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

where the letter r denotes an unspecified function of (x 1 , x 2 ), and where thecoefficient s ∗ of (y 1 ) 2 y 1,1 in the first equation satisfies (8.68).Proof. To get the second equation, we compute:279(8.70)y 1,2 = a 1 a 2 [2 + b x 1 x 1] + a1 a 1 a 2 [g x 1 x 2] + (a1 ) 3 R + (a 2 ) 2 R= 1 2 y 1y 1,1 + (y 1 ) 2 y 1,1 r + (y 1 ) 3 R + (y 1,1 ) 2 R.The third equation is got similarly from (8.63) 3 . To conclude, we developethe first two equations mod [ (y 1 ) 6 , (y 1,1 ) 2] and the third onemod [ (y 1 ) 4 , (y 1,1 ) 3] .This precise skeleton will be referred to as ∆ E5 in the sequel. With theletter r, the computation rules are cst.r = r + r = r + s ∗ = r · r = r;sometimes, s ∗ may be replaced plainly by r.8.71. Infinitesimal Lie symmetries of (E 5 ). Letting L = X 1 ∂X 2 ∂ + Y ∂ be a candidate infinitesimal Lie symmetry and applying∂x 2 ∂y(8.72)L (3) = X 1∂∂x + X 2 ∂1 ∂x + Y ∂2 ∂y + Y 1+ Y 1,1∂∂y 1,1+ Y 1,2∂∂y 1,2+ Y 2,1+ Y 1,1,1∂∂y 1,1,1+ · · · + Y 2,2,2∂+ Y 2∂y 1∂+ Y 2,2∂y 2,1∂∂y 2,2,2to the skeleton ∆ E5 , we obtain firstly, computing mod [ (y 1 ) 5 , y 1,1]:0 ≡ −Y 2 + 1 2 y 1 Y 1 +∂∂y 2+∂x 1 +∂∂y 2,2+(8.73)+ (y 1 ) 3 r X 1 + (y 1 ) 4 r X 1 + (y 1 ) 3 r X 2 + (y 1 ) 4 r X 2 ++ Y 1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ++ Y 1,1[(y1 ) 2 s ∗ + (y 1 ) 3 r + (y 1 ) 4 r ] ,secondly, computing mod [ (y 1 ) 5 , y 1,1]:0 ≡ −Y 1,2 + 1 2 y 1 Y 1,1 +(8.74)+ (y 1 ) 3 r X 1 + (y 1 ) 4 r X 1 + (y 1 ) 3 r X 2 + (y 1 ) 4 r X 2 ++ Y 1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ++ Y 1,1[(y1 ) 2 r + (y 1 ) 3 r + (y 1 ) 4 r ] ,