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

284by − 1 Y 2 x 1 2x1 and solving for Xx 2 x: 2(8.91)0 = X 1x 1 x 2 + 1 2 Y x 1 x 1,X 2x 2 x 2 = −(1 + s∗ x 2) Y x 1 x 1 + r Y x 1.This is (8.86) 7 . Differentiating (8.91) 2 with respect to x 1 , taking account of(8.83) 1 , we get (8.86) 16 :(8.92) X 2x 1 x 2 x 2 = r Y x 1 + r Y x 1 x 1.We then replace X 1x 1 from (8.90) in (8.81) 4 :(8.93)0 = X 2x 1 + 2 X 1y + r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y++ r Y x 1 x 1 + s∗ Y x 1 y + s ∗ X 1x 1 x 1.We differentiate this equation with respect to x 2 , knowing Y x 2 = 0:(8.94)0 = Xx 2 1 x + 2 X 1 2 x 2 y + r X 1 + r Xx 1 + r X 2 + r X 22 x + r X 2 2 x 2 x + r Y 2 x 1 + r Y y++ r Y x 1 x 1 + s∗ x 2 Y x 1 y + s ∗ x 2 X 1x 1 x 1 + s∗ X x 1 x 1 x 2.We replace: X 1xfrom (8.89); X 22 x 2 xfrom (8.91) 2 2 ; we differentiate (8.81) 2with respect to x 1 x 1 to replace X 1x 1 x 1 xby r Y 2 x 1 x 1, thanks to (8.83) 1; and wereorganize:(8.95)2 X 1x 2 y +s∗ x Y 2 x 1 y+s ∗ x X 1 2 x 1 x = −X 2 1 x 1 x 2+r X 1 +r X 2 +r X 2x 2+r Y x 1+r Y y+r Y x 1 x 1.We differentiate (8.81) 2 with respect to y and (8.81) 3 with respect to x 1 :(8.96)X 1x 2 y + 1 2 Y x 1 y = 0,Y x 1 y − 2 X 1x 1 x 1 = −X 2x 1 x 2 + r Y x 1 + r Y x 1 x 1.For the three unknowns X 1x 1 x, Y 1 x 1 y, X 1x 2 y, we solve the three equations(8.95), (8.96) 1 , (8.96) 2 , reminding s ∗ x(0) = 0: 2(8.97)X 1x 1 x = r X 1 + r X 2 + r X 21 x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,Y x 1 y = r X 1 + r X 2 + r X 2x + r Y 2 x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2,X 1x 2 y = r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.We get (8.86) 11 and (8.86) 9 .Thus, we may replace X 1x 1 x 1 and Y x 1 y in (8.81) 4 to get (8.86) 2 :(8.98)X 2x 1 = −2 X 1y + r X 1 + r X 2 + r X 2x 2 + r Y x 1 + r Y y + r Y x 1 x 1 + r X 2x 1 x 2.Next, we differentiate (8.83) 3 with respect to x 1 and we replace: X 1xfrom 1(8.90); X 2xfrom (8.98); Y 1 x 1 y from (8.97) 2 ; X 1x 1 xfrom (8.97) 1 1 ; Y x 1 x 1 x 1from (8.83) 1 ; and we compare with (8.83) 2 ; we differentiate (8.96) 1 with