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

also appears some derivatives Q l′x k ′1x k ′2x k ′3(we replace them by their valueobtained in the fourth equation of (7.28)), some derivatives Rx j′k ′ ···x k(we′1 κreplace them by their value obtained in the first equation of (7.28)) andsome derivatives R j′(we replace them by their value obtained in thex k ′1u i′ 1fifth equation of (7.28)). Consequently we may write:181(7.28) [Π(x, u, J)] xl = Π(x, u, J).It follows that any derivative with respect to x l (to any order) of the fourthand the fifth equations of (7.28) provides expressions of the form Π(x, u, J).In other words for any integer λ ≥ 3 and any integer µ ≥ 1 we have(7.28)⎧⎨ Π(x, u, J) = Q l x k1 x k2 x k3 ···x kλ,⎩ Π(x, u, J) = R j . x k1 ···x kµ u i 1We may replace then these values in the equation (7.28), replacing also thederivatives Q l′x k ′ x 1 kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ by their values obtainedin the sixth and the seventh equations of (7.28). This gives the second2equation of (7.28).We also remark that by a differentiation with respect to the variables x l ,the second equation Q l = Π(x, u, J) just obtained implies, using (7.28):u i 1(7.28) Π(x, u, J) = Q l x k1 u i 1.It remains finally to write (7.28) [4] first with the choice of indices l = k 1 =· · · = k κ , j = i 1 then with the choice of indices l = k 1 = · · · = k κ , j ≠ i 1 .We also write (7.28) [5] first with the choice of indices l = k 1 = · · · = k κ ,j = i 2 then with the choice of indices l = k 1 = · · · = k κ , j ≠ i 1 , j ≠ i 2 .We obtain four new equations: