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

and we then expand carefully the result by collecting somewhat in advancethe obtained terms with respect to the derivatives of T :(G yxy x= − Q b ∂∆ ∂a + Q )[a ∂− Q b∆ ∂b ∆ T a + Q ]a∆ T b(Qb Q ab∆ − Q )b Q b ∆ a∆ ∆ 2 T a + Q b Q b∆ ∆ T aa+=∆(+ − Q b∆(+ − Q a∆(Qa+∆Q aa∆ + Q )b Q a ∆ a∆ ∆ 2 T b − Q b∆Q bb∆ + Q )a Q b ∆ b∆ ∆ 2 T a − Q a∆Q ab∆ − Q a∆)Q a ∆ b∆ 2 T b + Q a∆Q a∆ T ab+Q b∆ T ab+Q a∆ T bb.The terms involving T aa , T ab , T bb are exactly the ones exhibited by the lemmafor the expression of G yxy x. In the four large parentheses which are coefficientsof T a , T b , T a , T b , we replace the occurences of ∆ a , ∆ a , ∆ b , ∆ b simplyby:∆ a = Q xb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa∆ b = Q xb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab ,and the total sum of terms coefficiented by T a in our expression now becomes:T(a [ ] [ ]∆ 3 Q b Q ab Qa Q xb − Q b Q xa − Qb Q b Qxb Q aa + Q a Q xab − Q xa Q ab − Q b Q xaa −[ ] [ ] )− Q a Q bb Qa Q xb − Q b Q xa + Qa Q b Qxb Q ab + Q a Q xbb − Q xa Q bb − Q b Q xab == T (a∆ 3 Q a Q b Q xb Q ab − Q b Q b Q xa Q ab1 − Q b Q b Q xb Q aa − Q a Q b Q b Q xab ++ Q b Q b Q xa Q ab1 + Q b Q b Q b Q xaa −− Q a Q a Q xb Q bb + Q a Q b Q xa Q bb2 + Q a Q b Q xb Q ab + Q a Q a Q b Q xbb −)− Q a Q b Q xa Q bb − Q 2a Q b Q b Q xab == T (a [ ] [ ]∆ 3 Q a Q a Qb Q xbb − Q xb Q bb − 2 Qa Q b Qb Q xab − Q xb Q ab ++ Q b Q b[Qb Q xaa − Q xb Q aa] ) ,so that we now have effectively reconstituted the three 2 × 2 determinantsappearing in the second line of the expression claimed by the lemma forthe transfer of G yxy xto the (x, a, b)-space. The treatment of the coefficientof T bmakes only a few differences, hence will be skipped here (but not∆ 3in the manuscript). Finally, the two remaining expressions for G yyx andfor G yy are obtained by performing entirely analogous algebrico-differentialcomputations.End of the proof of the Main Theorem. Applying the above formula forG yxy xwith x := z, with a := z, with b := w, with ∆ := Θ z Θ zw − Θ w Θ zz ,209