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

308variables y j′iis provided (in principle) by the following compact formulas′1([BK1989]):⎛ ⎛(1.17)⎜⎝⎞Y j XD1X 1 1 · · · D1X 1 n ⎞1⎟. ⎠ = ⎝. · · · .⎠Y j D 1 X n n X1 · · · Dn 1Xn −1 ⎛⎝D1Y 1 j ⎞.⎠ ,Dn 1Y jwhere, for i ′ = 1, . . ., n, the symbol Di 1 denotes the ′ i′ -th first order totaldifferentiation operator:(1.18) Di 1 := ∂ m∑ ∂+ y j′′∂x i′ i.′∂y j′Striclty speaking, these formulas (1.17) are not explicit, because an inversematrix is involved and because the terms Di 1 ′Xi , Di 1 ′Y j are not developed.However, it would be feasible and elementary to write down thecorresponding totally explicit complete formulas for the functions Y j = X i 1Y j X i 1().j ′ =1x i′ , y j′ , y j′i ′ 1Next, the second prolongation ϕ (2) is of the form(1.19)ϕ (2) :)(x i′ , y j′ , y j′i, y j′′1 i ′ 1 ,i′ 2for some functions Y j X i 1X i 2↦−→()(ϕ(x (1) i′ , y j′ , y j′i ′ 1)x i′ , y j′ , y j′i ′ 1, y j′i ′ 1 ,i′ 2, Y j X i 1X i 2))(x i′ , y j′ , y j′i, y j′′1 i,′1 ,i′ 2which depend on the purefirst and second jet variables. For i = 1, . . .,n, the expressions of Y j X i 1Xare given by the following compact formulas (again [BK1989]):i⎛ ⎞Y j ⎛DX⎜i 1X 11 1⎟X1 · · · D 1 ⎞ ⎛ ⎞1 Xn−1D1Y 2 j X(1.20) ⎝ . ⎠ = ⎝. · · · .⎠ ⎜ i 1⎟ ⎝ . ⎠ ,Y j D 1 X i 1X nn X1 · · · Dn 1Xn DnY 2 j X i 1where, for i ′ = 1, . . .,n, the symbol Di 2 denotes the ′ i′ -th second order totaldifferentiation operator:(1.21) Di 2 := ∂ m∑ ∂m∑ n∑ ∂+ y j′′∂x i′ i+.′∂y j′j ′ =1j ′ =1i ′ 1 =1 y j′i ′ ,i ′ 1∂y j′i ′ 1Again, these formulas (1.20) are not explicit in the sense that an inversematrix is involved and that the terms Di 1 ′Xi , Di 2 ′Y j are not developed. ItX i 1would already be a nontrivial computational task to develope these expressionsand to find out some nice satisfying combinatorial formulas.In order to present the general inductive non-explicit formulas for thecomputation of the κ-th prolongation ϕ (κ) , we need some more notation.