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

I 5 = ∑ i 1I 6 = ∑ i 1 ,i 2I 7 = ∑ i 1 ,i 2⎡∑⎣ ∑l 1 ,...,l κ−1σ∈S κ−1κδ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)R j x kσ(κ) u i 1 −⎛−δ j ⎝ ∑i 1σ∈S κ−2κ⎡∑⎣ ∑ δ l τ(1),...,l τ(κ)k 1 ,......,k κl 1 ,...,l κ τ∈C κ−δ j i 2⎛⎝ ∑σ∈S κ−1κ⎞δ l 1,......,l κ−2 l κ−1 ⎠k σ(1) ,...,k σ(κ−2) x kσ(κ−1) x kσ(κ)⎛R j u i 1u − i 2 δj ⎝ ∑i 1σ∈S κ−1κ⎤⎦U i 1165l 1 ,...,l κ−1,δ l 1,......,l κk σ(1) ,...,k σ(κ−1)Q l 1x kσ(κ) u i 2(δ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)Q lκx kσ(κ) u i 1 + · · · + δl 3,......,l 1k σ(1) ,...,k σ(κ−1)Q l 2x kσ(κ) u i 2× U i 1l 1U i 2l 2 ,...,l κ,∑ [ (−δ j i 1δ l 2,...,l κ+1k 1 ,...,k κQ l 1+ · · · + u i 2 δl κ+1,...,l 2k 1 ,...,k κQ l 1u i 2l 3 ,...,l κ+1⎛⎞⎤−δ j ⎝ ∑i 2δ l τ(1),...,l τ(κ)⎠⎦U i 1τ∈S 2 κk 1 ,......,k κQ l κ+1u i 1)−l 1 ,l 2U i 2l 3 ,...,l κ+1,⎡⎛⎞⎤I 8 = ∑ ∑⎣δ l 1,...,l κk 1 ,...,k κR j − u i 1 δj ⎝ ∑i 1δ l 1,......,l κ−1k σ(1) ,...,k σ(κ−1)Q lκ ⎠x kσ(κ)⎦U i 1i 1 l 1 ,...,l κ σ∈Sκκ−1I 9 = ∑ ∑ [(−δ j i 1δ l 2,...,l κ+1k 1 ,...,k κQ l 1− u i 2 δj i 2δ l 1,...,l κk 1 ,...,k κQ l κ+1+ · · · + δ l 3,...,l 1u i 1i 1 ,i 2 l 1 ,...,l κ+1l 1 ,...,l κ,k 1 ,...,k κQ l 2u i 1× U i 1l 1U i 2l 2 ,...,l κ+1and where the term Remainder denotes the remaining terms in the expansionof R j k 1 ,k 2 ,...,k κ.In I 6 the summation on the upper indices (l 1 , . . .,l κ ) gets on all the circularpermutations of {1, 2, . . ., κ} except the identity. In I 7 the summationgets on all the circular permutations of {2, 3, . . ., κ + 1}. In I 9 the summationgets on all the circular permutations of {1, 2, . . ., κ + 1} except theone transforming (l 1 , l 2 , . . .,l κ+1 ) into (l 2 , l 3 , . . .,l 1 ). For κ = 3, comparingwith (7.28), we see that the formula remains valid, with the same conventionsas in the case n = 1.3.4. Lie criterion and defining equations of Sym(E ). We recall the Liecriterion, presented in Subsection 2.6 (see Theorem 2.71 of [Ol1986]):A vector field X is an infinitesimal symmetry of the completely integrablesystem (E ) if and only if its prolongation X (κ) of order κ is tangent to theskeleton ∆ E in the jet space J κ n,m .⎞⎠ −) ⎞ ⎤⎠⎦ ×)]×