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

+ 1 ∑2 δj l 1Hl k 3M l2 ,kk+ 1 4 δj l 1L l 3l3 ,l 3Θ l 2w7+ 1 4 δj l 1L l 3l3 ,l 3L l 2l2 ,l 2− 1 ∑t 2 δj l 1L l3 ,l 2L k k,k +ku− 1 ∑2 δj l 1L k l 3 ,l 2Θ k x++ 1 4 δj l 1L l 2l2 ,l 2Θ l 3+ 1 2 δj l 1M l3 ,l 2Θ 0 y+ 1 4 δj l 1Θ l 3Θ l 2+ 1 ∑2 δj l 3Hl k 1M l2 ,kk+ 1 4 δj l 3L l 1l1 ,l 1Θ l 2k9v+zk+ 1 4 δj l 3L l 1l1 ,l 1L l 2l2 ,l 2− 1 ∑g 2 δj l 3L k l 1 ,l 2L k k,k +kh− 1 ∑2 δj l 3L k l 1 ,l 2Θ k n++ 1 4 δj l 3L l 2l2 ,l 2Θ l 1+ 1 2 δj l 3M l1 ,l 2Θ 0 q+ 1 4 δj l 3Θ l 1Θ l 2Simplifying and ordering, we obtain the family (III) of partial differentialrelations of Theorem 1.7 (3):sm.0 = L j l 1 ,l 2 ,y l 3 − Lj l 1 ,l 3 ,y l 2 + δj l 3M l1 ,l 2 ,x − δ j l 2M l1 ,l 3 ,x+k57(3.110)+ 1 2 Hj l 3M l1 ,l 2− 1 2 Hj l 2M l1 ,l 3++ 1 ∑2 δj l 1Hl k 3M l2 ,k − 1 ∑2 δj l 1Hl k 2M l3 ,k+k+ 1 ∑2 δj l 3Hl k 1M l2 ,k − 1 ∑2 δj l 2Hl k 1M l3 ,k+kk+ ∑ L k l 1 ,l 3L j l 2 ,k − ∑ L k l 1 ,l 2L j l 3 ,k .kkk3.111. Arguments for the proof of Theorem 1.7 (3): necessity and sufficiencyof (I), (II), (III), (IV). Let us summarize the implications that havebeen established so far, from the beginning of Section 3. Recall that m 2.• There exist functions X, Y j of (x, y) transforming the system y j xx =F j (x, y, y x ), j = 1, . . ., m, to the free particle system Y j XX = 0, j =1, . . ., m.⇓• There exist functions Π j l 1 ,l 2of (x, y), 0 j, l 1 , l 2 m, satisfying thefirst auxiliary system (3.38) of partial differential equations.⇓• There exist (principal unknowns) functions Θ 0 , Θ j satisfying the sixfamilies of partial differential equations (3.69), (3.86), (3.89), (3.91),(3.93) and (3.96).