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

256of K n x × K m y × K p a × K m bsolutionsthat maps to itself the associated submanifold of(6.11) M E = { (x, y, a, b) : y = Π(x, a, b) } .Proof. In fact, we know that the n-dimensional leaf {( x, Π(x, a, b) ) : x ∈K n} is sent {( x, Π(x, h(a, b)) ) : x ∈ K n} .Equivalently, setting c := (a, b) and writing (ϕ, h) = (φ, ψ, h), we haveψ = Π(φ, h) when y = Π(x, c), namely(6.12) ψ(x, Π(x, c)) ≡ Π ( φ(x, Π(x, c)), h(c) )Proposition 6.13. There exists a universal rational map Ĥ such that()(6.14) h(c) ≡ Ĥ J κ+1x,a,b Π(x, c), Jκ x,y ϕ(x, Π(x, c))This shows unique determination of h from ϕ, given (E ) or equivalently,given Π.Proof. Differentiating a function χ(x, Π(x, c)) with respect to x k , k =1, . . ., n, corresponds to applying to χ the vector field(6.15) L k := ∂∂x k+m∑j=1∂Π j(x, c) ∂ , k = 1, . . ., n.∂x k ∂yj Thus, applying L k to the m scalar equations (6.12), we get(6.16) L k ψ j =n∑l=1∂Π j∂x l L k φ l ,for 1 k n and 1 j m. It follows from the assumption that ϕ is alocal diffeomorphism that det ( L k φ l (0) ) 1ln≠ 0 also. So we may solve1knthe first derivatives Π x above: there exist universal polynomials S j lsuch that(∂Π j S j {Lk } ) 1i′lϕ i′ ′ n+m1k(6.17)=′ n∂x l det ( L k ′ φ l′) .1l ′ n1k ′ nAgain, we apply the L k to these equations, getting, thanks to the chain rule:( {Lk } )n∑R j 1i′l 1 ,k 1L k ′2ϕ i′ ′ n+m1k 1(6.18)′ ,k′ 2] n 2.l 2 =1∂ 2 Π j∂x l 1 xl 2L k φ l 2=[det ( L k ′ φ l′) 1l ′ n1k ′ n