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

⎧Θ l 1y l 2 = −Ll 1l 1 ,l 1 ,y l 2 + 2 M l 1 ,l 2 ,x+53(3.102)⎪⎨⎪⎩+ ∑ kH k l 1M l2 ,k + 1 2 Ll 1l1 ,l 1L l 2l2 ,l 2− ∑ k+ 1 2 Ll 1l1 ,l 1Θ l 2+ 1 2 Ll 2l2 ,l 2Θ l 1− ∑ k+ M l1 ,l 2Θ 0 + 1 2 Θl 1Θ l 2.L k l 1 ,l 2L k k,k +L k l 1 ,l 2Θ k +We notice that the right-hand side of (3.99) should be independent of l 1 ;this phenomenon will be explained in a while.Proof. For Θ 0 x in (3.99), it suffices to put j := l 1 in (3.69).To obtain Θ 0 y l 1 , we put j := l 2 and l 2 := l 1 in (3.86), which yields:(3.103)Θ l 1 x − 1 2 Θ0 y l 1 = −1 2 Hl 1l 1 ,y l 1 ++ G l 1M l1 ,l 1− 1 2+ 1 2− 1 4∑k∑k∑kH k l 1L l 1l1 ,k − 1 4H l 1kL k l 1 ,l 1+∑Hl k 1L k k,k −kH k l 1Θ k + 1 4 Ll 1l1 ,l 1Θ 0 + 1 4 Θ0 Θ l 1.We may easily solve Θ 0 y l 1 and Θl 1 x thanks to this equation (3.103) and thanksto (3.91): indeed, to obtain (3.100), it suffices to compute 4 3 · (3.91) + 2 3 ·(3.103); to obtain (3.101), it suffices to compute 2 3 · (3.91) + 4 3 · (3.103).Finally, (3.102) is a copy of (3.93). This completes the proof.3.104. Appearance of the crucial four families of first order partial differentialrelations (I), (II), (III) and (IV) of Theorem 1.7 (3). However,in solving Θ 0 x, Θ 0 , y l 1 Θl 1 x and Θ l 1from our six families of equations (3.69),y l 2(3.86), (3.89), (3.91), (3.93) and (3.96), only a subpart of these equations hasbeen used. We notice that the two families of equations (3.91) and (3.93)have been used completely and that the family of equations (3.96), whichdoes not involve Θ, coincides precisely with the system (IV) of Theorem1.7 (3). To insure that Θ 0 x , Θ0 , y l 1 Θl 1 x and Θ l 1as written in Proposi-y l 2tion 3.98 are true solutions, it is necessary and sufficient that they satisfythe remaining equations. Thus, we have to replace these solutions (3.99),(3.100), (3.101) and (3.102) in the three remaining families (3.69), (3.86)and (3.89).Firstly, let us insert inside (3.69) the value of Θ 0 x given by the equation(3.99), in which the index l 1 is replaced in advance by an arbitrary