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

− 1 3∑kL l 1l1 ,k,x Hk l 1+ 1 3∑kL k l 1 ,l 1 ,x Hl 1k+ 2 3∑L k k,k,x Hk l 1++ 2 ∑ L l 1l 1 ,k,y l 1 Gk −k− 2 3 M l 1 ,l 1 ,x G l 1− 10 ∑M k,l1 ,x G k −3k− ∑ ∑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 ,k Ll 1l1 ,l 1+ ∑ ∑G k L p k,l 1L p p,p−kk p− 1 ∑ ∑Hl k 31Hp k L p k,k + 1 ∑ ∑Hl k 31H p k Lk k,p −kp− 1 ∑ ∑Hl k 41H p k Lp p,p + 1 ∑4kpkkpkH k l 1H l 1kL l 1l1 ,l 1.We can now state the main technical lemma of this section and of this paper.Lemma 4.6. The second order partial differential relations (4.5) hold truefor l = 1, . . .,m, and they are a consequence, by differentiations and bylinear combinations, of the fundamental first order partial differential equations(3.106), (3.108), (3.110) and (3.96).4.7. Reconstitution of the appropriate linear combinations. The remainingof Section 4 is entirely devoted to the proof of this statement. Fromthe manual computational point of view, the difficulty of the task is dueto the fact that one has to manipulate formal expressions having from 10to 50 terms. So the real question is: how can we reconstitute the linearcombinations and the differentiations which lead to the goal (4.5) from thedata (3.106), (3.108), (3.110) and (3.96)?.The main trick is to first neglect the first order and the zero order terms inthe goal (4.5). Using the symbol “≡” to denote “modulo first order and thezero order terms”, we formulate the following sub-goal:(4.8) 0 ≡?≡ −2 G l 1y l 1y l 1 + 4 3 Hl 1l 1 ,xy l 1 − 2 3 Ll 1l1 ,l 1 ,xx ,for l 1 = 1, . . .,m. Before estabilishing that these partial differential relationsare a consequence of the data (3.106), (3.108), (3.110) and (3.96)(written with a similar sign ≡), let us check that they are a consequenceof the existence of the change of coordinates (x, y) ↦→ (X, Y ) (however,recall that, in establishing the reverse implications of §3.111, we still donot know that such a change of coordinates really exists); this will confirmthe coherence and the validity of our computations. Importantly, we have63