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

326the fourth lines of (3.18), we may permute freely certain indices in some ofthe terms inside the brackets. This yields the passage from lines 2, 3 and 4of (3.18) to lines 2, 3 and 4 of (3.20).It remains to explain how we pass from the fifth (last) line of (3.18) to thefifth (last) line of (3.20). The bracket in the fifth line of (3.18) contains threeterms: [−T 1 − T 2 − T 3 ]. The term T 3 involves the product δ k 1,k 3i 2 , i 1, which werewrite as δ k 3,k 1i 1 , i 2, in order that i 1 appears before i 2 . Then, we rewrite the threeterms in the new order [−T 2 − T 3 − T 1 ], which yields:(3.22)∑ [−δ k 1,k 3i 1 , i 2X k 2y − δ k 3,k 1i 1 , i 2X k 2y − δ k 2,k 3i 1 , i 2X k 1yk 1 ,k 2 ,k 3]y k1 y k2 ,k 3.It remains to observe that we can permute k 2 and k 3 in the first term −T 2 ,which yields the last line of (3.20). The detailed proof is complete.3.23. Final perfect expression of Y i1 ,i 2 ,i 3. Thanks to similar (longer) computations,we have obtained an expression of Y i1 ,i 2 ,i 3which we consider tobe in final harmonious shape. Without copying the intermediate steps, let uswrite down the result. The comments which are necessary to read it and tointerpret it start just below.Y i1 ,i 2 ,i 3= Y x i 1x i 2x i 3 + ∑ k 1[δ k 1i 1Y x i 2x i 3y + δ k 1i 2Y x i 1x i 3y + δ k 1i 3Y x i 1x i 2y − X k 1x i 1x i 2x i 3]y k1 +(3.24)+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y 2 + δ k 1,k 2i 3 , i 1Y x i 2y 2 + δ k 1,k 2i 2 , i 3Y x i 1y 2−]−δ k 1i 1X k 2x i 2x i 3y − δk 1i 2X k 2x i 1x i 3y − δk 1i 3X k 2yx i 1x i 2y k1 y k2 ++ ∑ []δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 3 − δ k 1,k 2i 1 , i 2X k 3− δ k 1,k 2x i 3y 2 i 1 , i 3X k 3− δ k 1,k 2x i 2y 2 i 2 , i 3X k 3yx i 1y 2 k1 y k2 y k3 +k 1 ,k 2 ,k 3+ ∑ [−δ k 1,k 2 ,k 3i 1 , i 2 , i 3X k 4]yy 3 k1 y k2 y k3 y k4 +k 1 ,k 2 ,k 3 ,k 4+ ∑ k 1 ,k 2[δ k 1,k 2i 1 , i 2Y x i 3y + δ k 1,k 2i 3 , i 1Y x i 2y + δ k 1,k 2i 2 , i 3Y x i 1y −]−δ k 1i 1X k 2x i 2x − i 3 δk 1i 2X k 2x i 1x − i 3 δk 1i 3X k 2yx i 1x i 2 k1 ,k 2++ ∑ [δ k 1,k 2 ,k 3i 1 , i 2 , i 3Y y 2 + δ k 3,k 1 ,k 2i 1 , i 2 , i 3Y y 2 + δ k 2,k 3 ,k 1i 1 , i 2 , i 3Y y 2−k 1 ,k 2 ,k 3−δ k 1,k 2i 1 , i 2X k 3x i 3y − δk 3,k 1i 1 , i 2X k 2x i 3y − δk 2,k 3i 1 , i 2X k 1x i 3y −−δ k 1,k 2i 1 , i 3X k 3x i 2y − δk 3,k 1i 1 , i 3X k 2x i 2y − δk 2,k 3i 1 , i 3X k 1x i 2y −−δ k 1,k 2i 2 , i 3X k 3x i 1y − δk 3,k 1i 2 , i 3X k 2x i 1y − δk 2,k 3i 2 , i 3X k 1x i 1y]y k1 y k2 ,k 3+