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

300by means of the analytic implicit function theorem; also, in the second lineof (11.14), we consider the (n+m) equations written for (j, 0), (j ∼ (l), δ(l))and we solve ϕ(z). We get:(11.17)⎧⎪⎨⎪⎩h(c) = Ĥ⎛⎜⎝φ(z),⎛ϕ(z) = ̂Φ ⎜⎝f(c),S j(1)β(1)( {Lβ ′ ϕ i′ (z) } )1i ′ n+m|β ′ ||β(1)|[ (Lkdet ′ φ l′ (z) ) ]1l ′ n 2|β(1)|+1, . . .1k ′ n( {LS j(p) β ′β(p)ϕ i′ (z) } ) ⎞1i ′ n+m|β. . .,′ ||β(p)| ⎟[ (Lkdet ′ φ l′ (z) ) ]1l ′ n 2|β(p)|+1⎠,1k ′ n( {L∗δ ′ h i′ (c) } )1i ′ p+mS ∗ j ∼ (1)δ(1)|δ ′ ||δ(1)|[det ( L ∗ qf ′ r′ (c) ) ]1r ′ p 2|δ(1)|+1, . . .. . .,S ∗ j ∼ (n)δ(n)[1q ′ p( {L∗δ ′ h i′ (c) } ) ⎞1i ′ p+m|δ ′ ||δ(n)| ⎟] 2|δ(n)|+1⎠ ,det ( L ∗ q ′ f r′ (c) ) 1r ′ p1q ′ pfor (z, c) ∈ M . The maps Ĥ and ̂Φ depend only on Π ′ , Π ′∗ .Lemma 11.18. For every β ′ ∈ N n , there exists a universal polynomial P β ′ inthe jet variables J |β′ |z having K-analytic coefficients in (z, c) which dependsonly on Π, Π ∗ such that, for i ′ = 1, . . .,n + m:( )(11.19) L β′ ϕ i′ (z) ≡ P β ′ z, c, J |β′ |z ϕ i′ (z) .A similar property holds for L ∗δ′ h i′ (c).We deduce that there exist two local K-analytic mappings Φ 0 0 and H0 0 suchthat we can write{ (ϕ(z) = Φ00 z, c, Jκ ∗c(11.20)h(c)) ,h(c) = H0( 0 z, c, Jκz ϕ(z) ) ,for (z, c) ∈ M . Concretely, this means that we have two equivalent pairs offormal identities(ϕ(z) ≡ Φ 0 0 z, a, Π ∗ (a, z), Jc κ∗h(a, Π∗ (a, z)) )ϕ ( x, Π(x, c) ) (≡ Φ 0 0 x, Π(x, c), c, Jκ ∗c(11.21)( h(c))h(c) ≡ H00 x, Π(x, c), c, Jκz ϕ(x, Π(x, c)) )h ( a, Π ∗ (a, z) ) (≡ H00 z, a, Π ∗ (a, z), Jz κ ϕ(z))