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

174equations:(7.28)⎧⎪⎨⎪⎩0 = C 1 κ Rj x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1,0 = R j x k1 u i 1 − C1 κ−1 δ j i 1Q k 2x k1 x k2, k 2 ≠ k 1 ,0 = − δ j i 1Q k 3x k1 x k2, k 3 ≠ k 1 , k 3 ≠ k 2 .We specify the indices in the second equation of (7.28) as follows: l =k κ = · · · = k 3 = k 2 = k 1 ; then l = k κ = · · · = k 3 = k 2 ≠ k 1 ; thenl = k κ = · · · = k 3 , k 3 ≠ k 2 , k 3 ≠ k 1 ; finally l = k κ = · · · = k 4 , l ≠ k 1 ,l ≠ k 2 , l ≠ k 3 . This gives the four following equalities:(7.28)⎧⎪⎨0 = C 2 κ R j x k1 x k1 u i 1 − C3 κ δ j i 1Q k 1x k1 x k1 x k1,0 = C 1 κ−1 R j x k1 x k2 u i 1 − C2 κ−1 δ j i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = R j − x k1 x k2 u i 1 C1 κ−2 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ 0 = − δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .Let us differentiate now the equations (7.28) with respect to the variables x las follows: we differentiate (7.28) 1 with respect to x k1 ; then we differentiate(7.28) 2 with respect to x k2 ; finally we differentiate (7.28) 3 with respect tox k3 . This gives the three following equations:(7.28)⎧⎪⎨⎪⎩0 = C 1 κ Rj x k1 x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1 x k1,0 = R j x k1 x k2 u i 1 − C1 κ−1 δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = − δ j i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 .The seven equations given by the systems (7.28) and (7.28) may be consideredas three systems of two equations (of two variables) with a nonzerodeterminant, to which we add the last equation (7.28) 4 . We get immediately:(7.28)⎧⎪⎨0 = R j x k1 x k1 u i 1 = δj i 1Q k 1x k1 x k1 x k1,0 = R j x k1 x k2 u i 1 = δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,0 = R j = x k1 x k2 u i 1 δj i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,⎪⎩ 0 = δ j i 1Q l x k1 x k2 x k3, l ≠ k 1 , l ≠ k 2 , l ≠ k 3 .It follows from these relations and from the relations Q l u i 1 = Rj u i 1u i 2 = 0obtained in (7.28) that all the third order partial derivatives of Q l vanishidentically, this being also satisfied by the third order partial derivatives of