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

Next, replacing plainly (3.64) in (3.65) 3 , we get:(3.87) (Π j l 1 ,l 2)y l 3)−(Π j l 1 ,l 3 y l 2== −L j l 1 ,l 2 ,y l 3 + 1 2 δj l 1L l 2l 2 ,l 2 ,y l 3 + 1 2 δj l 2L l 1l 1 ,l 1 ,y l 3 + 1 2 δj l 1Θ l 2y l 3 + 1 2 δj l 2Θ l 1y l 3 ++ L j l 1 ,l 3 ,y l 2 − 1 2 δj l 1L l 3l 3 ,l 3 ,y l 2 − 1 2 δj l 3L l 1l 1 ,l 1 ,y l 2 − 1 2 δj l 1Θ l 3y l 2 − 1 2 δj l 3Θ l 1y l 2 == −Π 0 l 1 ,l 2 · Π j l 3 ,0 − ∑ Π k l 1 ,l 2 · Π j l 3 ,k + Π0 l 1 ,l 3 · Π j l 2 ,0 + ∑kk= −M l1 ,l 2 ·(− 1 2 Hj l 3+ 1 )2 δj l 3Θ 0 − ∑ k)+ 1 2 δk l 2L l 1l1 ,l 1+ 1 2 δk l 1Θ l 2+ 1 2 δk l 2Θ l 1+ 1 2 δj k Ll 3l3 ,l 3+ 1 2 δj l 3Θ k + 1 2 δj k Θl 3+ M l1 ,l 3 ·)+(− 1 2 Hj l 2+ 1 )2 δj l 2Θ 0 + ∑ k)+ 1 2 δk l 3L l 1l1 ,l 1+ 1 2 δk l 1Θ l 3+ 1 2 δk l 3Θ l 1+ 1 2 δj k Ll 2l2 ,l 2+ 1 2 δj l 2Θ k + 1 2 δj k Θl 2).Π k l 1 ,l 3 · Π j l 2 ,k =(−L k l 1 ,l 2+ 1 2 δk l 1L l 2l2 ,l 2+(· −L j l 3 ,k + 1 2 δj l 3L k k,k +(−L k l 1 ,l 3+ 1 2 δk l 1L l 3l3 ,l 3+(· −L j l 2 ,k + 1 2 δj l 2L k k,k +Developing the products and ordering each monomial, we get:(3.88)= 1 2 Hj l 3M l1 ,l 2− 11 2 δj l 3M l1 ,l 2Θ 0 − ∑ L k l 1 ,l 2L j l 3 ,k+ 1 ∑162 δj l 3k+ 1 2 Lj l 1 ,l 2L l 3l3 ,l 3+ 1 ∑a 2 δj l 3L k l 1 ,l 2Θ kk136+ 1 2 Lj l 1 ,l 2Θ l 3bk47L k l 1 ,l 2L k k,k+ 1 2 Ll 2l2 ,l 2L j l 3 ,l 1c−− 1 4 δj l 3L l 2l2 ,l 2L l 1l1 ,l 1− 14 4 δj l 1L l 2l2 ,l 2L l 3l3 ,l 3− 1d 4 δj l 3L l 2l2 ,l 2Θ l 1− 1e 4 δj l 1L l 2l2 ,l 2Θ l 3+f+ 1 2 Ll 1l1 ,l 1L j l 3 ,l 2− 1g 4 δj l 3L l 1l1 ,l 1L l 2l2 ,l 2− 1h 4 δj l 2L l 1l1 ,l 1L l 3l3 ,l 3− 1i 4 δj l 3L l 1l1 ,l 1Θ l 2−j7+− 1 4 δj l 2L l 1l1 ,l 1Θ l 3k+ 1 2 Lj l 3 ,l 1Θ l 2l− 1 4 δj l 3L l 1l1 ,l 1Θ l 210− 1 4 δj l 1L l 3l3 ,l 3Θ l 2−m− 1 4 δj l 3Θ l 2Θ l 1n− 1 4 δj l 1Θ l 2Θ l 3o+ 1 2 Lj l 3 ,l 2Θ l 1p− 1 4 δj l 3L l 2l2 ,l 2Θ l 112−− 1 4 δj l 2L l 3l3 ,l 3Θ l 1q− 1 4 δj l 3Θ l 1Θ l 218− 1 4 δj l 2Θ l 1Θ l 3−r