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

For the two unknowns Xyy 1 and X 2x 1 y, we solve the two equations (8.108)and (8.112); we get:(8.113)Xyy 1 = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy, 2X 2x 1 y = r X 1 + r X 2 + r Xy 1 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x + r Y yy. 2287We get (8.86) 12 .Next, we differentiate (8.113) 1 with respect to x 1 and we replace: X 1x, 1X 2x, X 1 1 x 1 y , Y x 1 y, Y x 1 x 1 x 1, X 2x 1 x 1 x, Y 2 x 1 yy; we get:(8.114)X 1x 1 yy = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.Also, we differentiate (8.113) 2 with respect to x 1 and we replace:(8.115)X 2x 1 x 1 y = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.Also, we differentiate (8.83) 4 with respect to y; we replace X 1x 1 yy from(8.114), we replace X 2x 1 x 1 yfrom (8.115); and we achieve other evident replacements;we get:(8.116)Y yyy = r X 1 +r X 2 +r Xy 1 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1+r X 2x 1 x 2+r Y yy.This is (8.96) 22 , which completes the proof.Theorem 8.117. The bound dimSYM(E 5 ) 10 is attained if and only if(E 5 ) is equivalent, through a diffeomorphism (x 1 , x 2 , y) ↦−→ (X 1 , X 2 , Y ),to the model system(8.118) Y X 2 = 0, Y X 1 X 1 X 1 = 0.Proof. Firstly, setting r = s ∗ = 0 everywhere, the solution to (8.81), (8.82),(8.83) is(8.119)X 1 = k + (c + j) x 1 − bx 2 − h y + e x 1 x 1 − d x 1 x 2 + f x 1 y − e x 2 y,X 2 = g + 2h x 1 + 2j x 2 − d x 2 x 2 + 2e x 1 x 2 − f x 1 x 1 ,Y = a + 2bx 1 + 2c y + d x 1 x 1 + 2e x 1 y + f yy,