AbstractTwo generalized forms of the Pinney equation, recently derived by Rogers, Schief, and Winternitz from Lie group considerations, are here connected with two linearizable second-order ODE's belonging to the Gambier classification

Musette, Micheline

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Ž .Journal of Mathematical Analysis and Applications 234, 723]726 1999
Article ID jmaa.1999.6321, available online at http:rrwww.idealibrary.com on
NOTE
On the Links Between Some Generalized Pinney and
Nonlinear Gambier Equations*
Micheline Musette
Theoretische Natuurkunde, Vrije Uni¤ersiteit Brussel, B-1050 Brussel, Belgie¨
E-mail address: mmusette@vub.ac.be
Submitted by Colin Rogers
Received August 24, 1998
Two generalized forms of the Pinney equation, recently derived by Rogers,
Schief, and Winternitz from Lie group considerations, are here connected with two
linearizable second-order ODE’s belonging to the Gambier classification. Q 1999
Academic Press
I. INTRODUCTION
w xIn a paper of Rogers, Schief, and Winternitz 1 , two generalizations of
w xthe Ermakov]Pinney equation 2, 3 of the canonical forms
K
4 9u q v x u s y 6u u y u , 1Ž . Ž .x x x3u
K
2 5U q v x U s y 8U U y 4U , 2Ž . Ž .x x x3U
Ž .are constructed in adopting a Lie group approach. Equation 1 is found to
be related to a third-order linear equation
c q 4vc q 2v c s 8 Kc 3Ž .x x x x x
*This work was supported by IUAP contract P4r08 funded by the Belgian government.
723
0022-247Xr99 $30.00
Copyright Q 1999 by Academic Press
All rights of reproduction in any form reserved.
NOTE724
via the transformation
1r41
2u s › Log c dx , 4Ž .Hx ž /4
Ž .while the equation 2 occurs in a similarity reduction of a well-known
Ž .C-integrable nonlinear partial differential equation PDE of the Eckhaus
w xtype 4, 5 related to the time-dependent Schrodinger equation.¨
In this note, making algebraic transformations, we establish their links
with two second-order nonlinear differential equations which are lineariz-
w xable and which belong to the Gambier classification 6 of differential
equations possessing the Painleve property. A complementary interest of´
Ž .those transformations consists in the remarks that the first equation G25
w x w xdetermines 8 the Backlund transformation of the Kaup]Kupershmidt 7¨
Ž .nonlinear PDE, while the general solution of the second equation G27
w x Ž .represents 9 , when v x is a constant, the 2-soliton solution of the
potential Korteweg]de Vries equation.
II. LINK WITH THE GAMBIER CLASSIFICATION
In his thesis, Gambier gave a classification of the second-order nonlin-
ear ordinary differential equations of the form
y s F y , y ; x , 5Ž . Ž .x x x
where F is rational in y , algebraic in y, analytic in x, whose generalx
Ž .solution possesses the nice property known as the Painleve property of´
being single-valued around its movable singular points.
1r4 Ž .On the one hand, the transformation u s y made on 1 yields the
equation
3 2 2 2 4yy y y q 4v y q 6 y y q 4 y y 4Ky s 0, K constant, 6Ž .x x x x4
wwhich satisfies the Painleve property. Indeed, it can be identified with 6, p.´
Ž .x14, eq. 25 ,
Y y 3Y 2r 4Y q 3YY r2 q Y 3r4 y q Y q Y 2 r 2 q y rY y q s 0,Ž . Ž .Ž .x x x x x x
7Ž .
NOTE 725
where r and q are two arbitrary functions of x, which possesses the
linearizing transformation
q cx
Y s , z s ,2 c2 z q z y q rq z y rŽ .x x
8Ž .23q q q r q q rx x x x x x
c y c y r q y c y q y c s 0.x x x x x x2 ž /ž /2 q q 2 2 2 qq
The identification Y ’ 4 y, r ’ y4v, q ’ 16K, where K is a constant,
leads to the existence of the following first integral for the third-order
linear equation,
1
2 2 2c c y c q 2vc s 8 K c dx , 9Ž .Hx x x2
and implies for y the linearizing transformation
1
2y s › Log c dx . 10Ž .Hx ž /4
On the other hand, the transformation U s z1r2, which uniformizes the
Ž .general solution of 2 around its movable singular points, yields the
equation
1 2 2 2 4zz y z q 2v x z q 8 z z q 8 z y 2 K s 0, 11Ž . Ž .x x x x2
Ž .bearing the reference number 27 for the particular case n s 2 in the
Gambier list. This equation was found to be bilinearized in the way
1 C 1x 2 2 2z s , C C y C q 2vC y 32 K C s 0, 12Ž .x x x x x x x4 C 2
with the last equation being linearized by derivation into
C q 4vC q 2v C y 64K C s 0. 13Ž .x x x x x x x x
ACKNOWLEDGMENT
It is a pleasure to thank Colin Rogers for encouraging me to perform this work.
REFERENCES
1. C. Rogers, W. Schief, and P. Winternitz, Lie theoretical generalization and discretization
Ž .of the Pinney equation, J. Math. Anal. Appl. 216 1997 , 246]264.
NOTE726
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Ž < < 2 . < < 45. A. Kundu, Comments on the Eckhaus PDE ic q c q 2 c c q c c s 0, In¤erset x x x
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7. D. J. Kaup, On the inverse scattering problem for cubic eigenvalue problems of the class
Ž .c q 6Qc q 6Rc s lc , Stud. Appl. Math. 62 1980 , 189]216.x x x x
8. M. Musette and R. Conte, Backlund transformation of partial differential equations¨
obtained from the Painleve-Gambier classification, I. Kaup-Kupershmidt equation, J.´
Ž .Math. Phys. 39 1998 , 5617]5630.
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On the Links Between Some Generalized Pinney and Nonlinear Gambier Equations 

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On the Links Between Some Generalized Pinney and Nonlinear Gambier Equations

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