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

160Note that this identity involves only the troncature of D k to order λ, denotedby Dk λ , and defined by⎧Dk ⎪⎨λ := ∂ m∑+ U i ∂1k∂x k ∂u + ∑ m n∑U i ∂1i 1 k,l 1i 1 =1i 1∂U i + · · ·+1=1 l 1 =1 l 1(7.28)m∑ n∑+ U ⎪⎩i ∂1k,l 1 ,...,l λ−1∂U i .1l 1 ,...,l λ−1i 1 =1 l 1 ,...,l λ−1 =1According to Theorem 2.36 of [Ol1986], the prolongation of order κ of avector field X = ∑ nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j , denoted byX (κ) , is the unique vector field on the space Jn,m κ of the form(7.28) ⎧X ⎪⎨(κ) = X +⎪⎩m∑j=1+n∑k 1 =1m∑j=1R j k 1∂∂U j +k 1n∑k 1 ,...,k κ=1m∑j=1R j k 1 ,...,k κn∑k 1 ,k 2 =1R j k 1 ,k 2∂∂U j k 1 ,k 2 ,...,k κ,∂∂U j k 1 ,k 2+ · · ·+corresponding to the infinitesimal action of the flow of X on the jets of orderκ of the graphs of maps u = u(x), and whose coefficients are computedrecursively by the formulas⎧n∑R j k 1:= Dk 1 1(R j ) − Dk 1 1(Q l 1) U j l 1,(7.28)⎪⎨R j k 1 ,k 2:= D 2 k 2(R j k 1) −l 1 =1n∑Dk 1 2(Q l 1) U j k 1 ,l 1,l 1 =1· · · · · · · · · · · · · · · · · · · · · · · · · · ·n∑R ⎪⎩j k 1 ,k 2 ,...,k λ:= Dk λ λ(R j k 1 ,...,k λ−1) − Dk 1 λ(Q l 1) U j k 1 ,...,k λ−1 ,l 1.For a better comprehension of the general computation, let us start by computingR κ in the case n = m = 1.3.2. Computation of R κ when n = m = 1. A direct application of thepreceding formulas leads to the following classical expressions:(7.28) ⎧⎪⎨R 1 = R x + [R u − Q x ] U 1 + [−Q u ] (U 1 ) 2 .R 2 = R x 2 + [2R xu − Q x 2] U 1 + [R u 2 − 2Q xu ] (U 1 ) 2 + [−Q u 2] (U 1 ) 3 +⎪⎩+ [R u − 2Q x ] U 2 + [−3Q u ] U 1 U 2 .l 1 =1