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

359(III’) ⎧0 = ∑ (δ k 2)j 3H k σ(1)j 1 ,j 2 ,y − δk σ(2)j 2H k σ(1)j 1 ,j 3 ,y +σ∈S 2+ 1 2 δk σ(2)j 2L k σ(1)− 1j 1 ,x j 32 δk σ(2)j 3L k σ(1)+ j 1 ,x j 2⎪⎨⎪⎩+ 1 2 δk σ(2)j 1+δ k σ(2)j 2L k σ(1)− 1j 2 ,x j 32 δk σ(2)j 1L k σ(1)+ j 3 ,x j 2G j1 ,j 3M k σ(1)− δ k σ(2)j 3G j1 ,j 2M k σ(1)++δ k σ(1),k σ(2)j 1 , j 2n∑+ 1 2 δk σ(1)j 1+ 1 2 δk σ(1)j 2+ 1 2 δk σ(1)j 3k 3 =1n∑k 3 =1n∑k 3 =1n∑k 3 =1G k3 ,j 3M k 3− δ k σ(1),k σ(2)j 1 , j 3n∑H k σ(2)k 3 ,j 3L k 3j 2− 1 2 δk σ(1)j 1H k σ(2)k 3 ,j 3L k 3j 1− 1 2 δk σ(1)j 3H k 3j 1 ,j 2L k σ(2)k 3− 1 2 δk σ(1)j 2n∑k 3 =1n∑k 3 =1n∑k 3 =1k 3 =1G k3 ,j 2M k 3+H k σ(2)k 3 ,j 2L k 3j 3+H k σ(2)k 3 ,j 2L k 3j 1+)H k 3j 1 ,j 3L k σ(2)k 3.(IV’) ⎧0 = ∑ ( 12 δk σ(3),k σ(2)j 3 , j 1L k σ(1)j 2 ,y − 1 2 δk σ(3),k σ(2)j 2 , j 1L k σ(1)j 3 ,y +σ∈S 3⎪⎨⎪⎩+δ k σ(3),k σ(2)j 2 , j 1M k σ(1)− δ k σ(3),k σ(2)x j 3 j 3 , j 1M k σ(1)+x j 2+δ k σ(3),k σ(1)j 2 , j 1n∑k 4 =1+ 1 4 δk σ(1),k σ(3)j 1 , j 3n∑k 4 =1H k σ(2)k 4 ,j 3M k 4− δ k σ(3),k σ(1)n∑j 3 , j 1k 4 =1L k σ(2)k 4L k 4j 2− 1 4 δk σ(1),k σ(3)n∑j 1 , j 2k 4 =1H k σ(2)k 4 ,j 2M k 4+)L k σ(2)k 4L k 4j 3.(These systems (I’), (II’), (III’) and (IV’) should be distinguished from thesystems (I), (II), (III) and (IV) of Theorem 1.7 in [Me2004], although theyare quite similar.) Here, the indices j 1 , j 2 , j 3 , k 1 , k 2 , k 3 vary in {1, 2, . . ., n}.By S 2 and by S 3 , we denote the permutation group of {1, 2} and of{1, 2, 3}. To facilitate hand- and Latex-writing, partial derivatives are denotedas indices after a comma; for instance, G j1 ,j 2 ,x j 3 is an abreviation for∂G j1 ,j 2/∂x j 3. To deduce (I’), (II’), (III’) and (IV’) from equation (1.11), we