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

Thirdly, put j := l 2 in (3.108) with l 1 ≠ l 2 and differentiate with respect tox:(4.25)0 ≡ − 1 2 Hl 2l 1 ,y l 2x + 1 3 Hl 1l 1 ,y l 1x + Ll 2l1 ,l 2 ,xx − 2 3 Ll 1l1 ,l 1 ,xx ++ G l 2 x M l1 ,l 2+ G l 2 x M l1 ,l 2 ,x − 2 3 Gl 1 x M l1 ,l 1− 2 3 Gl 1M l1 ,l 1 ,x−− 1 ∑G k x3M l 1 ,k − 1 ∑G k M l1 ,k,x − 1 ∑H l 232 k,xL k l 1 ,l 2− 1 2+ 1 2− 1 3k∑k∑kH k l 1 ,x Ll 2l2 ,k + 1 2H k l 1 ,x Ll 1l1 ,k − 1 3k∑k∑kH k l 1L l 2l2 ,k,x + 1 3H k l 1L l 1l1 ,k,x .k∑kH l 1k,xL k l 1 ,l 1+ 1 3∑k∑k67H l 2kL k l 1 ,l 2 ,x +H l 1kL k l 1 ,l 1 ,x −Fourthly, put j := l 1 in (3.108) with l 2 ≠ l 1 , differentiate with respect to xand permute the indices (l 1 , l 2 ) ↦→ (l 2 , l 1 ):(4.26)0 ≡ − 1 2 Hl 2l 2 ,y l 1x + 1 6 Hl 1l 1 ,y l 1x + Ll 2l2 ,l 1 ,xx − 1 3 Ll 1l1 ,l 1 ,xx ++ G l 2 x M l2 ,l 1+ G l 2M l2 ,l 1 ,x − 1 3 Gl 1 x M l1 ,l 1− 1 3 Gl 1M l1 ,l 1 ,x++ 1 ∑G k x M l1 ,k + 1 ∑G k M l1 ,k,x − 1 ∑H l 2332 k,xL k l 2 ,l 1− 1 2+ 1 2− 1 6k∑k∑kH k l 2 ,x Ll 2l1 ,k + 1 2H k l 1 ,x Ll 1l1 ,k − 1 6k∑k∑kH k l 2L l 2l1 ,k,x + 1 6H k l 1L l 1l1 ,k,x .k∑kH l 1k,xL k l 1 ,l 1+ 1 6∑k∑kH l 2kL k l 2 ,l 1 ,x +H l 1kL k l 1 ,l 1 ,x −