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

322tiny detail. Without such a care, it would be impossible to claim that some ofour subsequent computations, for which we will not provide the intermediatesteps, may be redone and verified. Consequently, we will expose our rulesof formal computation thoroughly.Replacing the value of Y 1 just obtained in the induction formula (3.4) 2and developing, we may conduct the very first steps of the computation:Y i1 ,i 2= Di 2 2(Y i1 ) − ∑ )Di 1 2(X k 1y i1 ,k 1k 1=⎛⎝ ∂∂x i 2 + y i 2∂∂y + ∑ k 1y i2 ,k 1∂⎞ ⎛⎠∂y k1⎝Y x i 1 + ∑ k 1[δ k 1i 1Y y − X k 1x i 1]y k1 +⎞+ ∑ ] [−δ k 1i 1X k 2y y k1 y k2⎠ − ∑ ][X k 1x + y i 2 i 2X k 1y y i1 ,k 1k 1 ,k 2 k 1(3.10)( ∂=∂x i 2) ⎛ ⎞⎝Y x i 1 + ∑ [δ k 1i 1Y y − X k 1]yx i 1 k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠ +k 1 k 1 ,k 2( ) ⎛ ⎞∂+ y i2⎝Y∂y x i 1 + ∑ [δ k 1i 1Y y − X k 1]yx i 1 k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠+k 1 k 1 ,k 2⎛ ⎞ ⎛+ ⎝ ∑ ∂y i2 ,k 1⎠∂y k1k 1+ ∑ k 1[−X k 1x i 2⎝Y x i 1 + ∑ k 1[δ k 1i 1Y y − X k 1x i 1]y k1 ,i 1+ ∑ k 1[−X k 1y]y i2 y i1 ,k 1⎞]y k1 + ∑ [ ]−δ k 1i 1X k 2y y k1 y k2⎠ +k 1 ,k 2= Y x i 1x i 2 + ∑ [δ k 1i 1Y x i 2y − X k 1]yx i 1x i 2 k1 + ∑ [ ]−δ k 1i 1X k 2yx i 2y k1 y k2 +k 1 k 1 ,k 2+ Y x i 1y y i2 + ∑ [δ k 1i 1Y yy − X k 1]yx i 1y k1 y i2 + ∑ [ ]−δ k 1i 1X k 2yy y k1 y k2 y i2 +k 1 k 1 ,k 2+ ∑ [δ k 1i 1Y y − X k 1]yx i 1 i2 ,k 1+ ∑ [−δ k 1i 1X k 2y]y k2 y i2 ,k 1+ ∑ [ ]−δ k 1i 1X k 2y y k1 y i2 ,k 2+k 1 k 1 ,k 2 k 1 ,k 2+ ∑ [−X k 1]yx i 2 k1 ,i 1+ ∑ [k 1 k 1−X k 1y]y i2 y i1 ,k 1.Some explanations are needed about the computation of the last two termsof line 11, i.e. about the passage from line 7 of (3.10) just above to line 11.