NONLOCAL CONSTANTS OF MOTIONS OF EQUATIONS OF PAINLEVé--GAMBIER TYPE AND GENERALIZED SUNDMAN TRANSFORMATION Текст научной статьи по специальности «Математика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 1, pp. 103-118. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220107

MATHEMATICAL PROBLEMS OF NONLINEARITY

MSC 2010: 34C14, 34C20

Nonlocal Constants of Motions of Equations of Painleve —Gambier Type and Generalized Sundman

Transformation

P. Guha, A. G. Choudhury, B. Khanra, P. G. L. Leach

We describe a method to generate nonlocal constants of motion for a special class of nonlinear ODEs. We employ the method of the generalized Sundman transformation to obtain certain new nonlocal first integrals of autonomous second-order ordinary differential equations belonging to the classification scheme developed by Painleve and Gambier.

Keywords: Sundman transformation, Painleve-Gambier, symmetry, nonlocal first integrals, Jacobi equation

Received October 19, 2021 Accepted January 28, 2022

Partha Guha partha.guha@ku.ac.ae

Department of Mathematics, Khalifa University of Science and Technology Main Campus, Abu Dhabi, United Arab Emirates

A. Ghose Choudhury aghosechoudhury@gmail .com

Department of Physics, Diamond Harbour Women's University D.H.Road, Sarisha 743368, West Bengal, India

Barun Khanra

barunkhanra@rediffmail.com

Sailendra Sircar Vidyalaya

62A, Shyampukur Street, Kolkata-700004, India

Peter G.L. Leach leach@ucy.ac.cy

Department of Mathematics and Statistics, University of Cyprus Lefkosia 1678, Cyprus

School of Mathematical, Statistical, Acturial and Computer Sciences, University of KwaZulu-Natal Private Bag X54001, Durban 4000, Republic of South Africa

1. Introduction

In recent times there has been a fair amount of interest in the use of nonlocal transformations for studying various properties of ordinary differential equations (ODEs). Unlike linear ODEs, there are no formal methods for the construction of solutions of nonlinear ODEs. However, the process of constructing solutions is greatly aided by the existence of first integrals which invariably lead to a reduction of order of the differential equation under consideration. Although each nonlinear ODE bears a unique character and requires individual attention the various ad hoc methods for their solutions were systematized to a great extent as a result of Lie's seminal work based on continuous point transformations in the latter half of the nineteenth century. The fact that many nonlinear equations can be linearized through point transformations provided an efficient route for the construction of their solutions. For the case of second-order ODEs Lie himself solved the linearization problem by finding the general form of all second-order ODEs that can be reduced to a linear equation by a change of the independent and dependent variables of the equation. Lie showed that, if a second-order ODE is linearizable, then it is at most cubic in the first-order derivatives; moreover, he provided a test for linearization in terms of its coefficients [22]. Tresse also worked on the same problem, but expressed the criterion for linearization in terms of the relative invariants of the equivalence group of point transformations [31]. Cartan, on the other hand, studied the same problem from the point of view of differential geometry.

Nonlocal transformations have in recent times been used for studying the linearization problem. Duarte et al. have considered transformations of the form

X = F (t,x), dT = G(t,x)dt, (1.1)

where F and G are arbitrary smooth functions satisfying FxG = 0, which follows from the condition that the Jacobian J = d(T, X)/d(t, x) = 0. The transformation involves in general complex (or real) scalar variables and functions. The nonlocal character of the transformation is evident from the latter half of the transformation. Given that the term nonlocal can have a wide range of meanings, it is perhaps more appropriate to refer to the transformation (1.1) as a generalized Sundman transformation (GST). Sundman, who was a pioneering contributor to the field of celestial mechanics, introduced the transformation, dt = r dr, in his study of the 3-body problem, where r is the dependent variable (radial component) [29]. About a quarter of a century ago Sundman's method was revitalized by Szebehely and Bond [30], who also considered a transformation of the dependent variable r = F(p). In particular, these transformations of Sundman type are quite useful for the solution of several nonlinear ODEs. Many authors (for example, [5]) have studied Sundman transformations under the name of nonpoint transformations.

The simplest second-order linear ODE is

X" (T) = 0,

(here X1 = dX/dT) and the problem of deriving the most general condition under which a second-order ordinary differential equation is transformable to such a linear equation was analyzed by the authors of [8] using a generalized Sundman transformation. By exploiting the fundamental invariants of the linear equation, X"(T) = 0, they obtained first integrals of second-order ordinary differential equations which could be linearized. The case of the general anharmonic oscillator was studied by N. Euler and M. Euler in [9] who also investigated the Sundman symmetries of second-order and third-order nonlinear ODEs. These symmetries, which are in general the

generators of nonlocal transformations, can be obtained in a systematic manner and may be used to find first integrals of the equations. Euler et al. [10] have also used the generalized Sundman transformation to obtain a relation between a generalized Emden-Fowler equation and the first Painleve transcendent.

In fact, the study of the linearization problem via nonlocal transformations and computation of first integrals is still a very active field of research in nonlinear dynamics [18, 23, 27]. Of late there have been several papers concerned with linearization of third-order ODEs using generalized Sundman transformations and by other methods [2, 28]. Unlike second-order equations, third-order nonlinear ODEs can be effectively linearized through a wider class of transformations, viz., invertible point transformation [19], contact transformation [3, 19] and other methods [6]. In a recent survey [11] the conditions under which a nonlinear ordinary third-order differential equation can be linearized to the form d3x/dt3 = 0 are summarized using invertible point transformations.

In a previous article [17] we examined some of the equations belonging to the Painleve -Gambier classification as contained in Ince's book [16, 20] using the technique of Sundman transformations. In particular, we derived time-dependent first integrals for equations numbered XVII, XXXVII, XLL and XLII which were not included in Ince's list. Moreover, using this method, we could also reproduce the known time-independent first integrals for these equations. Further to the above, parametric solutions of these equations were also constructed. In arriving at these results there appeared the possibility of the existence of first integrals which were of a nonlocal character. Consequently, in this article we investigate the existence of nonlocal first integrals for a class of Painleve-Gambier equations. The existence of nonlocal first integrals is not completely without precedence. Nonlocal Noetherian constants of motion starting from nonlocal transformations were introduced by [15]. Bogdanov in [4] has dwelt on the meaning of such first integrals for polynomial vector fields. Recently, nonlocal constants of motion have been studied by Gorni and Zampieri in a series of papers [12-14]. The general form of the constants of motion they considered is given by

Such an integral is called nonlocal because its value at any time t depends not only on the value of position and velocity at time t, but also on the past history of the motion. Recently, nonlocal constants of motion in higher-order Lagrangian systems have been discussed in [26]. In addition, Crumey [7] obtained nonlocal conserved quantities or Hamiltonians for the generalized nonlinear Schrodinger (GNLS) equation and showed more than a quarter century ago that Hamiltonians form the Kac-Moody type algebraic structures under the Poisson bracket. In fact, the notion of nonlocal conserved quantities can be extended to even supersymmetric systems where nonlocal conservation laws (conserved quantities) are derived for the superfield version of the N = 1 SKdV equation [1]. Finally, the method of nonlocal generalized Sundman transformation has been employed to obtain the isochronicity condition of the Lienard equation in [18], which is indicative of the wide range of applicability of such transformations.

1.1. Results and plan

The results of our present investigations are briefly summarized in the following three tables.

t

Table 1. Summary of major features of GST for equations of the Painleve - Gambier classification (ß = const)

No. Painleve - Gambier Eq. Generalized Sundman Trans.

XVIII 1-2 nß _ _ 2x 4x2 = 0 V 2ix A - — dT = ßix1/2 dt

XXI 3-2 nß _ nß^ _ 4x 3x2 = 0 X = 2ix3/4 dT — X ^ dt

XXII 3-2 nß _ f- _ 4x f 1 = 0 X = 2ix-1/4 dT = - ^ ^ dijfy

Table 2. Nonlocal first integrals for equations of the Painleve - Gambier classification No. Painleve - Gambier Eq. Nonlocal First Integral

XVIII 1-2 2x 4x2 = 0 I2 1 ( X 2i \2x1/2 ■+ x) e- / 2x1/2 dt

XXI 32 nß _ nß^ _ 4x 3x2 = 0 I2 = 2 i -+- 2x3/4)e-lf*1/2 dt

XXII 32 nß _ '-* f- - 4x f 1 = 0 h = J, (-2L- + 2i Va,3/4 T 2x~1/4) ei f x~1/2 dt

Table 3. Solutions of some equations of the Painleve - Gambier classification (y, A and £ being arbitrary constants)

No. Painleve - Gambier Eq. Solution

XVIII x - ¿.i-2 - Ax? = 0 x(t) =

3-2 _ oJl _ n - _I_

XXI x - f^±2 - 3x = 0 x(t)

XXII x - ¿¿2 + 1 = 0 x{t) = (£ +1)2

In Section 2 we describe the generalized Sundman transformation and Sundman symmetry and apply this method to the reduced Jacobi equation. We obtain the (nonlocal) first integrals of the equations of Painleve Gambier type in Section 3.

2. The generalized Sundman transformation

In this section we briefly recollect some of the essential features of the generalized Sundman transformation (GST) [17]. For an nth-order ODE

x(n = w(t, x, x, x, ..., x(n-1)), (2.1)

where x = x(t) and x(k = dkx/dtk, a generalized Sundman transformation may be defined as follows:

Definition 1 (Generalized Sundman transformation). A coordinate transformation of the form

dF

X = F(t,x), dT = G(t,x)dt, — / 0, G + 0, (2.2)

is said to be a generalized Sundman transformation of (2.1) if there exist differentiable functions F and G such that (2.1) is transformed to the autonomous equation

X(n) = w0(X, X', ..., X(n-1)), (2.3)

where X' = dX/dT etc.

The nonlocal character arises from the fact that T = J G(t, x(t))dt. If (2.3) turns out to be a linear ODE, then we say that (2.1) is linearizable.

For a generalized Sundman transformation, one can introduce the concept of a Sundman symmetry similar in spirit to a Lie symmetry under point transformation. For a generalized Sundman transformation

X = F(t, x) and dT = G(t, x)dt,

which maps the equation

x(n) = w(t, x,x,..., x(n-1)) X(n) = w0(X, X',..., X(n-1)),

if there exists a transformation of the differentiable functions F(t, X) and G(t, X), considered as functions of F(t, x) and G(t, x), such that our original differential equation (2.1) remains invariant under the transformation, then the transformation defines a Sundman symmetry. The formal definition is as follows:

Definition 2 (Sundman symmetry). A Sundman symmetry [9] for Eq. (2.1) is a transformation of the form

F(t, x) = M(F(t, x), G(x, t)), G(t, x)dt = N(F(t, x), G(t,x))dt, (2.4)

where M and N are some differentiable functions such that the transformation keeps (2.1) invariant. In other words, (2.1) is transformed to

x(n = w(t; x,x,x, ..., x(n-1)). (2.5)

If M(F, G) = F and N(F, G) = G, the symmetry is trivial. The set of conditions on the differentiable functions F and G when (2.1) is mapped to (2.3) is referred to as the Sundman determining equations. In order to deduce the Sundman symmetry (2.4), one chooses M and N in such a manner that the Sundman determining equations remain invariant. If

X = F(t, x), dT = G(t, x)dt

transforms (2.5) to (2.3) and

X = M(F(t, x), G(t, x)), dT = N(F(t, x), G(t, x))dt

also transforms (2.1) to (2.3), then the composition of these two GSTs leads to the Sundman symmetry (2.4) for (2.1). In [17] we have constructed GSTs in which the linearized equation was that of a free particle, namely, X'' = 0. In this paper we consider its generalization where we instead map the original equation to that of a linear harmonic oscillator X'' + X = 0.

3. The generalized Sundman transformation for the reduced Jacobi equation

Consider the general second-order Levinson-Smith (or general Lienard) equation of the following type:

x + ¿(x)x2 + f (x, t)x + g(x, t) = 0, (3.1)

where is the independent variable, x is the dependent variable of the equation, and the overdot denotes the derivative with respect to t. By application of the Sundman transformation of the

/ x \

dT = G(x)dt and X = x, with G = exp ( — J 5(x)dx

we obtain the standard Lienard equation

x" + f(x, t)x' + g(x, t) = 0, (3.2)

where

x

f(x, t) = f (x, t)ef5(x)dx, g(x, t) = g(x, t)e2f5(x)dx. (3.3)

Equation (3.2) can be mapped to the Abel equation of the first kind with the modified coefficients

dv—'

— = g(x, t)v3 + f(x, t)v2, with x' = -. (3.4)

dx v

It is possible to obtain the first integrals of the form A(t, x)x + B(t, x) by using the procedure given by Muriel and Romeo [24].

It is apparent that the Jacobi equation, given by [21, 25],

x + ^<pxx2 + (ptx + B(t, x) = 0, (3.5)

is a special case of the Levinson-Smith equation (3.1) stated above. We consider below a reduced version of the Jacobi equation when 01 = 0 = Bt but Bx = 0.

There are several equations of the Painleve - Gambier classification which belong to this particular category. Consequently, the prototypical equation for this section has the generic form

x + p)XX2 + B(x) = 0. (3.6)

The objective is to construct a generalized Sundman transformation (2.2) such that (3.6) is mapped into

X'' + a0(X) = 0, (3.7)

where X' = dX/dT. The exact form of a0 (X) is specified below. This is true if the following conditions (i.e., the Sundman determining equations) on the coefficients of (3.6) hold good:

1 F G

= (3.8)

2Yx Fx G '

Fxt _ _ ;

Fx G Fx G

0 = (3.9)

F G F G^

B{x) = fx~^F-x+a°{F)Tx- (3-10)

These conditions follow from the fact that, as

/ dX Fx _ Ft

= HT=~gx+G'

x

one finds that

x" = ct2i+]G (§)T+ (§)t+h (§) J *+h(ji/t

Using Eq. (3.6) to eliminate the term x, we have upon equating coefficients of the different powers of x the above equations.

From (3.8) we have

lnFT-lnG = /i,M,-l„i,(t).

Here b(t) is an arbitrary function of integration. It follows that

G(t, x) = b(t)e-(p/2Fx. (3.11)

On substitution of G from (3.11) into (3.10) we have

If we set b(t) = f, i.e., a constant independent of t, and assume

(3.12)

then (3.12) implies

a0(F)f2 e-^ Fx = B(x). (3.14)

Instead of firstly trying to determine the form of F, it is more convenient to specify a0(F) and see whether with such a choice of a0(F) we can satisfy the remaining equation (3.9). To this end we choose

a0 (F) = ±F. (3.15)

Then (3.14) yields

F2 = J B^e4' dx + 7 (i), (3.16)

so that assuming 7(y) =0 F becomes a function of x only and, as a result, it is obvious that (3.13) is trivially satisfied. It remains to verify whether such an expression for F is consistent with (3.9). Because b(t) = f is a constant, we have from (3.11)

G(t, x) = Pe~«2Fx = --f^2 (3.17)

(±2 J B(xe dx)1/2

which is clearly independent of t and hence Gt = 0. Consequently, as F and G are functions of x, it follows that (3.9) is satisfied. In summary we have therefore the following form of the GST mapping (3.6) to the equation X'' ± X = 0, viz.,

X = F(x) = (±4 I B{x)e^x] dx] ' , dT =-B(x)eH2— dL (318)

V P2 J J (±2 f B(x)e^dx)

The latter is obviously a nonlocal transformation.

3.1. The Sundman symmetry

The Sundman symmetry associated with (3.6) may be easily deduced. For the sake of notational convenience let us denote

F = F(t,x) and G = G(t, x).

To ensure invariance of the Sundman determining equations, namely, (3.8)-(3.10), we assume

F = M(F) and G = G(t, x)$(F). (3.19)

The functional forms of M and ^ are determined by demanding the invariance of the Sundman determining equations. Invariance of (3.8) leads to

where K is a constant of integration, which may be set to unity, so that

■0(F) = M'(F). (3.20)

Invariance of (3.10) then leads to the equation

dM _ a0{F) ~~dF ~ a,0(M)'

whence it follows with a0(F) = ±F that

M = ±VF2 + c, (3.21)

where c is a constant of integration. Note that, if c = 0, then we have a trivial symmetry. The functional form of ^ is therefore given by

1>(F) = ±-f=|=. (3.22)

VF2 + c

With M and ^ given by (3.21) and (3.22), respectively, one can easily verify that the final Sundman determining equation (3.9) is identically satisfied. Thus, in summary we have the following Sundman symmetry for (3.6):

.--~ F

F(t, x) = itv F2(t, x) + c and G(t, x)dt = ±G(t, x) dt. (3.23)

VF2 + c

In the following we consider only the cases in which the GST maps equations of the Painleve-Gambier classification belonging to the class of (3.6) to a harmonic oscillator equation

X'' + X = 0. (3.24)

It is easily checked that (3.24) admits the first integrals I1 = X'2 + X2 and I2 = (X' — iX)eiT/2i along with its complex conjugate.

3.2. The Painleve — Gambier equation XVIII

Equation XVIII is

x - —x2 - 4x2 = 0. (3.25)

2x y '

Here 0x = —1/x, which implies that 0 = lnx-1 and B(x) = -Ax2. As a result, taking the positive square root from (3.16), we find that. F(x) = 2ix/f3 and it turns out that. G = f3is/x. Hence, the Sundman transformation has the explicit form

o jr.

X = —-, dT = pi-y/ccdt. (3.26)

(

A first integral for (3.24) is

I1 = X '2 + X2

and its evaluation in terms of the preceding transformation does indeed reproduce the result given in [20], namely,

X2 = 4x(Il + x2)

upon setting (3 = 2. A second first integral for (3.24) is

h = ^X'-iX)eiT and its evaluation in terms of the preceding transformation gives

1 ( x

/2 = ^l^+-T)eXP

- / 2x1/2 dt

(3.27)

Using Sundman transformations, one can easily construct a solution of the differential equation under consideration. Since the nonlocal transformation maps the ODE to the linear harmonic oscillator equation, namely, X" + X = 0, we may use the fact that X(T) = c1eiT (c1 being a constant) is a solution of this differential equation. Now, as X = F(t, x), we have from the first part of (3.18) with ( = 2

X = ix (3.28)

or, in other words,

X

x = —. (3.29)

On the other hand, since

G(t, x) = 2iy/x, (3.30)

we can use the second expression in (3.26) with ( = 2 and the definition dT = Gdt to obtain

dT = 2(iX)1/2 dt = 2(ic1 )1/2 exp

iT

dt,

2

whence the variables may be separated (use having been made of the solution X = c1elT):

From X = c1eiT, by using the expression above for eiT and also (3.29), we obtain finally a particular solution of the ODE as

/ x 4ic1

or, in other words, a solution of the form

x{t) = hTt]2' (3'33)

where y = c2/ (2(ic1 )1/2) is an arbitrary constant.

Applying a similar procedure to the other linearly independent solution X(T) = c3 exp[—iT] of (3.24) and using the general Sundman transformation (3.28) and (3.30) for this ODE and the definition dT = Gdt, one obtains

dT = 2(iX)1/2 dt = 2(ic3)1/2 exp

whence the variables may again be separated to obtain

iT

dt,

e'T = -\{c, + 2(lc3)1/H}2. (3.34)

By using the expression for e-iT from (3.34) and also using (3.28), we get finally another solution of the ODE in the form

/ x 4ic3

X * = , 1 Oi ' \l/'2+Y2 ' (3-35) (c4 +2(ic3 )1/21)2

In other words,

= jsTW (3'36)

Here 5 = c4 (2(ic3)1/^ is an arbitrary constant. As this particular solution is of exactly the same nature as (3.33), it appears that the final outcome is independent of the choice of the particular solution of the transformed ODE (3.24).

To deduce the Sundman symmetry of this equation, we use (3.23) (taking the positive sign). The Sundman symmetries are then given by

x = —is/c — x2 and t = A+ f , ^— . dt, (3.37)

J i1/2(c — x2)3/4

where A is an arbitrary constant.

3.3. The Painleve — Gambier equation XXI

Equation XXI is

3

x - —x2 - 3x2 = 0. (3.38)

4x

Here 1/20x = — 3/(4x), which implies that 0 = lnx-3/2 and B(x) = — 3x2. As a result, taking the negative square root from (3.18) and f = 1, we find that F(x) = 2ix3/4 and it turns out that. G = y.r1/2. Hence, the Sundman transformation has the explicit, form

X = F{x) = 2ixs/4, dT = ?^x1'2 dt. (3.39)

2

The first integral for (3.24) is clearly given by

I1 = X '2 + X2

(3.40)

and its evaluation in terms of the preceding transformation does indeed reproduce the result given in [20], namely,

i1

x3/2

— 4x3/2.

The evaluation of another first integral for (3.24),

1

in terms of the preceding transformation gives

! > i f 4 • 2,-" 1

2i

x 4

exp

-U'x^dt

(3.41)

Using Sundman transformations, one can easily construct a solution of the differential equation under consideration. As the nonlocal transformation maps the differential equation to that of a linear harmonic oscillator, viz., X" + X = 0, we can make use of the fact that X(T) = c1eiT is a solution of the latter, so that by definition we have

X = F(t, x) = 2ix3^4, which implies x =

4/3

On the other hand,

^ 3i 1/2 , 3i X2/3 3i (c1 eiT)2/3 1 dT = —xl/2 dt = ———77^ dt = — ' , ., dt.

2 (2i)2/3 2 (2i)2/3

This may be separated to obtain

e

- 2iT/3

2/3

(2i)2/3

where c2 is an arbitrary constant so that

t + c2,

2/3

-3/2

X = Cie*T = 2 ix3/4 = Cl -^-j-t +

(2i)2/3

c2

which finally yields

\ 4/3

x

2i

2/3 C1 (2i)2/3

t + c2

(A +1)2'

(3.42)

with A = (2i)2/3c2/c^'i being an arbitrary constant.

As in the previous example of the Painleve - Gambier equation XVIII, we can show that this particular solution is independent of the choice of the particular solution of the transformed equation (3.24).

2

1

c

1

1

2

To deduce the Sundman symmetry, we use (3.23) (and take the positive sign). The Sundman symmetry in this case is given by

(r_ 4r3/2)2/3 _ f (2i)5/3

X = 1 and t = B+ -1 j dt, (3.43)

J (c-4x2y/sVc-2xs/2

where B is an arbitrary constant.

It is interesting to note that one may obtain a parametric solution of the given ODE by integrating the first integral (3.40) and expressing its solution in a suitable parametric form and then subsequently making use of the Sundman transformation; this is explained below. Consider a first-order equation of the form

^f) =0. (3.44)

Let X(t) = f (t), dX/dT = g(r), t = t(T), where f and g satisfy the relation F(f (t), g(r)) = 0 and t is a parameter. Since

dX df dr

~dT = dídГ=g^T',

it follows that

T^ = iirW)dT + c' (»■«)

where C is a constant of integration. Hence, the general parametric solution of (3.44) is given by

X (t ) = f (t), (3.46)

T^=Iiw)dT+c (3-47)

together with F(f (t), g(T)) = 0. Employing the above method, we may integrate (3.40) para-metrically to obtain the general solution of (3.24) in the form

X(t) = ^I-, - t2, (3.48)

T(t) = Cl- arcsin (3.49)

where t is a real parameter and I1 and C1 are arbitrary constants.

The general solution of (3.38) is then obtained by using the transformation (3.39) together with the parametric solution (3.48) and (3.49). It is given by

t(r) = J —iL—j- + c„ (3.50)

where I- and C- are arbitrary constants.

3.4. The Painleve — Gambier equation XXII

Equation XXII is

3 2

x - —x2 + 1=0. 4x

(3.51)

Here 0x = —3/(2x) and this implies that 0 = lnx-3/2 and B(x) = 1. As a result, from (3.18), after taking the positive square root and /3 = 1, we find that F(x) = 2ix-1/4 and it turns out that. G = — Hence, the Sundman transformation has the explicit, form

X = F{x) = 2ix~l/4 and dT = dt.

(3.52)

The first integral for (3.24) is well known to be

I1 = X/2 + X2

and its evaluation in terms of the preceding transformation does indeed reproduce the result given in [20], namely,

i1

x

x3/2

4

x

1/2

The evaluation of the other first integral for (3.24),

I2 = —(X' - iX)e

iT

2i

in terms of the preceding transformation gives

1

Io = -

x

2i x3/4

+ 2x

-1/4

exp

{fX~

1/2 dt

(3.53)

Once again, using Sundman transformations, one can easily construct a solution of the differential equation under consideration. Since the nonlocal transformation maps the ODE to the linear harmonic oscillator equation, namely, X//+X = 0, we make use of the fact that X(T) = = c1 eiT is a solution of this differential equation (here c1 is a constant). By definition, X = = F(t, x). Consequently, using the results of (3.52), we may write

X = 2ix-1/4,

(3.54)

whence it follows that

On the other hand, using

16_ X*'

G(t, x) = —

2x1/2

together with (3.52) and the definition dT = Gdt, one obtains

dT = -^-dt = -UeMTdt, 8 8

(3.55)

(3.56)

x

whence the variables may be separated (use having been made of the solution X = c1eiT):

=-7~7~~-7179 • (3-57)

i ( — c

x 1/2-

8 1 C2 J

From X = c1 eiT, by using the above expression for eiT and (3.54), we get finally a solution of the ODE in the form

,, 64(fi-c2)2

x(t) =---(3.58)

c1

that is, in other words,

x(t) = (t + C )2. (3.59)

Here C = ~ 8c2/(c2i) is an arbitrary constant.

If we consider the other linearly independent solution, X(T) = c3e-iT, of (3.24) and use the general Sundman transformation (3.54) and (3.56) for this ODE along with the definition dT = = Gdt, we obtain

dT = -^dt = -Ue-'MTdt, 8 8

whence the variables may be separated to give

e*T = ±|(c4 + i)2. (3.60)

By using the expression for e-%T from (3.34) and also using (3.28), we finally obtain a particular solution of the ODE in the form

x(t) = (c4 +1)2. (3.61)

Here c4 is an arbitrary constant. Once again we notice that this solution is similar in structure to (3.59). This leads us to conclude that a solution obtained in the above manner is independent of the choice of the particular solution of the transformed ODE (3.24).

To deduce the Sundman symmetry, we use (3.23) (taking the positive sign). The Sundman symmetry in this case is given by

„16 ~ f 8ix-3/4

x = ---7— and t = C+ -:— = dt, (3.62)

(C-2.T-V2) J (C_ 2^-1/2)^-4Z-V2

where C is an arbitrary constant.

Acknowledgments

We wish to thank Basil Grammaticos for enlightening discussions regarding the Painlevé-Gambier equations. The work of the author PG was supported by the Khalifa University of Science and Technology under grant number FSU-2021-014. PGLL thanks the Department of Mathematics and Statistics of the University of Cyprus for its generous provision of facilities and furthermore thanks the University of KwaZulu-Natal and the National Research Foundation of South Africa for their continued support. The opinions expressed in this paper should not be construed as being those of either institution.

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