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

158such that every element (h, φ) ∈ Sym(M ), sufficiently close to theidentity mapping, may be represented by{ h(t) = Hκ0 (t, J κ 0t h(0)),(7.28)φ(τ) = Φ κ0 (τ, J κ 0τ φ(0)).Consequently, every element of Sym(M ) is uniquely determined byits κ 0 -th jet at the origin and the dimension d of the Lie algebraSym(M ) is bounded by the number of components of the vector(J κ 0t h(0), J κ 0τ φ(0)), namely we have(7.28)dim K Sym(E ) = dim K Sym(M ) ≤ (n+m) C κ 0n+m+κ 0+(m+p) C κ 0m+p+κ 0.(c) There exists ε ′ with 0 < ε ′ < ε and a K-algebraic or K-analytic mapping(H M , Φ M ) which may be constructed algorithmically by meansof the defining equations of M , defined in a neighbourhood of theorigin in K n+2m+p × K d with values in K n+2m+p and which satifies(H M (t, 0), Φ M (τ, 0)) ≡ (t, τ), such that every element (h, φ) ∈Sym(M ) defined on the set {(t, τ) ∈ K n+2m+p : |t| < ε ′ , |τ| < ε ′ },sufficiently close to the identity mapping and stabilizing M may berepresented as (h(t), φ(τ)) ≡ (H M (t, s h,φ ), Φ M (τ, s h,φ )) for a uniqueelement s h,φ ∈ K d depending on the mapping (h, φ).(d) The mapping (t, τ, s) ↦−→ (H M (t, s), Φ M (τ, s)) defines a local K-algebraic or K-analytic Lie group of local K-algebraic or K-analytictransformations stabilizing M .2.12. Applications. The proof of Theorem 6.4, which possesses strongsimilarities with the proof of Theorem 4.1 in [8], will not be presented.It seems that Theorem 6.4, together with the argumentation on the necessityof assumptions that M be solvable with respect to the variables andthat its fundamental pair of foliations be covering, is a new result aboutthe finite-dimensionality of a completely integrable system of partial differentialequations having an arbitrary number of independent and dependentvariables. The main interest lies in the fact that we obtain the algorithmicallyconstructible representation formula (7.28) together with the local Lie groupstructure mapping (H M , Φ M ). In particular, we get as a corollary that everytransformation (h(t), φ(τ)) given by a formal power series (not necessarilyconvergent) is as smooth as the applications (H κ0 , Φ κ0 ) are, namely everyformal element of Sym(M ) is necessarily K-algebraic or K-analytic. As acounterpart of its generality, Theorem 3 does not provide optimal bounds,as shows the following illustration.Example 2.46. Let n = m = 1, let κ ≥ 3 and let (E ) denote the ordinarydifferential equation u x κ(x) = F(x, u(x), u x (x), . . .,u x κ−1(x)). Thenthe submanifold of solutions M is of the form u = ν + xχ 1 + · · · +