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

Here, the sign ≡ means “modulo zero order terms”. Before proceding further,recall the correspondence between partial differential relations:(4.13)(I) = (3.106),(II) = (3.108),(III) = (3.110),(IV) = (3.96).However, these couples of equivalent identities are written slightly differently,as may be read by comparison. To fix ideas and to facilitate the eyecheckingof our subsequent computations, we shall only use and refer to theexact writing of (3.106), of (3.108), of (3.110) and of (3.96).4.14. Construction of a guide. So we want to show that the approximate relation(4.8) is a consequence, by differentiations and by linear combinations,of the approximate identities (4.12). The interest of working with approximateidentities is that formal computations are lightened substantially. Afterhaving discovered which linear combinations and which differentiations areappropriate, i.e. after having constructed a “guide”, in §4.22 below, we shallwrite down the complete computations, including all zero order terms, followingour guide.We shall use two indices l 1 and l 2 with 1 l 1 , l 2 m and, crucially,l 2 ≠ l 1 . Again, the assumption m 2 is used strongly.Firstly, put j := l 1 in (3.106) mod with l 2 ≠ l 1 and differentiate withrespect to y l 1:(4.15) 0 ≡ −2G l 1y l 1y l 1 + 2Gl 2y l 2y l 1 + Hl 1l 1 ,xy l 1 − Hl 2l 2 ,xy l 1 .Secondly, put j := l 2 in (3.106) mod with l 1 ≠ l 2 and differentiate withrespect to y l 2:(4.16) 0 ≡ −2G l 2y l 1y l 2 + Hl 2l 1 ,xy l 2 .Thirdly, put j := l 2 in (3.108) mod with l 1 ≠ l 2 and differentiate with respectto x:(4.17) 0 ≡ − 1 2 Hl 2l 1 ,y l 2x + 1 3 Hl 1l 1 ,y l 1x + Ll 2l1 ,l 2 ,xx − 2 3 Ll 1l1 ,l 1 ,xx .Fourthly, put j := l 1 in (3.108) mod with l 2 ≠ l 1 and differentiate with respectto x:(4.18) 0 ≡ − 1 2 Hl 1l 1 ,y l 2x + 1 6 Hl 2l 2 ,y l 2x + Ll 1l1 ,l 2 ,xx − 1 3 Ll 2l2 ,l 2 ,xx .Fithly, permute the indices (l 1 , l 2 ) ↦→ (l 2 , l 1 ):(4.19) 0 ≡ − 1 2 Hl 2l 2 ,y l 1x + 1 6 Hl 1l 1 ,y l 1x + Ll 2l2 ,l 1 ,xx − 1 3 Ll 1l1 ,l 1 ,xx .65