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

180Let us differentiate now the equations (7.28) with respect to the variablesx l as follows: first we differentiate (7.28) 1 with respect to x k1 ; then wedifferentiate (7.28) 2 with respect to x k2 ; finally we differentiate (7.28) 3 withrespect to x k3 . The arguments in the expressions Π in the equation (7.28)contain now the terms Rx j′k ′ ···x 1 k; we replace them by their value given in′ κthe first equation of (7.28) already obtained. The arguments also contain theterms R j′and Q l′x k ′ x kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ . We replace1 2x j ′1u i′ 1them by their value given by the fifth, the sixth and the seventh equationsof (7.28). We obtain three new equations in which the arguments of theexpressions Π are the desired ones: (x, u, J), where J is defined in (7.28):(7.28)⎧⎪⎨⎪⎩Π (x, u, J) = C 1 κ Rj x k1 x k1 u i 1 − C2 κ δj i 1Q k 1x k1 x k1 x k1,Π (x, u, J) = 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 ,Π (x, u, J) = −δ j i 1Q k 3x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 .The seven equations (7.28) and (7.28) may be considered as three systemsof two linear equations of two variables with a nonzero determinant, theseventh equation being the last equation in (7.28). We immediately obtain:(7.28) ⎧Π(x, u, J) = R j = x k1 x k1 u i 1 δj i 1Q k 1x k1 x k1 x k1,⎪⎨Π(x, u, J) = R j x k1 x k2 u i 1 = δj i 1Q k 2x k1 x k2 x k2, k 2 ≠ k 1 ,Π(x, u, J) = 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 ,⎪⎩ Π(x, u, J) = δ j i 1Q l x k1 x k2 x k3, k 3 ≠ k 1 , k 3 ≠ k 2 ,giving the fourth equation in (7.28).It remains now to obtain the second and the third equations in (7.28).Let us write firstly equation (7.28) [6] with the choice of the indices j = i 1 ,l = k 1 = · · · = k κ . This gives the equation:(7.28)⎧ (Q ⎪⎨l u i 1= Π x, u, Q l′ , Q l′x k, . . .,Q l′′ x 1 k ′ 1···x k, R j′ , R′ x j′κ−2k, . . ., R′ x j′1k ′ ···x 1 k,′κ−1)⎪⎩, R j′, R j′, . . .,R j′u i′ 1 x k ′ u i′ 1 x 1 k ′ 1···x k ′ u i′ 1κ−3We observe first that the differentiation with respect to the variables x l ofone of the expressions Π(x, u, J) remains an expression Π(x, u, J). Indeedwe see from (7.28) that there appears, in the partial derivative J xl , derivativesQ l′x k ′ x 1 kwith k′ 1 ′ ≠ k′ 2 or l′ ≠ k 1 ′ , l′ ≠ k 2 ′ . We may replace them2by their value obtained in the sixth and the seventh equations of (7.28). It.