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

60replaced by their values given in (3.99), (3.100), (3.101) and (3.102). In thisreplacement, some double sums appear. As before, we use the first indexk = 1, . . .,m for single summation and then the second index p = 1, . . .,mfor double summation. Finally, we put all the second order terms in the firstline, not disturbing the order of appearance of the 73 remaining terms. Weget:(4.2)0 = −2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx ++ 2 ∑ k− 1 2∑kG k y l 1 Ll 1l1 ,k4H k l 1 ,y l 1 Hl 1k10+ 2 ∑ k− 1 2G k L l 1l 1 ,k,y l 1∑k16H k l 1H l 1k,y l 111− ∑ k− ∑ kG k y l 1 Lk k,k5G k y l 1 Θk α+− ∑ kG k L k k,k,y l 1a−+ ∑ k− 1 2− 1 2G k L k k,k,y l 1∑k∑ka− 2 ∑ kG k L k k,k Ll 1l1 ,l 1+ ∑bkG k L l 1l1 ,l 1Θ k + ∑dkG k M k,l1 ,x18− ∑ k∑G k L p k,l 1L p p,pp∑G k H p k M l 1 ,p23∑G k L p k,l 1Θ p − ∑pkεp− 1 219−∑G k L k k,k Θl 1kG k M k,l1 Θ 0 e− 1 2c−∑G k Θ k Θ l 1kf++ 2 3 Ll 1l1 ,l 1 ,x Θ0 − 1g 3 Hl 1l 1 ,y l 1 Θ0 +h+ 2 3 Gl 1M l1 ,l 1Θ 0 + 4 ∑G k M l1 ,k Θ 0i 3ke− 1 ∑Hl k 2 1L k k,k Θ0 − 1 ∑2klk− 1 3∑kH l 1kL l 1l1 ,l 1Θ 0 j+ 1 3∑Hl k 1Θ k Θ 0 + 1 2 Ll 1l1 ,l 1Θ 0 Θ 0 + 1n 2 Θ0 Θ 0 Θ l 1−omkH k l 1L l 1l1 ,k Θ0 k−− 2 3 Gl 1 x M l1 ,l 11− 2 3 Gl 1M l1 ,l 1 ,x17+ 1 3+ 1 2∑k∑kH l 1k,xL k l 1 ,l 19H k l 1 ,x Lk k,k8+ 1 3+ 1 2∑k− 4 3H l 1kL k l 1 ,l 1 ,x∑Hl k 1L k k,k,xk∑G k x M l 1 ,kk1514− 1 3+ 1 2∑k2− 4 3∑G k M l1 ,k,xkH k l 1 ,x Ll 1l1 ,k∑Hl k 1 ,x Θk γ−k7− 1 3∑k18+H k l 1L l 1l1 ,k,x13+