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

Here, the sign ≡ precisely means: “modulo zero order terms”. We claimthat this approximated equation is a consequence of the existence of X, Y j .Indeed, according to the approximation (3.58), together with the definition(3.35) of the functions G j and H j l 1, we have45(3.81){ Gj= −□ j xx ∼ = −Y jH j l 1= −2 □ j xy l 1xx,∼= −2 Y jxy l 1 .Differentiation of the first line with respect to y l 1and of the second line withrespect to x yields:(3.82) G j y l 1∼ = −Yjxxy l 1and H j l 1 ,x ∼ = −2 Y jxy l 1x ,so that we indeed have 0 ≡ −2 G j + y l 1 Hj l 1 ,x, approximatively and modulothe derivatives of order 0, 1 and 2 of the functions X, Y j .Similar verifications have been effected constantly in our manuscript inorder to control the truth of the formal computations that we shall exposeuntil the end of Section 4.3.83. Continuation. From now on and up to the end of Section 4, the hardestcomputational core of the proof may — at last — be developed. Furtheramazing computational obstacles will be encountered.Replacing plainly (3.64) in (3.65) 2 , we get:(3.84)( )(Π j l 1 ,l 2)x − Π j l 1 ,0y l 2= −L j l 1 ,l 2 ,x + 1 2 δj l 1L l 2l2 ,l 2 ,x + 1 2 δj l 2L l 1l1 ,l 1 ,x + 1 2 δj l 1Θ l 2 x ++ 1 2 δj l 2Θ l 1 x + 1 2 Hj l 1 ,y l 2 − 1 2 δj l 1Θ 0 y l 2 == −Π 0 l 1 ,l 2 · Π j 0,0 − ∑ Π k l 1 ,l 2 · Π j 0,k + Π0 l 1 ,0 · Πj l 2 ,0 + ∑ Π k l 1 ,0 · Πj l 2 ,k =kk= M l1 ,l 2G j − ∑ (−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+ 1 2 δk l 2L l 1l1 ,l 1+ 1 2 δk l 1Θ l 2+ 1 )2 δk l 2Θ l 1·k·(− 1 2 Hj k + 1 )2 δj k Θ0 +( 1+2 Ll 1l1 ,l 1+ 1 )2 Θl 1·(− 1 2 Hj l 2+ 1 )2 δj l 2Θ 0 ++ ∑ (− 1 2 Hk l 1+ 1 )2 δk l 1Θ 0 ·k(· −L j l 2 ,k + 1 2 δj l 2L k k,k + 1 2 δj k Ll 2l2 ,l 2+ 1 2 δj l 2Θ k + 1 )2 δj k Θl 2.