Travaux sur les symétries de Lie des équations aux ... - DMA - Ens

270we insert it in (8.12) 5 ; we get:(8.13)0 = −Yx 2 + (2x + g1 )Yx 1 ,0 = −Y 2y − (2x + 1 g1 )Y 2y + (2 + 2 g1 x )X + (2 + 2g2 )Yx 1 + (2x + g1 )Y 1y 1++ [2x + g 1 ] 2 Yy 1 + r Y 1 + r Y 2 ,20 = −Y 2y − X 2 x + 2 Y 1y + (6x + 1 3g2 )Y 1y + r Y 1 + r Y 2 + s Y 12 x ++ g 2 x [1 + g2 ] −1 X ,0 = −X y 1 + (2 + s)Y 1y + r X + r Y 1 + r Y 2 + s X 2 x + s Yx 1 + s Y 1y + s Y 2 1 y 2,0 = −X y 2 + r X + r Y 1 + r Y 2 + s X x + s Yx 1 + s Y 1y + s Y 1 1 y + s Y 2 2 y 2.Similarly, developing the second equation (8.8) and computing mod (y 1 1 )2 ,we get:(8.14)0 ≡ −Y 1xx + [ − 2 Y 1xy + X 1 xx+ [ ]2h Yx1 y11 .]y11 + [ − (4x + 2g 1 )Y xy 2 − (2 + h)Y 1y 2 ]+Collecting the coefficients of the monomials cst., y1 1 , we get two more linearPDE’s:(8.15)0 = −Yxx,10 = −2 Y 1xy + X 1 xx − (4x + 2g 1 )Y 11xy2 − (2 + h)Yy + 2h Y 12Proposition 8.16. Setting as initial conditions the five specific differentialcoefficients(8.17) P := P ( X , Y 1 , Y 2 , Yx 1 , X )x = r X +r Y 1 +r Y 2 +r Yx 1 +r X x,it follows by cross differentiations and by linear substitutions from the sevenequations (8.13) i , i = 1, 2, 3, 4, 5, (8.15) j , j = 1, 2, that X y 1, X y 2, Y 1y, 1Y 1y, Y 2 2 x , Y 2y, Y 2 1 yand X 2 xx , X xy 1, X xy 2, Yxx, 1 Y 1xy, Y 1 1 xyare uniquely2determined as linear combinations of (X , Y 1 , Y 2 , Yx 1 , X x ), namely:⎧Yx⎪⎨2 1= P, X 2 xx = P, Yxx1 3= P,(8.18)⎪⎩X y 1X y 24= P, Y 1y 1 5 = P, Y 2y 1 6 = P, X xy 19= P, Y 1y 2 10= P, Y 2y 2 11= P, X xy 2Then the expressions P are stable under differentiation:(8.19)x .7= P, Y 1xy 1 8 = P,12= P, Y 1xy 2 13= P.P x = P + r Yx 2 + r Yxx 1 + r X xx = P,P y 1 = P + r X y 1 + r Y 1y + r Y 2 1 y + r Y 1 1 xy + r X 1 xy 1 = P,P y 2 = P + r X y 2 + r Y 1y + r Y 2 2 y + r Y 1 2 xy + r X 2 xy 2 = P,