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

Let (z 0 , c 0 ) = (x 0 , y 0 , a 0 , b 0 ) ∈ M be a fixed point, let x 1 := (x 1 1, . . ., x n 1) ∈K n and define the multiple flow map(10.9) {Lx1 (x 0 , y 0 , a 0 , b 0 ) := exp(x 1 L)(p 0 ) := exp ( x n 1 L n(· · ·(exp(x11 L 1 (z 0 , c 0 ))) · · ·) ):= ( x 0 + x 1 , Π(x 0 + x 1 , a 0 , b 0 ), a 0 , b 0).Similarly, for a 1 = (a 1 1, . . ., a p 1) ∈ K p , define the multiple flow map(10.10) L ∗ a 1(x 0 , y 0 , a 0 , b 0 ) := ( x 0 , y 0 , a 0 + a 1 , Π ∗ (a 0 + a 1 , x 0 , y 0 ) ) .Starting from the (z 0 , c 0 ) = (0, 0) and moving alternately along F v , F p , F v ,etc., we obtain⎧Γ 1 (x 1 ) := L x1 (0),⎪⎨ Γ 2 (x 1 , a 1 ) := L ∗ a(10.11)1(L x1 (0)),Γ 3 (x 1 , a 1 , x 2 ) := L x2 (L⎪⎩∗ a 1(L x1 (0))),Γ 4 (x 1 , a 1 , x 2 , a 2 ) := L ∗ a 2(L x2 (L ∗ a 1(L x1 (0)))),and so on. Generally, we get chains Γ k := Γ k ([xa] k ), where [xa] k :=(x 1 , a 1 , x 2 , a 2 , . . .) with exactly k terms, where each x l ∈ K n and eacha l ∈ K p .If, instead, the first movement consists in moving along F p , we start withΓ ∗ 1 (a 1) := L ∗ a 1(0), Γ ∗ 2 (a 1, x 1 ) := L x1 (L ∗ a 1(0)), etc., and generally we get dualchains Γ ∗ k ([ax] k), where [ax] k := (a 1 , x 1 , a 2 , x 2 , . . .), with exactly k terms.Both Γ k and Γ ∗ k have range in M .For k = 1, 2, 3, · · · , integers e k and e ∗ k are defined inductively by{e1 + e 2 + e 3 + · · · + e k = genrk K (Γ k ),(10.12)e ∗ 1 + e∗ 2 + e∗ 3 + · · · + e∗ k = genrk K(Γ ∗ k ).By (10.9) and (10.10), it is clear that e 1 = n, e 2 = p, e ∗ 1 = p, and e∗ 2 = n.Example 10.13. For y xx = 0, the submanifold of solutions M is simplyy = b + xa, whence⎧⎪⎨Γ 1 (x 1 ) = (x 1 , 0, 0, 0),(10.14)Γ 2 (x 1 , a 1 ) = (x 1 , 0, a 1 , −x 1 a 1 ),⎪⎩Γ 3 (x 1 , a 1 , x 2 ) = (x 1 + x 2 , x 2 a 1 , a 1 , −x 1 a 1 ).The rank at (0, 0, 0) of Γ 3 is equal to two, not more. However, its genericrank is equal to three. Similar observations hold for the two submanifoldsof solutions y = b + xxa + xaa and y = b + xa 1 + xxa 2 (in K 5 ).Lemma 10.15. If genrk K (Γ k+1 ) = genrk K (Γ k ), then for each positive integerl 1, we have genrk K (Γ k+l ) = genrk K (Γ k ). The same stabilizationproperty holds for Γ ∗ k .293