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

1623.3. Computation of R κ in the general case. Following the exact samescheme as in the case n = 1 we give the general partial formula forR κ . We start with the first three families of coefficients R j k 1, R j k 1 ,k 2andR j k 1 ,k 2 ,k 3. Let δp q be the Kronecker symbol, equal to 1 if p = q and to 0 ifp ≠ q. More generally, the generalized Kronecker symbols are defined byδ q 1,...,q kp 1 ,...,p k:= δ q 1p 1δ q 2p 2 · · ·δ q kp k.By convention, the indices j, i 1 , i 2 , . . . , i λ run in the set {1, . . ., m}, theindices k, k 1 , k 2 , . . . , k λ and l, l 1 , l 2 , . . . , l λ running in {1, . . ., n}. Hence wewill write ∑ m ∑ mi 1 =1 i 2 =1 · · ·∑mi λ =1 as ∑ i 1 ,...,i λand ∑ n ∑ nl 1 =1 l 2 =1 · · ·∑nl λ =1as ∑ l 1 ,...,l λ. The letters i 1 , i 2 , . . .,i λ and l 1 , l 2 , . . .,l λ will always be used forthe summations in the development of R j k 1 ,k 2 ,...,k λ. We will always use theindices j and k 1 , k 2 , . . ., k λ to write the coefficient R j k 1 ,k 2 ,...,k λ.We have:(7.28)⎧R ⎪⎨j k 1= Rx j k1+ ∑ i 1⎪⎩∑ []δ l 1k1R j − u i 1 δj i 1Q l 1xk1U i 1l 1+l 1+ ∑ ∑i 1 ,i 2l 1 ,l 2[−δji 2δ l 1k1Q l 2u i 1]Ui 1l 1U i 2l 2.For R j k 1 ,k 2we have:(7.28) ⎧R j k 1 ,k 2= Rx j k1 x k2+ ∑ i 1+ ∑ ∑ [(δ l 1,l 2k 1 ,k 2R j u i 1u − i 2 δj i 2δ l 1k1Q l 2+ x k2 u i 1 δl 1k2Q l 2x k1 u i 1i 1 ,i 2 l 1 ,l 2⎪⎨+ ∑⎪⎩i 1 ,i 2 ,i 3∑ [−δ j i 3δ l 1,l 2k 1 ,k 2Q l 3u i 1u i 2l 1 ,l 2 ,l 3∑l 1[δ l 1k2R j x k1 u i 1 + δl 1k1R j x k2 u i 1 − δj i 1Q l 1xk1 x k2]U i 1l 1+]U i 1l 1U i 2l 2U i 3l 3++ ∑ ∑ []δ l 1,l 2k 1 ,k 2R j − u i 1 δj i 1δ l 1k2Q l 2xk1− δ j i 1δ l 1k1Q l 2xk2U i 1l 1 ,l 2+i 1 l 1 ,l 2+ ∑ i 1 ,i 2∑l 1 ,l 2 ,l 3[−δ j i 2δ l 1,l 2k 1 ,k 2Q l 3u i 1 − δj i 2δ l 3,l 1k 1 ,k 2Q l 2u i 1 − δj i 1δ l 2,l 3k 1 ,k 2Q l 1u i 2)]U i 1l 1U i 2l 2+]U i 1l 1U i 2l 2 ,l 3.Since we also treat systems of order κ ≥ 3, it is necessary to computeR j k 1 ,k 2 ,k 3. We write this as follows:(7.28) R j k 1 ,k 2 ,k 3= I + II + III,