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

In terms of Sussmann’s approach [27], this means that the local orbit ofthe two systems of vector fields (L k ) 1≤k≤n and (Lq ∗)1≤q≤p is of maximal dimension.Reasoning as in [27] (using the so-called backward trick in ControlTheory, see also [Me2003]), it may be shown that there exists the smallesteven integer 2µ 0 such that the ranks of the two maps Γ 2µ0 and Γ ∗ 2µ 0at theorigin (not only their generic rank) in K nµ 0+pµ 0are both equal to dim K M .This means that Γ 2µ0 and Γ ∗ 2µ 0are submersive onto a neighborhood of theorigin in M . We call µ 0 the type of the pair of foliations (F p , F v ). It mayalso be established that µ 0 ≤ m + 2.Example 2.46. We give an example of a submanifold which is both 1-solvable with respect to the parameters and with respect to the variablesbut whose pair of foliations is not covering: with n = 1, m = 2 andp = 1, this is given by the two equations u 1 = ν 1 , u 2 = ν 2 + xχ 1 .Then Sym(M ) is infinite-dimensional since it contains the vector fieldsa(u 1 ) ∂/∂u 1 +a(ν 1 ) ∂/∂ν 1 , where a is an arbitrary K-analytic function. Forthis reason, we shall assume in the sequel that the pair of foliations (F p , F v )is covering at the origin.2.11. Estimate on the dimension of the local symmetry group of thesubmanifold of solutions. We may now formulate the main theorem ofthis section, which shows that, under suitable nondegeneracy conditions,Sym(M ) is a finite dimensional local Lie group of local transformations. Ift ∈ K n+m , we denote by |t| := max 1≤k≤n+m |t k |. If (h, φ) ∈ Sym(M )we denote by J k t h(0) the k-th order jet of h at the origin and by J k τ φ(0) thek-th order jet of φ at the origin. Also, we shall assume that M is eitherK-algebraic or K-analytic. Of course, the K-algebraicity of the submanifoldof solutions does not follow from the K-algebraicity of the right hand sidesF j α of the system of partial differential equations (E ).Theorem 6.4. Assume that the K-algebraic or K-analytic submanifold ofsolutions M of the completely integrable system of partial differential equations(E ) is both l 0 -sovable with respect to the parameters and l ∗ 0 -solvablewith respect to the variables. Assume that the fundamental pair of foliations(F p , F v ) is covering at the origin and let µ 0 be its type at the origin. Thenthere exists ε 0 > 0 such that for every ε with 0 < ε < ε 0 , the following fourproperties hold:(a) The (pseudo)group Sym(M ) of local K-analytic diffeomorphisms definedfor {(t, τ) ∈ K n+2m+p : |t| < ε, |τ| < ε} which are of theform (t, τ) ↦→ (h(t), φ(τ)) and which stabilize M is a local Lie pseudogroupof transformations of finite dimension d ∈ N.(b) Let κ 0 := µ 0 (l 0 + l ∗ 0). Then there exist two K-algebraic or K-analyticmappings H κ0 and Φ κ0 which depend only on M and which may beconstructed algorithmically by means of the defining equations of M157