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

62and θ also vanish:(4.3)0 =?= − ∑ k+ ∑ kG k y l 1 Θk + G l 1y l 1 Θl 1+ 1 2∑G k L p k,l 1Θ p − ∑pk∑kH k l 1 ,x Θk − 1 2 Hl 1l1 ,x Θl 1+G k L l 1l1 ,k Θl 1− 1 4∑ ∑Hl k 1H p k Θp + 1 4kp∑kH k l 1H l 1kΘ l 1.Indeed, it suffices to observe that this identity coincides with(4.4) 0 = 1 2∑kΘ k ( (3.106)| j:=k; l1 :=l 1 ; l 2 :=l 1).Simplifying then (4.2), we get the explicit formulation of the first family ofcompatibility conditions for the second auxiliary system:(4.5)0 =?= 2G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx −− 2 3 Gl 1 x M l1 ,l 1− 4 3∑+ 2 ∑ G k y l 1 Ll 1l1 ,k − ∑kk− 1 2 Hl 1l1 ,x Ll 1l1 ,l 1− 1 ∑3+ 1 3− 1 3∑kkkH l 1k,xL k l 1 ,l 1− 1 2∑H k k,yH k k l 1−kG k x M l1 ,k + G l 1y l 1 Ll 1l1 ,l 1+G k y l 1 Lk k,k −H k l 1 ,x Ll 1l1 ,k + 1 2∑k∑Hl k 1 ,x Lk k,k +kH k l 1 ,y l 1 Hl 1k− 1 2∑kH l 1k,y l 1 Hk l 1−