ω+⋅+ω+−

)3i()p2([8);C,C(X)p2128ip322i({p128

1 6/121

32/32/3

+−+ω+−+

16

221

3/22/5 z)};C,C(Z]2p)13i(4 ω−+

+ω+−+ );C,C(X)pi2640i23p2048({p3072

121

32/92/5

)i3(p2512)13i(2p128[ 46/13/22/5 +⋅+−+

321

6/1 z)};C,C(Z]p)3i(212 ω+⋅+

)z(O 4+ (A.7)

Expanding (3.41) we obtain similarly

+ω−⋅+ω });C,C(Z)3i1(p22);C,C(X{ 213/2

21

+

ω⋅−+ω⋅−+ z);C,C(Z)3i1(4

2);C,C(Xp16 21

3/2

212

+ω⋅−⋅+ω⋅−+ 221

23/221 z});C,C(Z)13i(p22);C,C(Xp3{

+

ω⋅−⋅−ω⋅

−+ 3

21

3/2

21

3z);C,C(Z)3i1(p

1225

);C,C(X121

3p16

0)z(O 4 =+ (A.8)

In all the higher order expansion terms, there appears the same linear combination

of the quantities );C,C(X 21 ω and );C,C(Z 21 ω . Therefore the compatibility

conditions (3.39) and (3.41) are true for every z , if and only if

0);C,C(X 21 =ω and 0);C,C(Z 21 =ω (A.9)

The two equations (A.9) hold simultaneously if

)(Ai)(Bi

C

C

2

1

ωω−= and

)(iA)(iB

C

C

2

1

ω′ω′

−= (A.10)

Equating the two different expressions of the ratio 21 C/C , we arrive at the

condition

0)(Bi)(iA)(iB)(Ai =ω⋅ω′−ω′⋅ω (A.11)

The above condition determines the constant p.

17

Appendix B.

In this Appendix we consider the cases ( Ba1 = , Ab1 −= ) and ( Ba1 −= ,

Ab1 = ). We show that in these two cases we obtain incompatible equations.

Case III. We consider the case

Ba1 = and Ab1 −= (B.1)


BB

21

a0′

= and AA

21

a0′

= (B.2)

respectively. Equating the two different expressions of 0a , and integrating, we

find that

BsA 2−= (B.3)

where s is a real constant. We can equally well consider the case BsA 2= .

From equation (3.10), we get

0zBs8B

B

2

3

B

B 222

=−−

′−

′′ (B.4)


0zBs8AA

23

AA 22

2

=−−

′−

′′ (B.5)


0)B(s12BB

zBB

21

BB 22

3

=′−

′−

′−

″

′ (B.6)

Equation (B.4) under the substitution

)z(u

1B

2= (B.7)

takes on the form

)z(u

s4)z(u

2z

)z(u3

2−=+′′ (B.8)

18

which is Ermakov's equation (Ermakov [8]). The equation 0)z(u2z

)z(u =+′′

admits two linearly independent solutions,

− 3/1)2(

zAi and

− 3/1)2(

zBi .

Therefore, using the standard procedure, the general solution of (B.8) is given by

−

++−×

−= ∫

2

2

3/1

122

2

3/12

1

)2(

zAi

dzCCs4

)2(

zAi)z(uC (B.9)

and

−

++−×

−= ∫

2

2

3/1

122

2

3/12

1

)2(

zBi

dzCCs4

)2(

zBi)z(uC (B.10)

Every solution of the equation (B.4) found previously has to be substituted in

(B.6) and thus to obtain a compatibility condition. Despite the fact that (B.4)

admits closed-form solution, equations (B.4) and (B.6) are incompatible. The

proof goes as follows: Equations (B.4) and (B.6) can be written in terms of

BB

G′

≡ as

zBs8G21

G 222 +=−′ (B.11)

and

0)B(s12GzG21

G 223 =′−−−′′ (B.12)

respectively. Multiplying equation (B.11) by G we derive the equation

19

GzBGs8GGG21 223 ++′−=− (B.13)

Combining (B.12) with (B.13) we get

0)B(s12BGs8GGG 2222 =′−+′−′′ (B.14)

Differentiating (B.11) with respect to z we obtain

1)B(s8GGG 22 +′=′−′′ (B.15)

Equations (B.14) and (B.15) give

0)B(s12BGs81)B(s8 222222 =′−++′

which is equivalent to the quite remarkable result 01= . We have thus proved that

equations (B.4) and (B.6) are incompatible. The case BsA 2= leads again to

incompatible equations.

Case IV. We consider next the case

Ba1 −= and Ab1 = (B.16)

Using the same reasoning as before, we obtain again incompatible equations.

20

References

[1] M. J. Ablowitz, A. Ramani and H. Segur: "A connection between

nonlinear evolution equations and ordinary differential equations of

P-type. I" J. Math. Phys. 21 (1980) 715-721

[2] M. J. Ablowitz, A. Ramani and H. Segur: "A connection between

nonlinear evolution equations and ordinary differential equations of

P-type. II" J. Math. Phys. 21 (1980) 1006-1015

[3] M. Abramowitz and I. A. Stegun: "Handbook of Mathematical

Functions". Dover 1972

[4] H. Airault: "Rational solutions of Painlevé equations"

Stud. Appl. Math. 61 (1979) 31-53

[5] S. Antoniou: “The Riccati equation method with variable expansion

coefficients. I. Solving the Burgers equation”. submitted for publication

[6] S. Antoniou: “The Riccati equation method with variable expansion

coefficients. II. Solving the KdV equation”. submitted for publication

[7] P. A. Clarkson: "Painlevé equations-nonlinear special functions"

J. Comp. Appl. Math. 153 (2003) 127-140

[8] V. P. Ermakov: “Second-Order Differential Equations: Conditions of

Complete Integrability”. Appl. Anal. Discr. Math. 2 (2008) 123-145

Translation from the original Russian article:

Universitetskiye Izvestiya Kiev No. 9 (1880) 1-25

[9] A. S. Fokas and M. J. Ablowitz: "On a unified approach to

transformations and elementary solutions of Painlevé equations"

J. Math. Phys. 23 (1982) 2033-2042

[10] E. L. Ince: "Ordinary Differential Equations". Dover 1956

[11] A. Ramani, B. Dorizzi and B. Grammaticos: "Painlevé conjecture

revisited". J. Math. Phys. 21 (1980) 715-721

21

[12] A. Ramani, B. Grammaticos and T. Bountis: "The Painleve Property and

Singularity Analysis of Integrable and Non-Integrable Systems".

Phys. Rep. 180 (1989) 159-245

[13] J. Weiss, M. Tabor and G. Carnevale: "The Painlevé property for partial

differential equations". J. Math. Phys. 24 (1983) 522-526

[14] J. Weiss: "The Painlevé property for partial differential equations. II:

Bäcklund transformation, Lax pairs, and the Schwarzian derivative"

J. Math. Phys. 24 (1983) 1405-1413

[15] J. Weiss: "On classes of integrable systems and the Painlevé property "

J. Math. Phys. 25 (1984) 13-24

[16] J. Weiss: "The Painlevé property and Bäcklund transformations for the

sequence of Boussinesq equations".

J. Math. Phys. 26 (1985) 258-269

[17] Y. Wenjum and Li Yezhou: "Rational Solutions of Painlevé Equations"

Canadian J. Math. 54 (2002) 648-670

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Some General Solutions to the Painlevé – PII Equation (New Version 11/07/2014)