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

defined by the graphed equations:((1.35) yα j = Fαj x, y, ( y j(q) )β(q)1qpfor ( j, α ) ≠ (j, 0) and ≠ ( j(q), β(q) ) with |α| κ + 1. Clearly, the naturalcoordinates on ∆ E are:(1.36) (x, y, ( y j(q)β(q))1qp)≡),(x, y, ( y j(q) ) )l 1 (q),...,l λq (q)∈ K n × K m × K p ,1qpwhere λ q := |β(q)| and ( l 1 (q), . . ., l λq (q) ) := β(q).Next, in view of the form (1.34) of the generators of (C T κ+1n,m )⊥ and inview of the equations of ∆ E , the intersection(1.37) (C T κ+1n,m )⊥ ∩ T∆ Eis a vector subbundle of T∆ E that is generated by n linearly independentvector fields obtained by restricting the D i to ∆ E , which yields:⎧D i = ∂ ∑ m ( )⎪⎨ ∂x i+ A j i x i 1, y j 1, y j(q 1) ∂β(q 1 )∂y + j j=1(1.38)⎪⎩+p∑q=1B q i(x i 1, y j 1, y j(q 1)β(q 1 )) ∂∂y j(q)β(q)i = 1, . . .,n, where the coefficients A j i and Bq i are given by:(1.39) {yjA j i := i if the variable yj i appears among the p variables yj(q 1)β(q 1 ) ;F ji otherwise;B q i := ⎧⎨⎩,239y j(q)i,l 1 (q),...,l λq (q)if yj(q)l 1 (q),...,l λq(q)appears among the p variables y j(q 1)β(q 1 ) ;F j(q)i,l 1 (q),...,l λq (q) otherwise.Example 1.40. For (E 1 ), we get D = ∂ + y ∂x 1 ∂ + F(x, y, y ∂x 1) ∂∂y 2; exercise:treat (E 2 ) and (E 3 ). For (E 4 ), we get D = ∂ + ∂x y1 1 ∂ + F ∂ + G ∂ . For∂y 1 ∂y 2 ∂y11(E 5 ), whose skeleton is written y 2 = F , y 1,1,1 = G, y 1,2 = H, y 1,1,2 = K,with F , G, H, K being functions of ( )x 1 , x 2 , y, y 1 , y 1,1 , we get(1.41)D 1 = ∂∂x 1 + y 1∂∂y + y 1,1∂+ G ∂ ,∂y 1 ∂y 1,1D 2 = ∂∂x 2 + F ∂ ∂y + H ∂∂y 1+ K∂∂y 1,1.Definition 1.42. The system (E ) is completely integrable if the n vectorfields (1.38) satisfy the Frobenius integrability condition, namely every Lie