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

72Fithly, put j := l 2 , l 1 := k and l 2 := l 1 in (3.108):(4.38)0 = − 1 2 Hl 2k,y l 1 + 1 6 δl 2kH l 1l 1 ,y l 1 + Ll 2k,l1 ,x − 1 3 δl 2kL l 1l1 ,l 1 ,x ++ G l 2M k,l1 − 1 3 δl 2kG l 1M l1 ,l 1+ 1 3 δl 2k∑+ 1 2∑pH p k Ll 2l1 ,p + 1 6 δl 2k∑ppG p M l1 ,p − 1 2H l 1 p L p l 1 ,l 1− 1 6 δl 2k∑p∑pH p l 1L l 1l1 ,p .H l 2 p L p k,l 1+Next, apply the operator ∑ k Hk l 2(·) to (4.38), namely compute ∑ k Hk l 2 ·(4.38). This yields:(4.39)0 = − 1 2∑kH l 2k,y l 1 Hk l 2+ 1 6 Hl 1l 1 ,y l 1 Hl 2l2+ ∑ k− 1 3 Ll 1l1 ,l 1 ,x Hl 2l2++ ∑ k− 1 ∑ ∑2− 1 6kG l 2H k l 2M k,l1 − 1 3 Gl 1H l 2l2M l1 ,l 1+ 1 3∑ppH k l 2H l 2 p L p k,l 1+ 1 2H l 2l2H p l 1L l 1l1 ,p .∑ ∑kpL l 2k,l1 ,x Hk l 2−∑pG p H l 2l2M l1 ,p−H k l 2H p k Ll 2l1 ,p + 1 6∑pH l 2l2H l 1 p L p l 1 ,l 1−Sixthly, we form the expression:(4.40) (3.108)| j:=k; l1 :=l 1 ; l 2 :=l 2− (3.108)| j:=k; l1 :=l 2 ; l 2 :=l 1.