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

for some two local analytic maps Π = (Π 1 , . . .,Π m ) and Ω = (Ω 1 , . . .,Ω p )and the next lemma is straightforward.Lemma 2.4. The general solution of (E ) is(2.5) y(x) := Π(x, a, b),where (a, b) varies in K p × K m . Furthermore, for q = 1, . . .,p:(2.6) Ω q (x, a, b) ≡ Π j(q)x β(q) (x, a, b).This leads to introducing a fundamental geometric object.Definition 2.7. The submanifold of solutions V S (E ) associated to (E ) isthe analytic submanifold of K n x × K m y × K p a × K m b defined by the Cartesianequations:(2.8) 0 = −y j + Π j (x, a, b), j = 1, . . .,m.There is a strong interplay between the study of (E ) and the geometry ofV S (E ). By construction, the diffeomorphism:(2.9)⎧(⎨ A : K n+p+m [coordinates (x i , a q , b j )] −→ K[coordinatesn+m+p()⎩A (x i , a q , b j ) := x i , Π j (x, a, b), Π j(q) (x, a, b), ,x β(q)sends the foliation F v by the variables x whose leaves are {a = cst., b =cst.} (see the diagram), to the previous foliation F ∆E .2.10. PDE system associated to a submanifold. Inversely, let M be a submanifoldof K n x × Km y × Kp a × Km b of the form(2.11) y j = Π j (x, a, b), j = 1, . . ., m.A necessary condition for it to be the complexification of a generic M ⊂C n+m is that p = n (answer to an exercise above).241x i , y j , y j(q)β(q)Definition 2.12. M is solvable with respect to the parameters ifb ↦→ Π(0, 0, b) of rank m at b = 0 and if there exist κ 1, multiindicesβ(1), . . ., β(p) ∈ N n with |β(q)| 1 for q = 1, . . .,pand max 1qp |β(q)| = κ, together with integers j(1), . . ., j(p) with1 j(q) m such that the local K-analytic map(2.13) ( (ΠK m+p ∋ (a, b) ↦−→ j (0, a, b) ) ( ) )1jm, Π j(q) (0, a, b) ∈ K m+px β(q) 1qpis of rank equal to m + p at (a, b) = (0, 0))]