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Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I
Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378 www.elsevier.com/locate/cnsns
Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I Nail H. Ibragimov
a,*
, Sergey V. Meleshko
b
a
Department of Mathematics and Science, Research Centre ALGA: Advances in Lie Group Analysis, Blekinge Institute of Technology, Valhallevegen, SE-371 79 Karlskrona, Sweden b School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 3000, Thailand Received 23 November 2005; accepted 23 December 2005 Available online 28 February 2006
Abstract Lie’s group classification of ODEs shows that the second-order equations can possess one, two, three or eight infinitesimal symmetries. The equations with eight symmetries and only these equations can be linearized by a change of variables. Lie showed that the latter equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. Our aim is to provide a similar description of the equations with three symmetries. There are four different types of such equations. We present here the candidates for all four types. We give an invariant test for existence of three symmetries for one of these candidates. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Jr; 02.20.Tw Keywords: Lie’s group classification; Candidates for equations with three symmetries; Invariants
1. Introduction According to Lie’s classification [1] in the complex domain, any ordinary differential equation of the second order y 00 ¼ f ðx; y; y 0 Þ
ð1Þ
admitting a three-dimensional Lie algebra belongs to one of four distinctly different types. Each of these four types is obtained by a change of variables from the following canonical representatives (see, e.g., [2, Section 8.4]):
*
Corresponding author. Tel.: +46 455 385 450; fax: +46 455 385 460. E-mail address: [email protected] (N.H. Ibragimov).
1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.12.012
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
y 00 þ Cy 3 ¼ 0;
ð2Þ
00
y0
00
0ðk2Þ=ðk1Þ
y þ Ce ¼ 0; y þ Cy y 00 þ 2
1371
ð3Þ ¼ 0;
ð4Þ
y 0 þ Cy 03=2 þ y 02 ¼ 0; xy
ð5Þ
where k 5 0, 1/2, 1, 2 in (4), and C = const. Eqs. (2)–(5) admit non-similar three-dimensional Lie algebras L3 spanned by the operators o ; ox o X1 ¼ ; ox o X1 ¼ ; ox
X1 ¼
X 2 ¼ 2x
o o þy ; ox oy
o o þ xy ; ox oy o o X 3 ¼ x þ ðy xÞ ; ox oy o o X 3 ¼ x þ ky ; ox oy
o ; oy o X2 ¼ ; oy X2 ¼
X 3 ¼ x2
ð6Þ ð7Þ ð8Þ
and X1 ¼
o o þ ; ox oy
X2 ¼ x
o o þy ; ox oy
X 3 ¼ x2
o o þ y2 ; ox oy
ð9Þ
respectively (see, e.g., [2, Section 8.4]). 2. Candidates for equations with three symmetries Let us subject each of Eqs. (2)–(5) to the arbitrary change of variables t ¼ uðx; yÞ;
u ¼ wðx; yÞ;
ð10Þ
where t is a new independent variable and u is a new dependent variable. Then we obtain from (2)–(5) the equations of the form u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 ¼ 0;
0
b9 u þ b10 ¼ 0; b11 u0 þ b12 ðk2Þ=ðk1Þ b9 u0 þ b10 00 03 02 0 03 02 0 ¼ 0; u þ b1 u þ 3b2 u þ 3b3 u þ b4 þ ðb5 u þ 3b6 u þ 3b7 u þ b8 Þ b11 u0 þ b12
u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þ exp
ð11Þ ð12Þ ð13Þ
and u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þ
b9 u0 þ b10 b11 u0 þ b12
3=2 ¼ 0;
ð14Þ
respectively, where bi = bi(t, u), i = 1, . . . , 12. Eqs. (11)–(14) are the candidates for the equations with three symmetries. All candidates can be encapsulated in the formula b9 u0 þ b10 u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þf ¼ 0. b11 u0 þ b12 Namely, Eqs. (11)–(14) are obtained by letting f ðzÞ ¼ 0;
f ðzÞ ¼ ez ;
f ðzÞ ¼ zðk2Þ=ðk1Þ ;
f ðzÞ ¼ z3=2 .
Using the usual formula for the transformation of derivatives under the change of variables (10), we obtain the following statement.
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Theorem 1. Any equation of the form u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þf
b9 u0 þ b10 b11 u0 þ b12
¼0
is transformed by the change of variables (10) into an equation of the same form: a9 y 0 þ a10 y 00 þ a1 y 03 þ 3a2 y 02 þ 3a3 y 0 þ a4 þ ða5 y 03 þ 3a6 y 02 þ 3a7 y 0 þ a8 Þf ¼ 0. a11 y 0 þ a12 Here ai and bi are functions of x, y and t, u, respectively, and are connected by a1 ¼ D1 ½uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ; a2 ¼ D1 ½b4 ux u2y þ b3 uy ð2ux wy þ uy wx Þ þ b2 wy ðux wy þ 2uy wx Þ þ b1 wx w2y þ ðux wyy uyy wx 2uxy wy þ 2uy wxy Þ=3; a3 ¼ D1 ½b4 u2x uy þ b3 ux ðux wy þ 2uy wx Þ þ b2 wx ð2ux wy þ uy wx Þ þ b1 w2x wy þ ðuy wxx uxx wy 2uxy wx þ 2ux wxy Þ=3; a4 ¼ D1 ½b4 u3x þ 3b3 u2x wx þ 3b2 ux w2x þ b1 w3x uxx wx þ ux wxx ; a5 ¼ D1 ½b8 u3y þ 3b7 u2y wy þ 3b6 uy w2y þ b5 w3y ; a6 ¼ D1 ½b8 ux u2y þ b7 uy ð2ux wy þ uy wx Þ þ b6 wy ðux wy þ 2uy wx Þ þ b5 wx w2y ; a7 ¼ D1 ½b8 u2x uy þ b7 ux ðux wy þ 2uy wx Þ þ b6 wx ð2ux wy þ uy wx Þ þ b5 w2x wy ; a8 ¼ D1 ½b8 u3x þ 3b7 u2x wx þ 3b6 ux w2x þ b5 w3x ; a9 ¼ b10 uy þ b9 wy ; a10 ¼ b10 ux þ b9 wx ; a11 ¼ b12 uy þ b11 wy ; a12 ¼ b12 ux þ b11 wx ; where D ¼ ðux wy uy wx Þ 6¼ 0 is the Jacobian of the change of variables (10). 3. Equations equivalent to Eq. (2) In this paper, we will dwell on the first candidate, i.e., on equations of the form (11). Other candidates will be considered elsewhere. We know that all equations obtained from Eq. (2), y 00 þ Ky 3 ¼ 0
ðK ¼ const. 6¼ 0Þ;
ð15Þ
by the change of variables (10) are contained in the family of the equations of the form (11): u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 ¼ 0. We also know from Theorem 1 that any Eq. (11) is transformed by the change of variables (10) into an equation of the same form: y 00 þ a1 y 03 þ 3a2 y 02 þ 3a3 y 0 þ a4 ¼ 0;
ð16Þ
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
1373
and that the coefficients of Eqs. (11) and (16) are related by the following equations: a1 ¼ D1 ½uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ; a2 ¼ D1 ½b4 ux u2y þ b3 uy ð2ux wy þ uy wx Þ þ b2 wy ðux wy þ 2uy wx Þ þ b1 wx w2y þ ðux wyy uyy wx 2uxy wy þ 2uy wxy Þ=3; a3 ¼ D1 ½b4 u2x uy þ b3 ux ðux wy þ 2uy wx Þ þ b2 wx ð2ux wy þ uy wx Þ
ð17Þ
þ b1 w2x wy þ ðuy wxx uxx wy 2uxy wx þ 2ux wxy Þ=3; a4 ¼ D1 ½b4 u3x þ 3b3 u2x wx þ 3b2 ux w2x þ b1 w3x uxx wx þ ux wxx . We will use the following information about invariants of Eqs. (11). Lie [1] showed that any second-order equation obtained from the linear equation y00 = 0 by the change of variables (10) belongs to the family of Eqs. (11) and obtained the necessary and sufficient conditions for Eqs. (11) to be equivalent to the linear equation. Lie’s linearization test can be expressed by means of the equations L1 = 0, L2 = 0 (see, e.g., [3]). These equations are invariant with respect to the change of variables (10). Therefore the quantities L1 and L2 are called relative invariants for Eq. (11). They involve the coefficients of Eq. (11) and their derivatives of up to second order and can be readily calculated by means of the infinitesimal method [4]. We will write them, using notation from [5], in the following form: oP11 oP12 þ b4 P22 b2 P11 þ 2b3 P12 ; ou ot oP12 oP22 þ b1 P11 b3 P22 þ 2b2 P12 ; L2 ¼ ou ot L1 ¼
ð18Þ
where P11 ¼ 2ðb23 b2 b4 Þ þ b3t b4u ; P22 ¼ 2ðb22 3b1 b3 Þ þ b1t b2u ; P12 ¼ b2 b3 b1 b4 þ b2t b3u .
ð19Þ
The change of variables (10) converts the quantities (18) into the following relative invariants for Eq. (11): e L 1 ¼ DðL1 ux þ L2 wx Þ;
e L 2 ¼ DðL1 uy þ L2 wy Þ.
ð20Þ
For Eq. (15), the relative invariants (20) are written e L 1 ¼ 12Ky 5 ;
e L 2 ¼ 0.
ð21Þ
Hence, the following statement is valid. Lemma 1. For all Eq. (11) obtained from Eq. (15) by a change of variables, at least one of the relative invariants L1, L2 does not vanish, and the corresponding change of the variables (10) obeys the equation L1 uy þ L2 wy ¼ 0.
ð22Þ
We will use the following relative invariants of higher order given in [5–7]: v5 ¼ L2 ðL1 L2t L2 L1t Þ þ L1 ðL2 L1u L1 L2u Þ b1 L31 þ 3b2 L21 L2 3b3 L1 L22 þ b4 L32 w1 ¼
3 L4 1 ½L1 ðP12 L1
P11 L2 Þ þ
R1 ðL21 Þt
L21 R1t
þ L1 R1 ðb3 L1 b4 L2 Þ;
ð23Þ ð24Þ
and I 2 ¼ 3R1 L1 1 þ L2t L1u ; where R1 ¼ L1 L2t L2 L1t þ b2 L21 2b3 L1 L2 þ b4 L22 .
ð25Þ
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If the relative invariant I2 5 0, there is the set of absolute invariants J 2m ¼ I 2m I m 2
ðm P 1Þ;
where I 2mþ2 ¼ L1
oI 2m oI 2m L2 þ 2mI 2m ðL2t L1u Þ. ou ot
e 1 , eI 2 and e J 4 . For Eq. (15), invoking Eqs. (21), The similar relative invariants for Eq. (16) are denoted by ev 5 , w we obtain: ev 5 ¼ 0;
eI 2 ¼ 60Ky 6 ;
e 1 ¼ 0; w
e J 4 ¼ 4=5.
ð26Þ
Hence, we have the following necessary conditions for Eqs. (11) obtained from Eq. (15) by a change of variables: v5 ¼ 0;
w1 ¼ 0;
I 2 6¼ 0;
J 4 ¼ 4=5.
ð27Þ
We will obtain now the necessary and sufficient conditions. One has to find the conditions for the coefficients b1(t, u), b2(t, u), b3(t, u) and b4(t, u) that guarantee the existence of the functions u(x, y) and w(x, y) such that the change of variables (10) transforms the coefficients of Eq. (11) into a1 ¼ 0;
a2 ¼ 0;
a4 ¼ Ky 3 ;
a3 ¼ 0;
where a1, a2, a3, a4 are defined by formulae (17). Thus, we have to investigate the consistency of the following over-determined system: uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ¼ 0; 3b4 ux u2y þ 6b3 ux uy wy þ 3b3 u2y wx þ þ
3b1 wx w2y
3b4 u2x uy þ
þ
ð28Þ
þ 6b2 uy wx wy
2uxy wy þ ux wyy uyy wx þ 2uy wxy ¼ 0;
3b3 u2x wy
3b1 w2x wy
3b2 ux w2y
þ 6b3 ux uy wx þ 6b2 ux wx wy þ
ð29Þ
3b2 uy w2x
2uxy wx uxx wy þ 2ux wxy þ uy wxx ¼ 0;
y 3 ðb4 u3x þ 3b3 u2x wx þ
3b2 ux w2x
þ
b1 w3x
ð30Þ
uxx wx þ ux wxx Þ KD ¼ 0.
ð31Þ
Remark 1. For Eq. (11) equivalent to Eq. (15) one of the values, either L1 or L2, is not equal to zero. Notice that if L1 = 0 and L2 5 0, then the change t ¼ y;
u ¼ x
leads to the change e L 1 ¼ L2 ;
e L 2 ¼ L1 ;
a1 ¼ b4 ;
a2 ¼ b4 ;
a3 ¼ b2 ;
a4 ¼ b1 .
Further without loss of generality it is assumed that L1 5 0. Lemma 2. Let Eq. (11) which is equivalent to Eq. (15) has L1 5 0. Then uy = 0 if and only if L2 = 0. Proof. The statement follows from Lemma 1. h Theorem 2. Eq. (11) with L2 = 0 is equivalent to Eq. (15) if and only if its coefficients satisfy the following equations: b1 ¼ 0;
b3u 2b2t ¼ 0; 6 3 54 25 3 2 ðL1t Þ þ b3 L1t 15b4 b2 L1 þ b4 I 2 5b4u L1 þ 6b3t L1 þ b23 L1 L ¼ 0; L1tt 5L1 5 5 3I 2 1
ð32Þ ð33Þ
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
6 6 I 2 L1t b3 I 2 ¼ 0; 5L1 5 6 2 I ¼ 0. I 2u 6b2 I 2 þ 5L1 2
I 2t
1375
ð34Þ ð35Þ
Let Eqs. (32)–(35) be satisfied. Then Eq. (11) is mapped to Eq. (15) by the change of variables (10) of the form t ¼ uðxÞ;
u ¼ wðx; yÞ;
ð36Þ
where u(x) is determined by the equation u2x ¼
12KI 2 5L21 y 4
ð37Þ
and w(x, y) by the following integrable system: wy ¼
5L1 ; I 2y
ð38Þ
wxx ¼
1 ½5ux w2x I 2 L21 y 4 ð2I 2 15b2 L1 Þ þ 5ux KL1 ð25L31 12b4 I 22 Þ 24wx KI 22 ðL1t þ 6L1 b3 Þ. 25ux I 2 L31 y 4
ð39Þ
Remark 2. The left-hand sides of Eqs. (32)–(35) are relative invariants with respect to the transformation (36). The equations v5 = 0, w1 = 0 (see (27)) are Eqs. (32), the equation J4 = 4/5 is Eq. (35). In these equations, the variable I2 is given by I2 = 3b2L1 L1u. Theorem 3. Eq. (11) with L2 5 0 is equivalent to Eq. (15) if and only if its coefficients satisfy the following equations: 5L1 I 2t 6I 2 ðL1t b4 L2 þ b3 L1 Þ ¼ 0;
ð40Þ
L21 L2u b4 L32 þ 3b3 L1 L22 3b2 L21 L2 þ b1 L31 þ L1t L22 L1u L1 L2 L2t L1 L2 ¼ 0;
ð41Þ
5L21 I 2u
ð42Þ
6I 2 ð4b4 L22
9b3 L1 L2 þ
5b2 L21
4L1t L2 þ 5L2t L1 I 2 L1 Þ ¼ 0;
15I 2 L1 L1tt 63b24 I 2 L22 þ 126b4 b3 I 2 L1 L2 225b4 b2 I 2 L21 þ 81b4 L1t I 2 L2 90b4 L2t I 2 L1 þ 15b4 I 22 L1 þ 162b23 I 2 L21 þ 9b3 L1t I 2 L1 15b4t I 2 L1 L2 75b4u I 2 L21 þ 90b3t I 2 L21 18L21t I 2 125L41 ¼ 0;
ð43Þ
L21 L2tt þ b24 L32 3b4 b3 L1 L22 þ 3b4 b2 L21 L2 b4 b1 L31 3b4 L1t L22 þ 3b4 L2t L1 L2 þ 3b3 L1t L1 L2 3b3 L2t L21 þ b4t L1 L22 þ b4u L21 L2 3b3t L21 L2 b3u L31 þ 2b2t L31 L1tt L1 L2 þ 2L21t L2 2L1t L2t L1 ¼ 0.
ð44Þ
Let Eqs. (40)–(44) be satisfied. Then Eq. (11) is mapped to Eq. (15) by the change of variables (10) with uy 5 0. The functions u(x, y) and w(x, y) are determined by the following integrable system: L1 uy þ L2 wy ¼ 0;
ð45Þ
5y 4 ðux L1 þ wx L2 Þ2 12KI 2 ¼ 0;
ð46Þ
wy ¼
5L1 ; I 2y
ð47Þ
5L21 wxx ¼ w2x ð15b4 L22 þ 30b3 L1 L2 15b2 L21 þ 10L1t L2 10L2t L1 þ 2I 2 L1 Þ 2 3 þ 24wx D1 y 5 Kð6b4 L2 6b3 L1 L1t Þ þ Ky 4 I 1 2 ð12b4 I 2 þ 25L1 Þ.
ð48Þ
Remark 3. The left-hand sides of Eqs. (40)–(44) are relative invariants with respect to the general change of variables (10). The equations v5 = 0, w1 = 0 (see (27)) are Eqs. (41) and (44), the equation J4 = 4/5 is Eq. (42). Remark 4. The conditions of Theorem 2 are particular cases of the conditions of Theorem 3 provided that L2 = 0.
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N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
4. Proof of Theorem 2 We use the method similar to that employed in [8,9]. Routine calculations were made by means of the system for symbolic calculations Reduce [10]. According to Lemma 2, L2 = 0 implies uy = 0. Since D 5 0, one obtains uxwy 5 0. Eqs. (28)–(31) yield b1 ¼ 0;
1
wyy ¼ 3w2y b2 ;
wxy ¼ ð2ux Þ ðwy uxx 3u2x wy b3 6ux wx wy b2 Þ;
2 2 3 wxx ¼ u1 x wx uxx ux b4 3ux wx b3 3wx b2 þ y wy K.
Equating the mixed derivatives (wxy)x = (wxx)y and (wxy)y = (wyy)x one has u4x ð4b4u 6b3t þ 12b4 b2 9b23 Þ þ 6u3x wx ðb3u 2b2t Þ þ 12u2x y 4 K þ 2ux uxxx 3u2xx ¼ 0
ð49Þ
b3u ¼ 2b2t .
ð50Þ
and
The derivative uxxx is found from Eq. (49). The equation (uxxx)y = 0 gives u2x wy L1 ¼ 12Ky 5 .
ð51Þ
Differentiating this equation with respect to x and y, one obtains u2x ð3b3 L1 2L1t Þ þ 2ux wx ð3b2 L1 L1u Þ 5uxx L1 ¼ 0;
ð52Þ
wy yð3b2 L1 L1u Þ 5L1 ¼ 0.
ð53Þ
Since L1 5 0, one has (3b2L1 L1u) 5 0. Using Eqs. (51) and (53), one finds wy ¼
5L1 ; yð3b2 L1 L1u Þ
u2x ¼
12Kð3b2 L1 L1u Þ 5L21 y 4
Substituting them into (52), one obtains 10ux wx L21 y 4 ð3b2 L1 L1u Þ þ 12ð3b2 L1 L1u ÞKð3b3 L1 2L1t Þ 25uxx L31 y 4 ¼ 0. Since uy = 0, the equation
ðu2x Þy
ð54Þ
¼ 0 gives
5L1 L1uu ¼ 3ð12b22 L21 þ 3b2 L1u L1 þ 5b2u L21 þ 2L21u Þ.
ð55Þ
Using Eq. (54) one can find the derivative uxx: uxx ¼ 2ð3b2 L1 L1u Þð5ux wx L21 y 4 þ 6Kð3b3 L1 2L1t ÞÞ=ð25L31 y 4 Þ. By considering the equations ðu2x Þxx 2ðu2xx þ ux uxxx Þ ¼ 0;
ð56Þ
ðu2x Þx
ð57Þ
2ux uxx ¼ 0;
one can obtain conditions for the coefficients b1, b2, b3, b4. For example, Eq. (57) gives 5L1 L1tu ¼ 3ð6b3 b2 L21 þ 2b3 L1u L1 b2 L1t L1 þ 5b2t L21 þ 2L1t L1u Þ. Invoking that (uxx)y 0 and using the equation 2ux uxx ðu2x Þx 0 one obtains L1tt ¼ 3L1t ð2L1t b3 L1 Þ=ð5L1 Þ þ 15b4 b2 L1 b4 I 2 þ 5b4u L1 6b3t L1 54b23 L1 =5 þ 25L31 =ð3I 2 Þ; where the relative invariant I2 (25) becomes I 2 ¼ 3b2 L1 L1u . Eqs. (56) and (wy)xx (wxx)y = 0 are satisfied. Summing up the above results, we complete the proof of Theorem 2. h
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
1377
5. Proof of Theorem 3 We deal now with the case L2 5 0, and hence uy 5 0. Using Eqs. (28)–(31), one finds the derivatives wyy, wxy, wxx, uxx: 3 2 2 wyy ¼ u1 y ðuyy wy wy b1 Þ 3wy b2 3wy b3 uy b4 uy ;
2wxy ¼ ðu2y Þ1 ð2uxy wy uy Duyy þ b1 w2y ðux wy 3wx uy ÞÞ 3b3 ðux wy þ wx uy Þ 2ux b4 uy 6wx wy b2 ; 2 2 wxx ¼ u3 y ð2uxy ux wy uy þ 2uxy wx uy þ uxx wy uy þ Dux uyy
b1 wy ðu2x w2y 3ux wx wy uy þ 3w2x u2y ÞÞ u2x b4 3ux wx b3 3w2x b2 ; 2 2 3 uxx ¼ ux u2 y ð2uxy uy ux uyy þ ux wy b1 2wx wy b1 uy Þ þ wx b1 þ uy Ky .
Furthermore, the equations (wxy)x = (wxx)y and (wxy)y = (wyy)x determine the derivatives uxyy and uyyy, respectively, 4 2 3 2 2 2 2uxyy ¼ u2 y ð4uxy uyy uy ux uyy þ ux wy b1 4wx wy b1 uy Þ þ 2ux wy ðb1t 3b3 b1 Þ
þ 4ux wy uy ð2b2t b3u 2b4 b1 Þ þ ux u2y ð6b3t 4b4u 12b4 b2 þ 9b23 Þ þ 2wx w2y ðb1u 6b2 b1 Þ þ 4wx wy uy ðb1t 3b3 b1 Þ þ 2wx u2y ð2b2t b3u 2b4 b1 Þ; 4 2 3 2 2 2uyyy ¼ u1 y ð3uyy 3wy b1 Þ þ 2wy ð6b2 b1 þ b1u Þ þ 6wy uy ð3b3 b1 þ b1t Þ
þ 6y u2y ð2b2t 2b4 b1 b3u Þ þ u3y ð6b3t 12b4 b2 þ 9b23 4b4u Þ. The equations (uxyy)y = (uyyy)x and (uxyy)y = (uxx)yy yield: y 5 DðL2 ðux wy þ wx uy Þ þ 2L1 ux uy Þ 12uy K ¼ 0.
ð58Þ
Eq. (22) and the condition D 5 0 yield that L2wy 5 0 and L1ux + L2wx 5 0. By virtue of Eq. (22), Eq. (58) becomes 2
y 5 wy ðux L1 þ wx L2 Þ 12KL1 ¼ 0.
ð59Þ
Differentiating (22) with respect to x and y, and substituting the derivatives wxx, wxy, wyy, and uxx, one obtains uyy L21 ðux L1 þ wx L2 Þ þ w2y ux ð2b4 L32 3b3 L1 L22 þ b1 L31 2L1t L22 þ 2L2t L1 L2 Þ þ w2y wx ð3b3 L32 6b2 L1 L22 þ 3b1 L21 L2 2L1u L22 þ 2L2u L1 L2 Þ ¼ 0; b4 L32 þ 3b3 L1 L22 3b2 L21 L2 þ b1 L31 þ L1t L22 L1u L1 L2 L2t L1 L2 þ L2u L21 ¼ 0.
ð60Þ ð61Þ
Eq. (60) yields: 1 2 3 2 3 2 uyy ¼ L2 1 ðux L1 þ wx L2 Þ ðux wy ð2b4 L2 3b3 L1 L2 þ b1 L1 2L1t L2 þ 2L2t L1 L2 Þ
þ wx w2y L2 ð3b3 L22 6b2 L1 L2 þ 3b1 L21 2L1u L2 þ 2L2u L1 ÞÞ. Furthermore, Eq. (61) determines L2u: 3 2 2 3 2 L2u ¼ L2 1 ðb4 L2 3b3 L1 L2 þ 3b2 L1 L2 b1 L1 L1t L2 þ L1u L1 L2 þ L2t L1 L2 Þ.
ð62Þ
Using Eq. (59), one obtains the equation (uxy)x (uxx)y = 0. Notice that the equations (uy)yy uyyy = 0 and (uy)xy uxyy = 0 are also satisfied. Now we find u2x from Eq. (59) and substitute it into the equation uxyy (uyy)x = 0. This leads to the following expression for L2tt: 2 3 2 2 3 2 L2tt ¼ L2 1 ðb4 L2 þ 3b4 b3 L1 L2 3b4 b2 L1 L2 þ b4 b1 L1 þ 3b4 L1t L2 3b4 L2t L1 L2 3b3 L1t L1 L2
þ 3b3 L2t L21 b4t L1 L22 b4u L21 L2 þ 3b3t L21 L2 þ b3u L31 2b2t L31 þ L1tt L1 L2 2L21t L2 þ 2L1t L2t L1 Þ.
ð63Þ
1378
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
Differentiating (59) with respect to x and y, one finds uxy and wy, respectively, 1
wy ¼ 5L1 ðI 2 yÞ ; uxy ¼ 12Kð5L21 y 5 ðux L1 þ wx L2 ÞÞ1 ½6L1t L2 6b4 L22 þ 6b3 L1 L2 5L2t L1 þ wx ðI 2 L21 yÞ1 ½10b4 L32 15b3 L1 L22 þ 5b1 L31 10L1t L22 þ 10L2t L1 L2 I 2 L1 L2 . The equations (wy)y = wyy and ðu2x Þx ¼ 2ux uxx yield: 5L21 I 2u ¼ 6I 2 ð4b4 L22 9b3 L1 L2 þ 5b2 L21 4L1t L2 þ 5L2t L1 I 2 L1 Þ; 5L1 I 2t ¼ 6I 2 ðL1t b4 L2 þ b3 L1 Þ.
ð64Þ ð65Þ
Now the equations (wy)x = wxy and (uxy)y = (uyy)x are satisfied. The equation (uxy)x = (uxx)y yields: L1tt ¼ ð63b24 I 2 L22 126b4 b3 I 2 L1 L2 þ 225b4 b2 I 2 L21 81b4 L1t I 2 L2 þ 90b4 L2t I 2 L1 15b4 I 22 L1 162b23 I 2 L21 9b3 L1t I 2 L1 þ 15b4t I 2 L1 L2 þ 75b4u I 2 L21 90b3t I 2 L21 þ 18L21t I 2 þ 125L41 Þ=ð15I 2 L1 Þ. Notice that the equations (wy)x = wxy and (wy)y = wyy are satisfied. Summing up the above results, we complete the proof of Theorem 3.
ð66Þ
h
Acknowledgement One of the authors (SVM) is thankful to R. Conte for fruitful discussions. References [1] Lie S. Klassifikation und Integration von gewo¨nlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestaten. III, Archiv for Matematik og Naturvidenskab, 8, Heft 4:371–458, 1883. Reprinted in Lie’s Ges Abhandl, vol. 5, 1924, paper XIV, p. 362–427. [2] Ibragimov NH, editorCRC Handbook of Lie group analysis of differential equations. Symmetries, exact solutions and conservation laws, vol. 1. Boca Roton: CRC Press Inc.; 1994. [3] Ibragimov NH. Elementary Lie group analysis and ordinary differential equations. Chichester: John Wiley & Sons; 1999. [4] Ibragimov NH. Invariants of a remarkable family of nonlinear equations. Nonlinear Dyn 2002;30(2):155–66. [5] Liouville R. Sur les invariants de certaines equations differentielles et sur leurs applications. J Ecole Polytech 1889;59:7–76. ´ quation Diffe´rentielle Ordinaire du Second Ordre y00 = x(x, y, y 0 ), [6] Tresse, AM. De´termination des Invariants Ponctuels de lE Preisschriften der fu¨rstlichen Jablonowski’schen Geselschaft XXXII, Leipzig, S. Herzel, 1896. [7] Babich MV, Bordag LA. Projective differential geometrical structure of the Painleve equations. J Differen Equat 1999;157:452–85. [8] Ibragimov NH, Meleshko SV. Linearization of third-order ordinary differential equations by point transformations. Arch ALGA 2004;1:71–93. [9] Ibragimov NH, Meleshko SV. Linearization of third-order ordinary differential equations by point and contact transformations. J Math Anal Appl 2005;308(1):266–89. [10] Hearn AC. REDUCE Users Manual, ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987.
Invariants and invariant description of second-order ODEs with three infinitesimal symmetries. I Nail H. Ibragimov
a,*
, Sergey V. Meleshko
b
a
Department of Mathematics and Science, Research Centre ALGA: Advances in Lie Group Analysis, Blekinge Institute of Technology, Valhallevegen, SE-371 79 Karlskrona, Sweden b School of Mathematics, Suranaree University of Technology, Nakhon Ratchasima, 3000, Thailand Received 23 November 2005; accepted 23 December 2005 Available online 28 February 2006
Abstract Lie’s group classification of ODEs shows that the second-order equations can possess one, two, three or eight infinitesimal symmetries. The equations with eight symmetries and only these equations can be linearized by a change of variables. Lie showed that the latter equations are at most cubic in the first derivative and gave a convenient invariant description of all linearizable equations. Our aim is to provide a similar description of the equations with three symmetries. There are four different types of such equations. We present here the candidates for all four types. We give an invariant test for existence of three symmetries for one of these candidates. Ó 2006 Elsevier B.V. All rights reserved. PACS: 02.30.Jr; 02.20.Tw Keywords: Lie’s group classification; Candidates for equations with three symmetries; Invariants
1. Introduction According to Lie’s classification [1] in the complex domain, any ordinary differential equation of the second order y 00 ¼ f ðx; y; y 0 Þ
ð1Þ
admitting a three-dimensional Lie algebra belongs to one of four distinctly different types. Each of these four types is obtained by a change of variables from the following canonical representatives (see, e.g., [2, Section 8.4]):
*
Corresponding author. Tel.: +46 455 385 450; fax: +46 455 385 460. E-mail address: [email protected] (N.H. Ibragimov).
1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2005.12.012
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
y 00 þ Cy 3 ¼ 0;
ð2Þ
00
y0
00
0ðk2Þ=ðk1Þ
y þ Ce ¼ 0; y þ Cy y 00 þ 2
1371
ð3Þ ¼ 0;
ð4Þ
y 0 þ Cy 03=2 þ y 02 ¼ 0; xy
ð5Þ
where k 5 0, 1/2, 1, 2 in (4), and C = const. Eqs. (2)–(5) admit non-similar three-dimensional Lie algebras L3 spanned by the operators o ; ox o X1 ¼ ; ox o X1 ¼ ; ox
X1 ¼
X 2 ¼ 2x
o o þy ; ox oy
o o þ xy ; ox oy o o X 3 ¼ x þ ðy xÞ ; ox oy o o X 3 ¼ x þ ky ; ox oy
o ; oy o X2 ¼ ; oy X2 ¼
X 3 ¼ x2
ð6Þ ð7Þ ð8Þ
and X1 ¼
o o þ ; ox oy
X2 ¼ x
o o þy ; ox oy
X 3 ¼ x2
o o þ y2 ; ox oy
ð9Þ
respectively (see, e.g., [2, Section 8.4]). 2. Candidates for equations with three symmetries Let us subject each of Eqs. (2)–(5) to the arbitrary change of variables t ¼ uðx; yÞ;
u ¼ wðx; yÞ;
ð10Þ
where t is a new independent variable and u is a new dependent variable. Then we obtain from (2)–(5) the equations of the form u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 ¼ 0;
0
b9 u þ b10 ¼ 0; b11 u0 þ b12 ðk2Þ=ðk1Þ b9 u0 þ b10 00 03 02 0 03 02 0 ¼ 0; u þ b1 u þ 3b2 u þ 3b3 u þ b4 þ ðb5 u þ 3b6 u þ 3b7 u þ b8 Þ b11 u0 þ b12
u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þ exp
ð11Þ ð12Þ ð13Þ
and u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þ
b9 u0 þ b10 b11 u0 þ b12
3=2 ¼ 0;
ð14Þ
respectively, where bi = bi(t, u), i = 1, . . . , 12. Eqs. (11)–(14) are the candidates for the equations with three symmetries. All candidates can be encapsulated in the formula b9 u0 þ b10 u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þf ¼ 0. b11 u0 þ b12 Namely, Eqs. (11)–(14) are obtained by letting f ðzÞ ¼ 0;
f ðzÞ ¼ ez ;
f ðzÞ ¼ zðk2Þ=ðk1Þ ;
f ðzÞ ¼ z3=2 .
Using the usual formula for the transformation of derivatives under the change of variables (10), we obtain the following statement.
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N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
Theorem 1. Any equation of the form u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 þ ðb5 u03 þ 3b6 u02 þ 3b7 u0 þ b8 Þf
b9 u0 þ b10 b11 u0 þ b12
¼0
is transformed by the change of variables (10) into an equation of the same form: a9 y 0 þ a10 y 00 þ a1 y 03 þ 3a2 y 02 þ 3a3 y 0 þ a4 þ ða5 y 03 þ 3a6 y 02 þ 3a7 y 0 þ a8 Þf ¼ 0. a11 y 0 þ a12 Here ai and bi are functions of x, y and t, u, respectively, and are connected by a1 ¼ D1 ½uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ; a2 ¼ D1 ½b4 ux u2y þ b3 uy ð2ux wy þ uy wx Þ þ b2 wy ðux wy þ 2uy wx Þ þ b1 wx w2y þ ðux wyy uyy wx 2uxy wy þ 2uy wxy Þ=3; a3 ¼ D1 ½b4 u2x uy þ b3 ux ðux wy þ 2uy wx Þ þ b2 wx ð2ux wy þ uy wx Þ þ b1 w2x wy þ ðuy wxx uxx wy 2uxy wx þ 2ux wxy Þ=3; a4 ¼ D1 ½b4 u3x þ 3b3 u2x wx þ 3b2 ux w2x þ b1 w3x uxx wx þ ux wxx ; a5 ¼ D1 ½b8 u3y þ 3b7 u2y wy þ 3b6 uy w2y þ b5 w3y ; a6 ¼ D1 ½b8 ux u2y þ b7 uy ð2ux wy þ uy wx Þ þ b6 wy ðux wy þ 2uy wx Þ þ b5 wx w2y ; a7 ¼ D1 ½b8 u2x uy þ b7 ux ðux wy þ 2uy wx Þ þ b6 wx ð2ux wy þ uy wx Þ þ b5 w2x wy ; a8 ¼ D1 ½b8 u3x þ 3b7 u2x wx þ 3b6 ux w2x þ b5 w3x ; a9 ¼ b10 uy þ b9 wy ; a10 ¼ b10 ux þ b9 wx ; a11 ¼ b12 uy þ b11 wy ; a12 ¼ b12 ux þ b11 wx ; where D ¼ ðux wy uy wx Þ 6¼ 0 is the Jacobian of the change of variables (10). 3. Equations equivalent to Eq. (2) In this paper, we will dwell on the first candidate, i.e., on equations of the form (11). Other candidates will be considered elsewhere. We know that all equations obtained from Eq. (2), y 00 þ Ky 3 ¼ 0
ðK ¼ const. 6¼ 0Þ;
ð15Þ
by the change of variables (10) are contained in the family of the equations of the form (11): u00 þ b1 u03 þ 3b2 u02 þ 3b3 u0 þ b4 ¼ 0. We also know from Theorem 1 that any Eq. (11) is transformed by the change of variables (10) into an equation of the same form: y 00 þ a1 y 03 þ 3a2 y 02 þ 3a3 y 0 þ a4 ¼ 0;
ð16Þ
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
1373
and that the coefficients of Eqs. (11) and (16) are related by the following equations: a1 ¼ D1 ½uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ; a2 ¼ D1 ½b4 ux u2y þ b3 uy ð2ux wy þ uy wx Þ þ b2 wy ðux wy þ 2uy wx Þ þ b1 wx w2y þ ðux wyy uyy wx 2uxy wy þ 2uy wxy Þ=3; a3 ¼ D1 ½b4 u2x uy þ b3 ux ðux wy þ 2uy wx Þ þ b2 wx ð2ux wy þ uy wx Þ
ð17Þ
þ b1 w2x wy þ ðuy wxx uxx wy 2uxy wx þ 2ux wxy Þ=3; a4 ¼ D1 ½b4 u3x þ 3b3 u2x wx þ 3b2 ux w2x þ b1 w3x uxx wx þ ux wxx . We will use the following information about invariants of Eqs. (11). Lie [1] showed that any second-order equation obtained from the linear equation y00 = 0 by the change of variables (10) belongs to the family of Eqs. (11) and obtained the necessary and sufficient conditions for Eqs. (11) to be equivalent to the linear equation. Lie’s linearization test can be expressed by means of the equations L1 = 0, L2 = 0 (see, e.g., [3]). These equations are invariant with respect to the change of variables (10). Therefore the quantities L1 and L2 are called relative invariants for Eq. (11). They involve the coefficients of Eq. (11) and their derivatives of up to second order and can be readily calculated by means of the infinitesimal method [4]. We will write them, using notation from [5], in the following form: oP11 oP12 þ b4 P22 b2 P11 þ 2b3 P12 ; ou ot oP12 oP22 þ b1 P11 b3 P22 þ 2b2 P12 ; L2 ¼ ou ot L1 ¼
ð18Þ
where P11 ¼ 2ðb23 b2 b4 Þ þ b3t b4u ; P22 ¼ 2ðb22 3b1 b3 Þ þ b1t b2u ; P12 ¼ b2 b3 b1 b4 þ b2t b3u .
ð19Þ
The change of variables (10) converts the quantities (18) into the following relative invariants for Eq. (11): e L 1 ¼ DðL1 ux þ L2 wx Þ;
e L 2 ¼ DðL1 uy þ L2 wy Þ.
ð20Þ
For Eq. (15), the relative invariants (20) are written e L 1 ¼ 12Ky 5 ;
e L 2 ¼ 0.
ð21Þ
Hence, the following statement is valid. Lemma 1. For all Eq. (11) obtained from Eq. (15) by a change of variables, at least one of the relative invariants L1, L2 does not vanish, and the corresponding change of the variables (10) obeys the equation L1 uy þ L2 wy ¼ 0.
ð22Þ
We will use the following relative invariants of higher order given in [5–7]: v5 ¼ L2 ðL1 L2t L2 L1t Þ þ L1 ðL2 L1u L1 L2u Þ b1 L31 þ 3b2 L21 L2 3b3 L1 L22 þ b4 L32 w1 ¼
3 L4 1 ½L1 ðP12 L1
P11 L2 Þ þ
R1 ðL21 Þt
L21 R1t
þ L1 R1 ðb3 L1 b4 L2 Þ;
ð23Þ ð24Þ
and I 2 ¼ 3R1 L1 1 þ L2t L1u ; where R1 ¼ L1 L2t L2 L1t þ b2 L21 2b3 L1 L2 þ b4 L22 .
ð25Þ
1374
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
If the relative invariant I2 5 0, there is the set of absolute invariants J 2m ¼ I 2m I m 2
ðm P 1Þ;
where I 2mþ2 ¼ L1
oI 2m oI 2m L2 þ 2mI 2m ðL2t L1u Þ. ou ot
e 1 , eI 2 and e J 4 . For Eq. (15), invoking Eqs. (21), The similar relative invariants for Eq. (16) are denoted by ev 5 , w we obtain: ev 5 ¼ 0;
eI 2 ¼ 60Ky 6 ;
e 1 ¼ 0; w
e J 4 ¼ 4=5.
ð26Þ
Hence, we have the following necessary conditions for Eqs. (11) obtained from Eq. (15) by a change of variables: v5 ¼ 0;
w1 ¼ 0;
I 2 6¼ 0;
J 4 ¼ 4=5.
ð27Þ
We will obtain now the necessary and sufficient conditions. One has to find the conditions for the coefficients b1(t, u), b2(t, u), b3(t, u) and b4(t, u) that guarantee the existence of the functions u(x, y) and w(x, y) such that the change of variables (10) transforms the coefficients of Eq. (11) into a1 ¼ 0;
a2 ¼ 0;
a4 ¼ Ky 3 ;
a3 ¼ 0;
where a1, a2, a3, a4 are defined by formulae (17). Thus, we have to investigate the consistency of the following over-determined system: uy wyy uyy wy þ b4 u3y þ 3b3 u2y wy þ 3b2 uy w2y þ b1 w3y ¼ 0; 3b4 ux u2y þ 6b3 ux uy wy þ 3b3 u2y wx þ þ
3b1 wx w2y
3b4 u2x uy þ
þ
ð28Þ
þ 6b2 uy wx wy
2uxy wy þ ux wyy uyy wx þ 2uy wxy ¼ 0;
3b3 u2x wy
3b1 w2x wy
3b2 ux w2y
þ 6b3 ux uy wx þ 6b2 ux wx wy þ
ð29Þ
3b2 uy w2x
2uxy wx uxx wy þ 2ux wxy þ uy wxx ¼ 0;
y 3 ðb4 u3x þ 3b3 u2x wx þ
3b2 ux w2x
þ
b1 w3x
ð30Þ
uxx wx þ ux wxx Þ KD ¼ 0.
ð31Þ
Remark 1. For Eq. (11) equivalent to Eq. (15) one of the values, either L1 or L2, is not equal to zero. Notice that if L1 = 0 and L2 5 0, then the change t ¼ y;
u ¼ x
leads to the change e L 1 ¼ L2 ;
e L 2 ¼ L1 ;
a1 ¼ b4 ;
a2 ¼ b4 ;
a3 ¼ b2 ;
a4 ¼ b1 .
Further without loss of generality it is assumed that L1 5 0. Lemma 2. Let Eq. (11) which is equivalent to Eq. (15) has L1 5 0. Then uy = 0 if and only if L2 = 0. Proof. The statement follows from Lemma 1. h Theorem 2. Eq. (11) with L2 = 0 is equivalent to Eq. (15) if and only if its coefficients satisfy the following equations: b1 ¼ 0;
b3u 2b2t ¼ 0; 6 3 54 25 3 2 ðL1t Þ þ b3 L1t 15b4 b2 L1 þ b4 I 2 5b4u L1 þ 6b3t L1 þ b23 L1 L ¼ 0; L1tt 5L1 5 5 3I 2 1
ð32Þ ð33Þ
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
6 6 I 2 L1t b3 I 2 ¼ 0; 5L1 5 6 2 I ¼ 0. I 2u 6b2 I 2 þ 5L1 2
I 2t
1375
ð34Þ ð35Þ
Let Eqs. (32)–(35) be satisfied. Then Eq. (11) is mapped to Eq. (15) by the change of variables (10) of the form t ¼ uðxÞ;
u ¼ wðx; yÞ;
ð36Þ
where u(x) is determined by the equation u2x ¼
12KI 2 5L21 y 4
ð37Þ
and w(x, y) by the following integrable system: wy ¼
5L1 ; I 2y
ð38Þ
wxx ¼
1 ½5ux w2x I 2 L21 y 4 ð2I 2 15b2 L1 Þ þ 5ux KL1 ð25L31 12b4 I 22 Þ 24wx KI 22 ðL1t þ 6L1 b3 Þ. 25ux I 2 L31 y 4
ð39Þ
Remark 2. The left-hand sides of Eqs. (32)–(35) are relative invariants with respect to the transformation (36). The equations v5 = 0, w1 = 0 (see (27)) are Eqs. (32), the equation J4 = 4/5 is Eq. (35). In these equations, the variable I2 is given by I2 = 3b2L1 L1u. Theorem 3. Eq. (11) with L2 5 0 is equivalent to Eq. (15) if and only if its coefficients satisfy the following equations: 5L1 I 2t 6I 2 ðL1t b4 L2 þ b3 L1 Þ ¼ 0;
ð40Þ
L21 L2u b4 L32 þ 3b3 L1 L22 3b2 L21 L2 þ b1 L31 þ L1t L22 L1u L1 L2 L2t L1 L2 ¼ 0;
ð41Þ
5L21 I 2u
ð42Þ
6I 2 ð4b4 L22
9b3 L1 L2 þ
5b2 L21
4L1t L2 þ 5L2t L1 I 2 L1 Þ ¼ 0;
15I 2 L1 L1tt 63b24 I 2 L22 þ 126b4 b3 I 2 L1 L2 225b4 b2 I 2 L21 þ 81b4 L1t I 2 L2 90b4 L2t I 2 L1 þ 15b4 I 22 L1 þ 162b23 I 2 L21 þ 9b3 L1t I 2 L1 15b4t I 2 L1 L2 75b4u I 2 L21 þ 90b3t I 2 L21 18L21t I 2 125L41 ¼ 0;
ð43Þ
L21 L2tt þ b24 L32 3b4 b3 L1 L22 þ 3b4 b2 L21 L2 b4 b1 L31 3b4 L1t L22 þ 3b4 L2t L1 L2 þ 3b3 L1t L1 L2 3b3 L2t L21 þ b4t L1 L22 þ b4u L21 L2 3b3t L21 L2 b3u L31 þ 2b2t L31 L1tt L1 L2 þ 2L21t L2 2L1t L2t L1 ¼ 0.
ð44Þ
Let Eqs. (40)–(44) be satisfied. Then Eq. (11) is mapped to Eq. (15) by the change of variables (10) with uy 5 0. The functions u(x, y) and w(x, y) are determined by the following integrable system: L1 uy þ L2 wy ¼ 0;
ð45Þ
5y 4 ðux L1 þ wx L2 Þ2 12KI 2 ¼ 0;
ð46Þ
wy ¼
5L1 ; I 2y
ð47Þ
5L21 wxx ¼ w2x ð15b4 L22 þ 30b3 L1 L2 15b2 L21 þ 10L1t L2 10L2t L1 þ 2I 2 L1 Þ 2 3 þ 24wx D1 y 5 Kð6b4 L2 6b3 L1 L1t Þ þ Ky 4 I 1 2 ð12b4 I 2 þ 25L1 Þ.
ð48Þ
Remark 3. The left-hand sides of Eqs. (40)–(44) are relative invariants with respect to the general change of variables (10). The equations v5 = 0, w1 = 0 (see (27)) are Eqs. (41) and (44), the equation J4 = 4/5 is Eq. (42). Remark 4. The conditions of Theorem 2 are particular cases of the conditions of Theorem 3 provided that L2 = 0.
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N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
4. Proof of Theorem 2 We use the method similar to that employed in [8,9]. Routine calculations were made by means of the system for symbolic calculations Reduce [10]. According to Lemma 2, L2 = 0 implies uy = 0. Since D 5 0, one obtains uxwy 5 0. Eqs. (28)–(31) yield b1 ¼ 0;
1
wyy ¼ 3w2y b2 ;
wxy ¼ ð2ux Þ ðwy uxx 3u2x wy b3 6ux wx wy b2 Þ;
2 2 3 wxx ¼ u1 x wx uxx ux b4 3ux wx b3 3wx b2 þ y wy K.
Equating the mixed derivatives (wxy)x = (wxx)y and (wxy)y = (wyy)x one has u4x ð4b4u 6b3t þ 12b4 b2 9b23 Þ þ 6u3x wx ðb3u 2b2t Þ þ 12u2x y 4 K þ 2ux uxxx 3u2xx ¼ 0
ð49Þ
b3u ¼ 2b2t .
ð50Þ
and
The derivative uxxx is found from Eq. (49). The equation (uxxx)y = 0 gives u2x wy L1 ¼ 12Ky 5 .
ð51Þ
Differentiating this equation with respect to x and y, one obtains u2x ð3b3 L1 2L1t Þ þ 2ux wx ð3b2 L1 L1u Þ 5uxx L1 ¼ 0;
ð52Þ
wy yð3b2 L1 L1u Þ 5L1 ¼ 0.
ð53Þ
Since L1 5 0, one has (3b2L1 L1u) 5 0. Using Eqs. (51) and (53), one finds wy ¼
5L1 ; yð3b2 L1 L1u Þ
u2x ¼
12Kð3b2 L1 L1u Þ 5L21 y 4
Substituting them into (52), one obtains 10ux wx L21 y 4 ð3b2 L1 L1u Þ þ 12ð3b2 L1 L1u ÞKð3b3 L1 2L1t Þ 25uxx L31 y 4 ¼ 0. Since uy = 0, the equation
ðu2x Þy
ð54Þ
¼ 0 gives
5L1 L1uu ¼ 3ð12b22 L21 þ 3b2 L1u L1 þ 5b2u L21 þ 2L21u Þ.
ð55Þ
Using Eq. (54) one can find the derivative uxx: uxx ¼ 2ð3b2 L1 L1u Þð5ux wx L21 y 4 þ 6Kð3b3 L1 2L1t ÞÞ=ð25L31 y 4 Þ. By considering the equations ðu2x Þxx 2ðu2xx þ ux uxxx Þ ¼ 0;
ð56Þ
ðu2x Þx
ð57Þ
2ux uxx ¼ 0;
one can obtain conditions for the coefficients b1, b2, b3, b4. For example, Eq. (57) gives 5L1 L1tu ¼ 3ð6b3 b2 L21 þ 2b3 L1u L1 b2 L1t L1 þ 5b2t L21 þ 2L1t L1u Þ. Invoking that (uxx)y 0 and using the equation 2ux uxx ðu2x Þx 0 one obtains L1tt ¼ 3L1t ð2L1t b3 L1 Þ=ð5L1 Þ þ 15b4 b2 L1 b4 I 2 þ 5b4u L1 6b3t L1 54b23 L1 =5 þ 25L31 =ð3I 2 Þ; where the relative invariant I2 (25) becomes I 2 ¼ 3b2 L1 L1u . Eqs. (56) and (wy)xx (wxx)y = 0 are satisfied. Summing up the above results, we complete the proof of Theorem 2. h
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
1377
5. Proof of Theorem 3 We deal now with the case L2 5 0, and hence uy 5 0. Using Eqs. (28)–(31), one finds the derivatives wyy, wxy, wxx, uxx: 3 2 2 wyy ¼ u1 y ðuyy wy wy b1 Þ 3wy b2 3wy b3 uy b4 uy ;
2wxy ¼ ðu2y Þ1 ð2uxy wy uy Duyy þ b1 w2y ðux wy 3wx uy ÞÞ 3b3 ðux wy þ wx uy Þ 2ux b4 uy 6wx wy b2 ; 2 2 wxx ¼ u3 y ð2uxy ux wy uy þ 2uxy wx uy þ uxx wy uy þ Dux uyy
b1 wy ðu2x w2y 3ux wx wy uy þ 3w2x u2y ÞÞ u2x b4 3ux wx b3 3w2x b2 ; 2 2 3 uxx ¼ ux u2 y ð2uxy uy ux uyy þ ux wy b1 2wx wy b1 uy Þ þ wx b1 þ uy Ky .
Furthermore, the equations (wxy)x = (wxx)y and (wxy)y = (wyy)x determine the derivatives uxyy and uyyy, respectively, 4 2 3 2 2 2 2uxyy ¼ u2 y ð4uxy uyy uy ux uyy þ ux wy b1 4wx wy b1 uy Þ þ 2ux wy ðb1t 3b3 b1 Þ
þ 4ux wy uy ð2b2t b3u 2b4 b1 Þ þ ux u2y ð6b3t 4b4u 12b4 b2 þ 9b23 Þ þ 2wx w2y ðb1u 6b2 b1 Þ þ 4wx wy uy ðb1t 3b3 b1 Þ þ 2wx u2y ð2b2t b3u 2b4 b1 Þ; 4 2 3 2 2 2uyyy ¼ u1 y ð3uyy 3wy b1 Þ þ 2wy ð6b2 b1 þ b1u Þ þ 6wy uy ð3b3 b1 þ b1t Þ
þ 6y u2y ð2b2t 2b4 b1 b3u Þ þ u3y ð6b3t 12b4 b2 þ 9b23 4b4u Þ. The equations (uxyy)y = (uyyy)x and (uxyy)y = (uxx)yy yield: y 5 DðL2 ðux wy þ wx uy Þ þ 2L1 ux uy Þ 12uy K ¼ 0.
ð58Þ
Eq. (22) and the condition D 5 0 yield that L2wy 5 0 and L1ux + L2wx 5 0. By virtue of Eq. (22), Eq. (58) becomes 2
y 5 wy ðux L1 þ wx L2 Þ 12KL1 ¼ 0.
ð59Þ
Differentiating (22) with respect to x and y, and substituting the derivatives wxx, wxy, wyy, and uxx, one obtains uyy L21 ðux L1 þ wx L2 Þ þ w2y ux ð2b4 L32 3b3 L1 L22 þ b1 L31 2L1t L22 þ 2L2t L1 L2 Þ þ w2y wx ð3b3 L32 6b2 L1 L22 þ 3b1 L21 L2 2L1u L22 þ 2L2u L1 L2 Þ ¼ 0; b4 L32 þ 3b3 L1 L22 3b2 L21 L2 þ b1 L31 þ L1t L22 L1u L1 L2 L2t L1 L2 þ L2u L21 ¼ 0.
ð60Þ ð61Þ
Eq. (60) yields: 1 2 3 2 3 2 uyy ¼ L2 1 ðux L1 þ wx L2 Þ ðux wy ð2b4 L2 3b3 L1 L2 þ b1 L1 2L1t L2 þ 2L2t L1 L2 Þ
þ wx w2y L2 ð3b3 L22 6b2 L1 L2 þ 3b1 L21 2L1u L2 þ 2L2u L1 ÞÞ. Furthermore, Eq. (61) determines L2u: 3 2 2 3 2 L2u ¼ L2 1 ðb4 L2 3b3 L1 L2 þ 3b2 L1 L2 b1 L1 L1t L2 þ L1u L1 L2 þ L2t L1 L2 Þ.
ð62Þ
Using Eq. (59), one obtains the equation (uxy)x (uxx)y = 0. Notice that the equations (uy)yy uyyy = 0 and (uy)xy uxyy = 0 are also satisfied. Now we find u2x from Eq. (59) and substitute it into the equation uxyy (uyy)x = 0. This leads to the following expression for L2tt: 2 3 2 2 3 2 L2tt ¼ L2 1 ðb4 L2 þ 3b4 b3 L1 L2 3b4 b2 L1 L2 þ b4 b1 L1 þ 3b4 L1t L2 3b4 L2t L1 L2 3b3 L1t L1 L2
þ 3b3 L2t L21 b4t L1 L22 b4u L21 L2 þ 3b3t L21 L2 þ b3u L31 2b2t L31 þ L1tt L1 L2 2L21t L2 þ 2L1t L2t L1 Þ.
ð63Þ
1378
N.H. Ibragimov, S.V. Meleshko / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1370–1378
Differentiating (59) with respect to x and y, one finds uxy and wy, respectively, 1
wy ¼ 5L1 ðI 2 yÞ ; uxy ¼ 12Kð5L21 y 5 ðux L1 þ wx L2 ÞÞ1 ½6L1t L2 6b4 L22 þ 6b3 L1 L2 5L2t L1 þ wx ðI 2 L21 yÞ1 ½10b4 L32 15b3 L1 L22 þ 5b1 L31 10L1t L22 þ 10L2t L1 L2 I 2 L1 L2 . The equations (wy)y = wyy and ðu2x Þx ¼ 2ux uxx yield: 5L21 I 2u ¼ 6I 2 ð4b4 L22 9b3 L1 L2 þ 5b2 L21 4L1t L2 þ 5L2t L1 I 2 L1 Þ; 5L1 I 2t ¼ 6I 2 ðL1t b4 L2 þ b3 L1 Þ.
ð64Þ ð65Þ
Now the equations (wy)x = wxy and (uxy)y = (uyy)x are satisfied. The equation (uxy)x = (uxx)y yields: L1tt ¼ ð63b24 I 2 L22 126b4 b3 I 2 L1 L2 þ 225b4 b2 I 2 L21 81b4 L1t I 2 L2 þ 90b4 L2t I 2 L1 15b4 I 22 L1 162b23 I 2 L21 9b3 L1t I 2 L1 þ 15b4t I 2 L1 L2 þ 75b4u I 2 L21 90b3t I 2 L21 þ 18L21t I 2 þ 125L41 Þ=ð15I 2 L1 Þ. Notice that the equations (wy)x = wxy and (wy)y = wyy are satisfied. Summing up the above results, we complete the proof of Theorem 3.
ð66Þ
h
Acknowledgement One of the authors (SVM) is thankful to R. Conte for fruitful discussions. References [1] Lie S. Klassifikation und Integration von gewo¨nlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestaten. III, Archiv for Matematik og Naturvidenskab, 8, Heft 4:371–458, 1883. Reprinted in Lie’s Ges Abhandl, vol. 5, 1924, paper XIV, p. 362–427. [2] Ibragimov NH, editorCRC Handbook of Lie group analysis of differential equations. Symmetries, exact solutions and conservation laws, vol. 1. Boca Roton: CRC Press Inc.; 1994. [3] Ibragimov NH. Elementary Lie group analysis and ordinary differential equations. Chichester: John Wiley & Sons; 1999. [4] Ibragimov NH. Invariants of a remarkable family of nonlinear equations. Nonlinear Dyn 2002;30(2):155–66. [5] Liouville R. Sur les invariants de certaines equations differentielles et sur leurs applications. J Ecole Polytech 1889;59:7–76. ´ quation Diffe´rentielle Ordinaire du Second Ordre y00 = x(x, y, y 0 ), [6] Tresse, AM. De´termination des Invariants Ponctuels de lE Preisschriften der fu¨rstlichen Jablonowski’schen Geselschaft XXXII, Leipzig, S. Herzel, 1896. [7] Babich MV, Bordag LA. Projective differential geometrical structure of the Painleve equations. J Differen Equat 1999;157:452–85. [8] Ibragimov NH, Meleshko SV. Linearization of third-order ordinary differential equations by point transformations. Arch ALGA 2004;1:71–93. [9] Ibragimov NH, Meleshko SV. Linearization of third-order ordinary differential equations by point and contact transformations. J Math Anal Appl 2005;308(1):266–89. [10] Hearn AC. REDUCE Users Manual, ver. 3.3, The Rand Corporation CP 78, Santa Monica, 1987.