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

Proof. Let l = l ( x i , y j , yβ(q)) j(q) be any function defined on ∆E . Composingwith A yields the function Λ := l ◦ A, i.e.((2.24) Λ(x, a, b) ≡ lx i , Π j (x, a, b), Π j(q)x β(q) (x, a, b)Differentiating with respect to x i , we get, dropping the arguments:(2.25)∂Λ∂x = ∂li ∂x + ∑ m ij=1Π j x i∂lp∑∂y + jq=1Π j(q)x i x β(q)).∂l.∂y j(q)x β(q)Replacing the appearing Π j xα for which (j, α) ≠ (j, 0) and ≠ (j(q), β(q))by Fα j, we recover D i as defined by (1.38), whence ∂Λ = D∂x i i l.2.26. Transfer of algebrico-differential expressions. The diffeomorphismA may be used to translate algebrico-differential expressions from M to (E )and vice-versa:(2.27) I M(Jλ+κ+1x,a,bΠ ) ←→ I (E )(Jλx,y,y1 F ) .Here, λ ∈ N, the letter J is used to denote jets, and I = I M or = I (E) is apolynomial or more generally, a quotient of polynomials with respect to itsjet arguments. Notice the shift by κ + 1 of the jet orders.Example 2.28. Suppose n = m = 1 and κ = 1. Then F = Π xx . As anexercise, let us compute F x , F y , F y1 in terms of Jx,a,b 3 Π. We start with theidentity(2.29) F(x, y, y 1 ) ≡ Π xx(x, A(x, y, y1 ), B(x, y, y 1 ) ) ,that we differentiate with respect to x, to y and to y 1 :F x = Π xxx + Π xxa A x + Π xxb B x ,(2.30)F y = Π xxa A y + Π xxb B y ,F y1 = Π xxa A y1 + Π xxb B y1 .Thus, we need to compute A x , A y , A y1 , B x , B y , B y1 . This is easy: it sufficesto differentiate the two identities that define A and B as implicit functions,namely:y ≡ Π ( x, A(x, y, y 1 ), B(x, y, y 1 ) )and(2.31)(y 1 ≡ Π x x, A(x, y, y1 ), B(x, y, y 1 ) )with respect to x, to y and to y 1 , which gives six new identities:0 = Π x + Π a A x + Π b B x , 0 = Π xx + Π xa A x + Π xb B x ,(2.32)1 = Π a A y + Π b B y , 0 = Π xa A y + Π xb B y0 = Π a A y1 + Π b B y1 , 1 = Π xa A y1 + Π xb B y1 ,and to solve each of the three linear systems of two equations located ina line, noticing that their common determinant Π b Π xa − Π a Π xb does not243