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

262where 1 q p, 1 l n, 1 k n and γ ∈ N p . Then F q definedby (6.48) and G j defined by (6.45) are independent of x.This provides a second algorithm, essentially equivalent to Sophus Lie’s.Example 6.55. For y xx (x) = F(x, y(x), y x (x)), the first line of (6.54) is (thesecond one is redundant):(6.56) ⎧0 ≡ X [−Π xa Π xxx Π b + Π a Π xb Π xxx − Π x Π xxa Π xb + Π xa Π x Π xxb +⎪⎨⎪⎩+ Y [−Π xa Π xxb + Π xxa Π xb ] ++Π xxa Π xx Π b − Π a Π xxb Π xx ] ++ X x [−2Π xx Π xa Π b + 2Π xx Π a Π xb + Π x Π b Π xxa − Π x Π a Π xxb ] ++ Y x [−Π b Π xxa + Π a Π xxb ] ++ X y[−3Πx Π xx Π xa Π b + 3Π x Π a Π xx Π xb + (Π x ) 2 Π b Π xxa − (Π x ) 2 Π a Π xxb]++ Y y [Π xx Π b Π xa − Π xx Π a Π xb − Π x Π b Π xxa + Π x Π a Π xxb ] ++ X xx [−Π x Π b Π xa + Π x Π a Π xb ] +[ ]+ X xy −2(Πx ) 2 Π b Π xa + 2(Π x ) 2 Π a Π xb +[ ]+ X y 2 −(Πx ) 3 Π b Π xa + (Π x ) 3 Π a Π xb ++ Y xx [Π b Π xa − Π a Π xb ]++ Y xy [2Π x Π b Π xa − 2Π x Π a Π xb ]+[ ]+ Y y 2 (Πx ) 2 Π b Π xa − (Π x ) 2 Π a Π xb .We observe the similarity with (4.19): the expression is linear in the partialderivatives of X , Y of order 2, but the coefficients in the equation aboveare more complicated. In fact, after dividing by −Π b Π xa + Π a Π xb , thisequation coincides with (4.21), thanks to Π x = y 1 and to the formulas (2.34)for F x , F y , F y1 .6.57. Infinitesimal CR automorphisms of generic submanifolds. If thesystem (E ) is associated to the complexification M = (M) c of a genericM ⊂ C n+m as in §1.16, then a = (¯z) c = ζ, b = ( ¯w) c = ξ, and the vectorfield L ∗ associated to an infinitesimal Lie symmetry(6.58) L =n∑i=1X i (z, w) ∂∂z i + m∑i=1j=1Y j (z, w)∂∂w jof (E ) is simply the complexification L of its conjugate L , namelyn∑(6.59) L ∗ = L = X i (ζ, ξ) ∂∂ζ + ∑ mY j (ζ, ξ) ∂i ∂ξ . jj=1