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

functions w(z)). The coefficients R 1 l and R 2 k 1 ,k 2of the second prolongation135(7.7) X (2) =∑n−1k=1Q k∂+R ∂n−1∂z k ∂w + ∑l=1R 1 l∂∂W 1l+∑n−1k 1 ,k 2 =1R 2 k 1 ,k 2∂∂W 2 k 1 ,k 2,are completely determined by the following universal formulas(cf. [Ol1986], [Su2001a,b], [GM2001a]):(7.8) ⎧⎪⎨⎪⎩R 1 l = ∂ zl R + ∑ m 1[δ m 1l∂ w R − ∂ zl Q m 1] W 1 m 1+ ∑[−δ m 1l∂ w Q m 2] Wm 1 1Wm 1 2.m 1 ,m 2]R 2 k 1 ,k 2= ∂ 2 z k1 z k2R + ∑ m 1[δ m 1k 1∂ 2 z k2 w R + δm 1k 2∂ 2 z k1 w R − ∂2 z k1 z k2Q m 1W 1 m 1++ ∑ []δ m 1,m 2k 1 , k 2∂ 2 w 2R − δm 1k 1∂z 2 k2 w Qm 2− δ m 1k 2∂z 2 k1 w Qm 2Wm 1 1Wm 1 2+m 1 ,m 2+ ∑ [−δm 1 ,m 2k 1 , k 2∂ ] 2 w 2Qm 3Wm 1 1Wm 1 2Wm 1 3+m 1 ,m 2 ,m 3+ ∑ []δ m 1,m 2k 1 , k 2∂ w R − δ m 1k 1∂ zk2 Q m 2− δ m 1k 2∂ zk1 Q m 2Wm 2 1 ,m 2+m 1 ,m 2+ ∑ [−δm 1 ,m 2k 1 , k 2∂ w Q m 3− δ m 2,m 3k 1 , k 2∂ w Q m 1− δ m 3,m 1k 1 , k 2∂ w Q ] m 2Wm 1 1Wm 2 2 ,m 3.m 1 ,m 2 ,m 3In these formulas, by δl m we denote the Kronecker symbol equal to 1 if l =m and to 0 otherwise. The multiple Kronecker symbol δ m 1,m 2l 1 , l 2is defined tobe the product δ m 1l 1·δ m 2l 2. Finally, in the sums ∑ m 1, ∑ m 1 ,m 2and ∑ m 1 ,m 2 ,m 3,the integers m 1 , m 2 , m 3 run from 1 to n − 1. We would like to mentionthat in [GM2001a], we also provide some explicit expression of the k-thprolongation X (k) for k ≥ 3.Then the Lie criterion states that a holomorphic vector field X belongsto Sym(E M ) if and only if its second prolongation X (2) is tangent to ∆ M([Ol1986, Ch 2]). This gives the following equations:(7.9) R 2 k 1 ,k 2−∑n−1k=1Q k ∂ zk F k1 ,k 2− R ∂ w F k1 ,k 2−∑n−1l=1R 1 l ∂ W 1lF k1 ,k 2≡ 0,where 1 ≤ k 1 , k 2 ≤ n −1 and where each occurence of Wl 21 ,l 2is replaced byits value F l1 ,l 2on ∆ M . By developping (7.9) in power series with respect tothe variables Wl 1 , we get an expression of the form(7.10)∑l 1 ,...,l n−1 ≥0W 1l 1 · · ·W 1l n−1Φ l1 ,...,l n−1≡ 0,