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

following equations for j = 1, . . ., m and k 1 , . . .,k κ = 1, . . .,n:(7.28)⎧ [ n∑R j k⎪⎨1 ,...,k κ− Q ∂F j m∑l k 1 ,...,k κ+ R ∂F j i k 1 ,...,k κ+∂x l ∂u i l=1i=1⎤⎪⎩+ ∑ ∑ ∂F jR i 1 k 1 ,...,k κl1i 1∂U i + · · · + ∑ ∑ ∂F jR i 1 k 1 ,...,k κ1l1⎦,...,l κ−1l 1 l 1 i 1∂U i ≡ 0,1l 1 ,...,l κ−1 l 1 ,...,l κ−1on ∆ E , by replacing the variables U i 1l 1 ,...,l κby F i 1l 1 ,...,l κwherever they appear.Let us expand F j k 1 ,...,k κand their partial derivatives and use the fact thatR j k 1 ,...,k λare polynomials expressions of the jets variables (U i 1l 1, . . .,U i 1l 1 ,...,l λ),with coefficients being linear expressions of the partial derivatives of order≤ λ+1 of Q l and R j . We obtain for j = 1, . . ., m and k 1 , . . .,k κ = 1, . . .,nsome identities of the form(7.28) ⎧ ∑⎪⎨⎪⎩i 1 , ...,l 1 ,...)Φ j;i 1,......k 1 ,...,k κ; l 1 ,......(x,u,(Q l x α u) β 1≤l≤n, |α|+|β|≤κ+1 ,(R j )x α u β 1≤j≤m, |α|+|β|≤κ+1 ×171×U i 1l 1... U iµ 1l µ1× U iµ 1 +1l µ1 +1,l µ1 +2 · · · Ui µ 1 +µ 2 −1l µ1 +2µ 2 −1 Ui µ 1 +µ 2l µ1 +2µ 2× · · · · · · ≡ 0,satisfied if and only if the functions Q l and R j are solutions of the followingsystem of partial differential equations(7.28)Φ j,i (1,......k 1 ,...,k κ; l 1 ,...... x, u, (Qlx α u) β 1≤l≤n, |α|+|β|≤κ+1 , (R j ))x α u β 1≤j≤m, |α|+|β|≤κ+1 = 0.5.2. Homogeneous system. We start by giving the general form of thesymmetries of the homogeneous system in the case κ = 2. Then we provethe equality dim K (Sym(E 0 )) = n 2 +2n+m 2 +m Cn+κ−1 in the case κ ≥ 3.In the case κ = 2 we obtain:⎧n∑m∑Q l (x, u) = A l + Bk l 1x k1 + Ci l 1u i 1+(7.28)⎪⎨⎪⎩R j (x, u) = F j +k 1 =1n∑k 1 =1+n∑k 1 =1G j k 1x k1 ++n∑k 1 =1i 1 =1D k1 x l x k1 +m∑E i1 x l u i 1,i 1 =1m∑H j i 1u i 1+i 1 =1D k1 x k1 u j +m∑E i1 u i 1u j .i 1 =1