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

69correct. We get:(4.29)0 =?== − ∑ k+ 2 ∑ kG k y l 1 Lk k,k + Gl 1y l 1 Ll 1l1 ,l 1+ 1 2G k y l 1 Ll 2l2 ,k − ∑ k∑kH k l 1 ,x Ll 2l2 ,k −H k l 1 ,x Lk k,k − 1 2 Hl 1l1 ,x Ll 1l1 ,l 1+− 2 ∑ kG k y l 2 Ll 2l1 ,k + ∑ kH k l 2 ,x Ll 2l1 ,k ++ 1 2− 1 2+ 1 2+ 2 ∑ k∑k∑k∑kH l 2k,y l 2 Hk l 1− 1 3H l 2k,y l 1 Hk l 2+ ∑ kH k l 1 ,y l 2 Hl 2k− 1 2∑H k k,yH k l k 1− ∑kkL l 2l1 ,k,x Hk l 2+∑L l 2l 2 ,k,y l 1 Gk − 2 ∑ kkH k l 2 ,y l 1 Hl 2k+L l 2l 1 ,k,y l 2 Gk − 2 ∑ kL l 2l2 ,k,x Hk l 1+ 2 3M k,l1 ,x G k −∑L k k,k,x Hk l 1−− ∑ ∑G k H p k M l 1 ,p + 2 ∑G k Hl k 31M k,k + 1 ∑ ∑G p Hl k 31M k,p −k pkk p− ∑ G k L l 1l1 ,l 1L l 1l1 ,k + ∑ ∑G k L p k,l 1L p p,p − 1 ∑ ∑Hl k 31Hp k Lp k,k +kk pk p+ 1 ∑ ∑Hl k 31H p k Lk k,p − 1 ∑ ∑Hl k 41H p k Lp p,p + 1 ∑Hl k 4 1H l 1kL l 1l1 ,l 1.kpkWe have underlined plainly the first order terms appearing in lines 1, 2, 3, 4,5, 6 and 7.4.30. Reconstitution of the subgoal (4.29) from (3.106), from (3.108) andfrom (3.110). Now, it suffices to establish that the first order partial differentialrelations (4.29) for 1 l 1 , l 2 m and l 2 ≠ l 1 (crucial assumption)are a consequence of (3.106), of (3.108) and of (3.110) by linear combinations.The auxiliary index l 2 , which is absent in the goal (4.5), will disappearat the end. Differentiations will not be applied anymore. Also, the partialdifferential relations (3.96), which were not used above, will neither be usedin the sequel. However, they are strongly used in the treatment of the remainingthree compatibility conditions (3.112) 2 , (3.112) 3 and (3.112) 4 , thedetail of which we do not copy in the typesetted paper. Finally, the constructionof a guide for the subgoal (4.29) may be guessed similarly as in §4.14above. We shall provide the final computations directly, without any guide:pkk