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

conditions, totally equivalent to the compact identities (3.61):(3.65) ⎧( )Π j 0,0 −(Π j 0,l 1)x m = −Π0 0,0 Π j l 1 ,0 − ∑m Π k 0,0 Π j l 1 ,k + Π0 0,l 1Π j 0,0 + ∑Π k 0,l 1Π j 0,k ,⎪⎨⎪⎩y l 1( )(Π j l 1 ,l 2)x − Π j l 1 ,0y l 2(Π j l 1 ,l 2)y l 3)−(Π j l 1 ,l 3 y l 2k=1k=1m= −Π 0 l 1 ,l 2Π j 0,0 − ∑m Π k l 1 ,l 2Π j 0,k + Π0 l 1 ,0 Πj l 2 ,0 + ∑Π k l 1 ,0 Πj l 2 ,k ,k=141k=1m= −Π 0 l 1 ,l 2Π j l 3 ,0 − ∑m Π k l 1 ,l 2Π j l 3 ,k + Π0 l 1 ,l 3Π j l 2 ,0 + ∑Π k l 1 ,l 3Π j l 2 ,k ,k=1(Π00,0)y − ( Π 0 l 1 0,l 1)x = −Π0 0,0 Π0 l 1 ,0 − ∑m Π k 0,0a Π0 l 1 ,k + Π0 0,l 1Π 0 0,0 + ∑m Π k 0,la 1Π 0 0,k ,k=1( )Π0l1 ,l 2 x − ( Π 0 l 1 ,0)y m = l −Π0 2 l 1 ,l 2Π 0 0,0 − ∑m Π k l 1 ,l 2Π 0 0,k + Π0 l 1 ,0 Π0 l 2 ,0 + ∑Π k l 1 ,0 Π0 l 2 ,k ,k=1(Π0l1 ,l 2)y − ( Π 0 l 3 l 1 ,l 3)y m = l −Π0 2 l 1 ,l 2Π 0 l 3 ,0 − ∑m Π k l 1 ,l 2Π 0 l 3 ,k + Π0 l 1 ,l 3Π 0 l 2 ,0 + ∑Π k l 1 ,l 3Π 0 l 2 ,k .k=1k=1k=1k=1k=13.66. Convention about sums. Up to the end of Section 4, we shall abbreviateany sum ∑ mk=1 or ∑ mp=1 as ∑ k or ∑ p. Such sums will appear veryfrequently. For all other sums, we shall precisely write down the domain ofvariation of the summation index.3.67. Continuation. Thus, we have to replace (3.64) in the six identities(3.65). Firstly, let us expose all the intermediate steps in dealing with