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

Here, R j l 1 ,kare universal polynomials.Π j , we getx l 1x l 2(6.19)∂ 2 Π j∂x l 1 xl 2=By induction, for every β ∈ N n :(6.20)∂ |β| Π j∂x β =257Solving the second derivatives( {Lk } )S j 1i′l 1 ,l 2 1L k ′2ϕ i′ ′ n+m1k 1 ′ ,k′ 2[n det ( L k ′ φ l′) ]1l ′ n 3.1k ′ n[S j β( {L } )β ′ 1iϕ i′ ′ n+m|β ′ ||β|det ( L k ′ φ l′) ]1l ′ n 2|β|+1,1k ′ nwhere S j β are universal polynomials. Here, for β′ ∈ N n , we denote by L β′the derivation of order |β ′ | defined by (L 1 ) β′ 1 · · ·(Ln ) β′ n .Next, thanks to the assumption that M is solvable with respect to theparameters, there exist integers j(1), . . ., j(p) with 1 j(q) m and multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 and max 1qp |β(q)| = κsuch that the local K-analytic map(6.21) ((ΠK p+m ∋ c ↦−→ j (0, c) ) ( ) )1jm ∂ |β(q)| Π j(q), (0, c) ∈ K p+m∂x β(q) 1qphas rank p + m at c = 0. We then consider in (6.20) only the (p + m)equations written for (j, 0), (j(q), β(q)) and we solve h(c) by means of theanalytic implicit function theorem:(6.22)h = Ĥ⎛⎜⎝φ,( {LS j(1) } ) (β ′ 1iβ(1)ϕ i′ ′ n+m{LS j(p) } ) ⎞β ′ 1i|β ′ ||β(1)|β(p)ϕ i′ ′ n+m|β[ (Lkdet ′ φ l′) ]1l ′ n 2|β(1)|+1, . . .,′ ||β(p)| ⎟[ (Lkdet ′ φ l′) ]1l ′ n 2|β(p)|+1⎠ .1k ′ n1k ′ nFinally, by developping every derivative L β′ ϕ i′ (including L k ′φ l′ as a specialcase), taking account of the fact that the coefficients of the L k ′ depend(|βdirectly on Π, we get some universal polynomial P β ′ J ′ |+1x Π, J |β′ |)x,y ϕ i′ .Inserting above, we get the map Ĥ.6.23. Pseudogroup of twin transformations. The previous considerationslead to introducing the following.Definition 6.24. By G v,p , we denote the infinite-dimensional (pseudo)groupof local K-analytic diffeomorphisms(6.25) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )that respect the separation between the variables and the parameters.