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

order derivatives of X and of the Y j :(3.26) ⎧⎪⎨⎪⎩R j 1 = Y jx + m∑l 1 =1R j 2 = Y jxx + m∑++++m∑l 1 =1m∑l 1 =1 l 2 =1m∑m∑l 1 =1 l 2 =1 l 3 =1m∑l 1 =1m∑y l 1xx ·m∑l 1 =1 l 2 =1[ ]y l 1x · Y j − y l 1 δj l 1X x +m∑[]y l 1x · 2 Y j − xy l 1 δj l 1X xx +m∑l 1 =1 l 2 =1y l 1x y l 2x · [−δ j l 1X y l 2],[]y l 1x y l 2x · Y jy l 1y − l 2 δj l 1X xy l 2 − δ j l 2X xy l 1 +m∑y l 1x y l 2x y l 3x · [−δ j l 1X y l 2y l 3]+[Y jy l 1 − 2 δj l 1X x]+y l 1x y l 2xx · [−δ j l 1X y l 2 − 2 δ j l 2X y l 1].However, since the notations in [GM2003] are different and since the generalcase of n 1 independent variables and m 1 dependent variables isconsidered there, it is certainly easier to reconstiture formulas (3.26) directlyby means of the inductive formulas described in [Ol1986], [BK1989]).Analogously to the observation made in Section 2, we guess that thereexists a formal correspondence between the terms of R j 2 not involving yl xxand the explicit form of the equation yxx j = F j (x, y, y x ) equivalent to Y j XX =0. In the case m = 2, we claim that this formal correspondence also holdstrue. Indeed, it suffices to write formula (3.26) for R j 2 modulo the yxx, l whichyields two expressions in total analogy with the two explicit polynomialsappearing in the right-hand side of (3.23):(3.27)⎧⎪⎨⎪⎩R 1 2 (mod y l xx) ≡ Y 1xx + y 1 x · {2Y 1xy 1 − X xx31}+ y2x · {2Y 1xy 2 }+ y1x y 1 x · {Y 1y 1 y 1 − 2 X xy 1 }++ yx 1 y2 x · {2Y 1y 1 y − 2 X } 2 xy 2 + y2x yx 2 · {Y } 1y 2 y + 2+ yx 1 y1 x y1 x · {−X y 1 y 1} + y1 x y1 x y2 x · {−2 X y 1 y 2} + y1 x y2 x y2 x · {−X y 2 y 2} ,R 2 2 (mod y l xx) ≡ Y 2xx + y 1 x · {2Y 2xy 1 }+ y2x · {2Y 2xy 2 − X xx}+ y1x yx 1 · {Y } 2y 1 y + 1+ y 1 x y2 x · {2Y 2y 1 y 2 − 2 X xy 1 }+ y2x y 2 x · {Y 2y 2 y 2 − 2 X xy 2 }++ y 1 x y1 x y2 x · {−X y 1 y 1} + y1 x y2 x y2 x · {−2 X y 1 y 2} + y2 x y2 x y2 x · {−X y 2 y 2} ,Except for inductive inspiration (see the formulation of Lemma 3.32 below),this observation will not be used further. At this stage, it helps at least