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

Next, apply the operator ∑ k Ll 2l1 ,k (·) to (4.34), namely compute ∑ k Ll 2l1 ,k ·(4.34). This yields:(4.35)0 = −2 ∑ k+ 2 ∑ k+ 1 ∑2pG k y l 2 Ll 2l1 ,k + 2Gl 1y l 1 Ll 2l1 ,l 2+ ∑ k∑pG p L k l 2 ,p Ll 2l1 ,k − 2 ∑ pH p l 1H l 1 p L l 2l1 ,l 2.H k l 2 ,x Ll 2l1 ,k − Hl 1l1 ,x Ll 2l1 ,l 2+G p L l 1l1 ,p Ll 2l1 ,l 2− 1 2Fourthly, put j := l 2 , l 1 := k and l 2 := l 2 in (3.108):(4.36)0 = − 1 2 Hl 2k,y l 2 + 1 6 δl 2kH l 2l 2 ,y l 2 + 1 3 Hk k,y k + L l 2k,l2 ,x −− 1 3 δk l 2L l 2l2 ,l 2 ,x − 2 3 Lk k,k,x ++ G l 2M k,l2 − 1 3 δk l 2G l 2M l2 ,l 2− 2 3 Gk M k,k + 1 3 δl 2k∑− 1 ∑G p M k,p − 1 ∑32p− 1 6 δl 2k∑ppH p l 2L l 2l2 ,p + 1 3H l 2 p L p k,l 2+ 1 2p∑pp∑ ∑kpG p M l2 ,p−H p k Ll 2l2 ,p + 1 6 δl 2k∑∑Hp k L p k,k − 1 ∑H p3k Lk k,p .p71H p l 2H k p L l 2l1 ,k +pH l 2 p L p l 2 ,l 2−Next, apply the operator − ∑ k Hk l 1(·) to (4.36), namely compute − ∑ k Hk l 1·(4.36). This yields:(4.37)0 = 1 2∑k− ∑ k− ∑ k− 1 ∑3− 1 2H l 2k,y l 2 Hk l 1− 1 6 Hl 2l 2 ,y l 2 Hl 2l1− 1 3pL l 2k,l2 ,x Hk l 1+ 1 3 Ll 2l2 ,l 2 ,x Hl 2l1+ 2 3G l 2H k l 1M k,l2 + 1 3 Gl 2H l 2l1M l2 ,l 2+ 2 3∑ ∑kG p H l 2l1M l2 ,p + 1 3pkH k l 1H p k Ll 2l2 ,p − 1 6p∑H k k,yH k k l 1−k∑L k k,k,x Hk l 1−k∑G k Hl k 1M k,k −∑ ∑G p Hl k 1M k,p + 1 ∑ ∑2∑pkH l 2l1H l 2 p L p l 2 ,l 2+ 1 6− 1 ∑ ∑Hl k 31Hp k L p k,k + 1 ∑ ∑Hl k 31H p k Lk k,p .kpkpk∑ppH k l 1H l 2 p L p k,l 2−H l 2l1H p l 2L l 2l2 ,p −