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

3.12. Principal unknowns. As there are (m + 1) more square (or Pi) functionsthan the functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1, we cannot invert directlythe linear system (3.2). To quasi-inverse it, we choose the (m + 1) specificsquare functions(3.13) Θ 1 := □ 1 x 1 x 1, Θ2 := □ 2 x 2 x 2, · · · · · · , Θn+1 := □ n+1x n+1 x n+1 ,calling them principal unknowns, and we get the quasi-inversion:(3.14) ⎧Π k 1j 1 ,j 2= □ k 1x j 1x = j 2 Hk 1j 1 ,j 2− 1 2 δk 1j 1H j 2j 2 ,j 2− 1 2 δk 1j 2H j 1j 1 ,j 1+ 1 2 δk 1j 1Θ j 2+ 1 2 δk 1j 2Θ j 1,⎪⎨⎪⎩Π k 1j 1 ,n+1 = □k 1x j 1y = 1 2 Lk 1j 1+ 1 2 δk 1j 1Θ n+1 ,Π k 1n+1,n+1 = □k 1yy = Mk 1,Π n+1j 1 ,j 2= □ n+1x j 1x j 2 = −G j 1 ,j 2,Π n+1j 1 ,n+1 = □n+1 x j 1y = −1 2 Hj 1j 1 ,j 1+ 1 2 Θj 1.3.15. Second auxiliary system. Replacing the five families of functionsΠ k 1j 1 ,j 2, Π k 1j 1 ,n+1 , Πk 1n+1,n+1 , Πn+1 j 1 ,j 2, Π n+1j 1 ,n+1 by their values obtained in (3.14)just above together with the principal unknowns(3.16) Π j 1j1 ,j 1= Θ j 1, Π n+1n+1,n+1 = Θn+1 ,in the six equations (3.11) 1 , (3.11) 2 , (3.11) 3 , (3.11) 4 , (3.11) 5 and (3.11) 6 ,after hard computations that we will not reproduce here, we obtain six familiesof equations. From now on, we abbreviate every sum ∑ nk=1 as ∑ k 1.Firstly:(3.17) 0 = G j1 ,j 2 ,x j 3 − G j1 ,j 3 ,x j 2 + ∑ k 1G j3 ,k 1H k 1j 1 ,j 2− ∑ k 1G j2 ,k 1H k 1j 1 ,j 3.This is (I’) of Theorem 1.7. Just above and below, we plainly underline themonomials involving a first order derivative. Secondly:(3.18) ⎧Θ j 1= −2 G x j 2 j 1 ,j 2 ,y + H j 1+ j 1 ,j 1 ,x j 2373⎪⎨+ ∑ k 1G j2 ,k 1L k 1j 1+ 1 2 Hj 1j 1 ,j 1H j 2j 2 ,j 2− ∑ k 1H k 1j 1 ,j 2H k 1k 1 ,k 1−− G j1 ,j 2Θ n+1 − 1 2 Hj 1j 1 ,j 1Θ j 2− 1 2 Hj 2j 2 ,j 2Θ j 1+ ∑ k 1H k 1j 1 ,j 2Θ k 1+⎪⎩+ 1 2 Θj 1Θ j 2.