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

2667.22. Complete normal forms. The Moser theory of normal forms may betransferred with minor modifications to submanifolds of solutions associatedto (E 1 ) and to (E 2 ).Theorem 7.23. ([CM1974, Ja1990], [∗]) A local K-analytic submanifold ofsolutions associated to (E 1 ):(7.24) y ′ = b ′ + x ′ a ′ + O 3 = ∑ ∑Π ′ k ′ ,l ′(b′ ) x ′ k ′ a ′ l ′k ′ 0 l ′ 0can be mapped, by a transformation (x ′ , y ′ , a ′ , b ′ ) ↦→ (x, y, a, b) belongingto G v,p , to a submanifold of solutions of the specific form(7.25)y = b + xa + Π 2,4 (b) x 2 a 4 + Π 4,2 (b) a 2 x 4 + ∑ ∑ ∑Π k,l (b) x k a l .k2 l2 k+l7Solving (a, b) from y = Π and y x = Π x with Π as above, we deduce thefollowing.Corollary 7.26. Every y x ′ ′ x = F ′ (x ′ , y ′ , y ′ ′ x ′) is equivalent to(7.27)y xx = (y x ) [ 2 x 2 F 2,2 (y) + x 3 r(x, y) ] + (y x ) [ 4 F 0,4 (y) + x r(x, y) ] ++ ∑ ∑ ∑F k,l (y) x k (y x ) l .k0 l0 k+l5For the completely integrable system (E 2 ) having several dependent variables(x 1 , . . .,x n ), n 2, we have the following.Theorem 7.28. ([CM1974], [∗]) A local K-analytic submanifold of solutionsassociated to (E 2 ):(7.29) y ′ = b ′ + ∑x ′k a ′k + O 31kncan be mapped, by a transformation (x ′ , y ′ , a ′ , b ′ ) ↦→ (x, y, a, b) belongingto G v,p , to a submanifold of solutions of the specific form:(7.30) y = b + ∑x k a k + ∑ ∑Π k,l (x, a, b)1kn k2 l2where(7.31)Π k,l (x, a, b) :=∑∑k 1 +···+k n=k l 1 +···+l n=lwith the terms Π 2,2 , Π 2,3 and Π 3,3 satisfying:Π k1 ,...,k n,l 1 ,...,l n(b) (x 1 ) k1 · · ·(x n ) kn (a 1 ) l1 · · ·(a n ) ln(7.32) 0 = ∆ Π 2,2 = ∆∆ Π 2,3 = ∆∆ Π 3,2 = ∆∆∆ Π 3,3 ,