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

248Definition 4.2. ([Ol1986, Ol1995, BK1989]) ϕ is a (local) Lie symmetryof (E ) if it transforms the graph of every solution of (E ) into the graph ofanother solution.To explain, we must pass to jet spaces. Denote the components of the(κ + 1)-th prolongation ϕ (κ+1) : Jn,m κ+1 → Jn,m κ+1 by(4.3) ϕ (κ+1) = ( φ i 1, ψ j 1, Φ j i 1, Φ j i 1 ,i 2, . . .. . . , Φ j i 1 ,i 2 ,...,i κ+1).The restriction ϕ (κ+1)∣ ∣∆Eis obtained by replacing each jet variable yα j byFα j , whenever (j, α) ≠ (j, 0) and ≠ (j(q), β(q)), and wherever it appears32in the Φ j i 1 ,...,i λ.Let π κ,p denote the projection from Jn,m κ+1 to ∆ E ≃ K m+n+p defined by((4.4) π κ,p x i , y j , y j i 1, . . .,y j ) ( )i 1 ,...,i κ+1 := x i , y j , y j(q) ,and introduce the map(4.5) ϕ ∆E := π κ,p ◦ ( ϕ (κ+1)∣ ∣∆E)≡(ϕ(x i , y j ), Φ j(q)β(q)β(q)(x i , y j , y j(q 1)β(q 1 )) ) .Lemma 4.6. ([Ol1986, Ol1995, BK1989], [∗]) The following three conditionsare equivalent:(1) the diffeomorphism ϕ is a Lie symmetry of (E );(2) ϕ (κ+1)∣ ∣∆Esends ∆ E to ∆ E ;(3) ϕ (κ+1)∣ ∣∆Esends ∆ E to ∆ E and ϕ ∆E = π κ,p(ϕ(κ+1) ∣ ∣∆E)is a symmetryof the foliation F ∆E , namely it sends every leaf to some other leaf.Then the set of Lie symmetries of (E ) constitutes a local Lie(pseudo)group.4.7. Infinitesimal Lie symmetries of (E ). Letn∑(4.8) L = X i (x, y) ∂∂x + ∑ mY j (x, y) ∂i ∂y , ji=1be a (local) vector field on K n+m having analytic coefficients. Denote itsflow by ϕ t (x, y) := exp(t L )(x, y), t ∈ K. As in Section 1(II), by differentiatingthe prolongation (ϕ t ) (κ+1) with respect to t at t = 0, we get theprolonged vector field L (κ+1) on Jn,m κ+1 , having the general form (Part II):(4.9)L (κ+1) = L +m∑j=1n∑i 1 =1Y j i 1∂∂y j i 1+· · ·+j=1m∑j=1with known explicit expressions for the Y j i 1 ,...,i λ.n∑i 1 ,...,i κ+1 =1Y j i 1 ,...,i κ+1∂∂y j i 1 ,...,i κ+1,32 Remind from Section 1(II) that we have not (open problem) provided a complete explicitexpression of Φ j i 1,...,i λfor general n 1, m 1 and λ 1.