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

+ 1 ∑4 δj l 1Hl k 2L k k,kk− 2 3 δj l 2G l 1M l1 ,l 19+ 1 ∑4 δj l 2Hl k 1L k k,kke+ 1 4 δj l 1∑− 1 ∑3 δj l 2G k M l1 ,kakk+ 1 4 δj l 2∑+ 1 3 δj l 1G l 2M l2 ,l 2+ 2 ∑8 3 δj l 1G k M l2 ,k− 1 ∑4 δj l 1Hl k 2L k k,kkekk− 1 4 δj l 1∑k55Hl k 2Θ k − 1 4 δj l 1L l 2l2 ,l 2Θ 0 − 1g 4 δj l 1Θ 0 Θ l 2−hf11+ 1 3 δj l 2∑kH l 1kL k l 1 ,l 116− 1 3 δj l 2∑Hl k 1Θ k − 1 4 δj l 2L l 1l1 ,l 1Θ 0 − 1c 4 δj l 2Θ 0 Θ 0 d+b10− 1 6 δj l 1∑kH l 2kL k l 2 ,l 214k+ 1 6 δj l 1∑Hl k 2Θ k + 1 4 δj l 1L l 2l2 ,l 2Θ 0 + 1g 4 δj l 1Θ 0 Θ l 2.hfkH k l 1L l 1l1 ,kH k l 2L l 2l2 ,k1715+−Simplifying and ordering, we obtain the family (II) of partial differentialrelations of Theorem 1.7 (3):(3.108)0 = − 1 2 Hj l 1 ,y l 2 + 1 6 δj l 1H l 2l 2 ,y l 2 + 1 3 δj l 2H l 1l 1 ,y l 1 ++ L j l 1 ,l 2 ,x − 1 3 δj l 1L l 2l2 ,l 2 ,x − 2 3 δj l 2L l 1l1 ,l 1 ,x ++ G j M l1 ,l 2− 1 3 δj l 1G l 2M l2 ,l 2− 2 3 δj l 2G l 1M l1 ,l 1+ 1 ∑3 δj l 1G k M l2 ,k−− 1 ∑3 δj l 2G k M l1 ,k − 1 ∑H j k2Lk l 1 ,l 2+ 1 ∑Hl k 2 1L j l 2 ,k +k+ 1 6 δj l 1∑k+ 1 3 δj l 2∑kH l 2kL k l 2 ,l 2− 1 6 δj l 1∑H l 1kL k l 1 ,l 1− 1 3 δj l 2∑kkkH k l 2L l 2l2 ,k +H k l 1L l 1l1 ,k .kkThirdly, let us insert inside (3.89) the values of Θ l 2, of y l 3 Θl 3, of y l 2 Θl 1and y l 3of Θ l 1given by (3.102). We place all the terms in the right-hand side of they l 2equality and we place the first order terms in the beginning (first four lines