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

340The induction formulas are:(4.4)⎧⎪⎨Y j 1 := D 1 ( Y j) − D 1 (X ) y j 1,Y j 2 := D2 ( Y j 1)− D 1 (X ) y j 2 ,· · · · · · · · · · · ·⎪⎩Y j λ := ( Dλ Yλ−1) j − D 1 (X )y j λ ,where the total differentiation operators D λ are denoted by (insteadof (1.22)):(4.5) D λ := ∂∂x + m∑l=1y l 1∂∂y + ∑ m ll=1∂y2l ∂y1l+ · · · +m∑l=1∂yλl ∂yλ−1l.Applying the definitions in the first two lines of (4.4), we compute, we simplifyand we organize the results in a harmonious way, using in an essentialway the Kronecker symbol. Here, the computations are more elementarythan the computations of Y i1 and of Y i1 ,i 2achieved thoroughly in the previousSection 3, so that we do not provide a Latex track of the details. Firstlyand secondly:(4.6) ⎧⎪⎨⎪⎩Y j 1 = Y jx +Y j 2 = Y jx 2 +Thirdly:(4.7)Y j 3 = Y jx 3 +m∑l 1 =1m∑l 1 =1[Y jy l 1 − δj l 1X x]y l 11 +m∑l 1 ,l 2 =1[]2Y jxy − l 1 δj l 1X x 2 y l 11 +[−δ j l 1X y l 2]y l 11 y l 21 ,m∑l 1 ,l 2 =1[Y jy l 1y l 2 − δj l 12X xy l 2]y l 11 y l 21 ++ ∑ [−δ j l 1X y l 2y l 3]y l 11 y l 21 y l 31 + ∑ [ ]Y jy − l 1 δj l 12X x y l 12 +l 1 ,l 2 ,l 3 l 1m∑]+[−δ j l 1X y l 2 − δ j l 22X y l 1 y l 11 y l 22 .l 1 ,l 2 =1m∑l 1 =1[]3Y jx 2 y − l 1 δj l 1X x 3 y l 11 +m∑l 1 ,l 2 =1+ ∑l 1 ,l 2 ,l 3[Y jy l 1y l 2y l 3 − δj l 13X xy l 2y l 3]y l 11 y l 21 y l 31 ++ ∑l 1 ,l 2 ,l 3 ,l 4[−δ j l 1X y l 2y l 3y l 4]y l 11 y l 21 y l 31 y l 41 +[]3Y jxy l 1y − l 2 δj l 13X x 2 y l 2 y l 11 y l 21 +m∑l 1 =1[3Y jxy l 1 − δj l 13X x 2]y l 12 +