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

are a consequence of (I’), (I”), (III’), (IV’).For instance, in (3.29) 1 , replacing Θ j 1by its expression (3.24), differentiatingit with respect to x j 3x j 2, replacing Θ j 1by its expression (3.24), differentiatingit with respect to x j 2x j 3and substracting, we get:(3.30) ⎧0 = −2 G j1 ,j 2 ,yx j 3 + 2 G j1 ,j 3 ,yx j 2 + H j 1j 1 ,j 1− H j 1,x j 2x j 3 j a 1 ,j 1+,x j 3x j 2 a+ 1 2 Θj 1x j 3 Θj 2+ 1 2 Θj 1Θ j 2x j 3 − 1 2 Θj 1x j 2 Θj 3− 1 2 Θj 1Θ j 3x j 2 −377− 1 2 Hj 1j 1 ,j 1 ,x j 3 Θj 2− 1 2 Hj 1j 1 ,j 1Θ j 2x j 3 + 1 2 Hj 1j 1 ,j 1 ,x j 2 Θj 3+ 1 2 Hj 1j 1 ,j 1Θ j 3x j 2 −⎪⎨− 1 2 Hj 2j 2 ,j 2 ,x j 3 Θj 1− 1 2 Hj 2j 2 ,j 2Θ j 1x j 3 + 1 2 Hj 3j 3 ,j 3 ,x j 2 Θj 1+ 1 2 Hj 3j 3 ,j 3Θ j 1x j 2 −− G j1 ,j 2 ,x j 3 Θ n+1 − G j1 ,j 2Θ n+1x j 3+ G j1 ,j 3 ,x j 2 Θ n+1 + G j1 ,j 3Θ n+1x j 2 ++ ∑ lH l j 1 ,j 2 ,x j 3Θ l + ∑ lH l j 1 ,j 2Θ l x j 3− ∑ lH l j 1 ,j 3 ,x j 2Θ l − ∑ lH l j 1 ,j 3Θ l x j 2++ 1 2 Hj 1j 1 ,j 1 ,x j 3 Hj 2j 2 ,j 2+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2 ,x j 3 − 1 2 Hj 1j 1 ,j 1 ,x j 2 Hj 3j 3 ,j 3− 1 2 Hj 1j 1 ,j 1H j 3j 3 ,j 3 ,x j 2 −− ∑ lH l j 1 ,j 2 ,x j 3H l l,l − ∑ lH l j 1 ,j 2H l l,l,x j 3+ ∑ lH l j 1 ,j 3 ,x j 2H l l,l + ∑ lH l j 1 ,j 3H l l,l,x j 2+⎪⎩+ ∑ lG j2 ,l,x j 3 L l j 1+ ∑ lG j2 ,l L l j 1 ,x j 3− ∑ lG j3 ,l,x j 2 L l j 1− ∑ lG j3 ,l L l j 1 ,x j 2.Next, replacing the twelve first order partial derivatives underlined justabove:(3.31)⎧⎨⎩Θ j 1x j 3 , Θj 2x j 3 , Θj 1x j 2 , Θj 3x j 2 , Θj 2x j 3 , Θj 3x j 2 ,Θ j 1x j 3 , Θj 1x j 2 ,Θn+1 x j 3 ,Θn+1 x j 2 , Θl x j 3, Θ l x j 2.by their values issued from (3.24), (3.26) and adapting the summation indices,we get the explicit developed form of the first family of compatibility