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

66Finally, compute the linear combination (4.15)+(4.16)+2·(4.17)−2·(4.19):0 ≡ −2G l 1+ 2G l y l 1y l 2+ H l 1 y l 1y l2 1− H l l 1 a 1 ,xy l 2−1 l 2 2 ,xy l1 b(4.20)− 2G l 2+ H l y l 1y l2 2−la 1 ,xy l2 c− H l 2+ 2 l 1 ,xy l2 c 3 Hl 1+ 2L l 2l 1 ,l 1 ,xy l 1 l1 ,l 2 ,xx − 4 2 d 3 Ll 1l1 ,l 1 ,xx+3+ H l 2− 1 l 2 ,xy l1 b 3 Hl 1− 2L l 2l 1 ,xy l 1 l2 ,l 1 ,xx + 2 2 d 3 Ll 1l1 ,l 1 ,xx.3We indeed get the desired approximate identity:(4.21) 0 ≡ −2 G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx .4.22. Complete computation. Now that the guide is constructed, we canachieve the complete computations.Firstly, put j := l 1 in (3.106) with l 2 ≠ l 1 and differentiate with respectto y l 1:(4.23)0 ≡ −2G l 1y l 1y + l 1 2Gl 2y l 2y + l 1 Hl 1l 1 ,xy − l 1 Hl 2l 2 ,xy + l 1+ 2 ∑ k− 1 ∑2kG k y l 1 Ll 1l1 ,k + 2 ∑ kH k l 1 ,y l 1 Hl 1k− 1 2∑kG k L l 1l 1 ,k,y l 1 − 2 ∑ kH k l 1H l 1k,y l 1 + 1 2∑kG k y l 1 Ll 2l2 ,k − 2 ∑ kH k l 2 ,y l 1 Hl 2k+ 1 2G k L l 2l 2 ,k,y l 1 −∑kH k l 2H l 2k,y l 1 .Secondly, put j := l 2 in (3.106) with l 1 ≠ l 2 and differentiate with respectto y l 2:(4.24)0 ≡ −2G l 2y l 1y + l 2 Hl 2l 1 ,xy + l 22 ∑ kG k y l 2 Ll 2l1 ,k + 2 ∑ kG k L l 2l 1 ,k,y l 2 − 1 2∑kH k l 1 ,y l 2 Hl 2k− 1 2∑kH k l 1H l 2k,y l 2 .