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

Lemma 7.11. ([CM1974, BER1999, Me2005a], [∗]) In coordinates x ′ =(x ′1 , . . .,x ′n ) and y ′ = (y ′1 , . . .,y ′m ) an arbitrary submanifold M ′ definedby y ′ = Π ′ (x ′ , a ′ , b ′ ) or dually by b ′ = Π ′∗ (a ′ , x ′ , y ′ ) is equivalent to(7.12) y = Π(x, a, b) or dually to b = Π ∗ (a, x, y)with(7.13)Π(0, a, b) ≡ Π(x, 0, b) ≡ b or dually Π ∗ (0, x, y) ≡ Π ∗ (a, 0, y) ≡ y,namely Π = b + O(xa) and Π ∗ = y + O(ax).Proof. We develope(7.14) y ′ = Π ′ (0, a ′ , b ′ ) + Λ ′ (x ′ ) + O(x ′ a ′ ).Since b ′ ↦→ Π ′ (0, a ′ , b ′ ) has rank m at b ′ = 0, the coordinate change(7.15) b ′′ := Π ′ (0, a ′ , b ′ ), a ′′ := a ′ , x ′′ := x ′ , y ′′ := y ′ ,transforms M ′ to M ′′ defined by(7.16) y ′′ = Π ′′ (x ′′ , a ′′ , b ′′ ) := b ′′ + Λ ′ (x ′′ ) + O(x ′′ a ′′ ).Solving b ′′ by means of the implicit function theorem, we get(7.17) b ′′ = Π ′′∗ (a ′′ , x ′′ , y ′′ ) = y ′′ − Λ ′ (x ′′ ) + O(a ′′ x ′′ ),and it suffices to set y := y ′′ − Λ ′ (x ′′ ), x := x ′′ and a := a ′′ , b := b ′′ .Taking account of solvability with respect to the parameters, finer normalizationsholds.Lemma 7.18. With n = m = κ = 1, every submanifold of solutions y ′ =b ′ + x ′ a ′[ ]1 + O 1 of y′x ′ x = F ′ (x ′ , y ′ , y ′ ′ x ′) is equivalent to(7.19) y xx = b + xa + O(x 2 a 2 ).Proof. Writing y ′ = b ′ +x ′[ a ′ +a ′ Λ ′ (a ′ , b ′ )+O(x ′ a ′ ) ] , where Λ ′ = O 1 , weset a ′′ := a ′ + a ′ Λ ′ (a ′ , b ′ ), b ′′ := b ′ , x ′′ := x ′ , y ′′ := y ′ , whence y ′′ = b ′′ +x ′′[ a ′′ +O(x ′′ a ′′ ) ] . Dually b ′′ = y ′′ −a ′′[ x ′′ +x ′′ x ′′ Λ ′′ (x ′′ , y ′′ )+O(x ′′ x ′′ a ′′ ) ] ,so we set x := x ′′ + x ′′ x ′′ Λ ′′ (x ′′ , y ′′ ), y := y ′′ , a := a ′′ , b := b ′′ .Corollary 7.20. Every second order ordinary differential equation y x ′ ′ x = ′F ′ (x ′ , y ′ , yx ′ ′) is equivalent to(7.21) y xx = (y x ) 2 R(x, y, y x ).265