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

⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 ·⎩ −2 x 2 yX 1 xX 1 2 y ∣∣∣∣∣ ⎬X 2 x 2 yX 2 xX 2 2 y⎭ +Y x 2 y Y x 2 Y y⎧∣ ⎨X 1 xX 1+ y x 2 y x 2 ·1 xX 2 yy1 ∣∣∣∣∣ ⎩X 2 xX 2∣1 xX 2 2 yy − 2Y x 1 Y x 2 Y yy∣⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 1 y x 2 y x 2 ·⎩ − yy X 1 xX 1 2 y ∣∣∣∣∣ ⎬Xyy 2 X 2 xX 2 2 y⎭ +Y yy Y x 2 Y y⎧ ∣ ∣⎫ ⎨ ∣∣∣∣∣ X 1+ y x 2 y x 2 y x 2 ·⎩ − xX 1 1 yy Xy1 ∣∣∣∣∣ ⎬X 2 xX 2 1 yy Xy2 ⎭ .Y x 1 Y yy Y yX 1 x 1 X 1 x 2 yX 1 yX 2 x 1 X 2 x 2 yX 2 yY x 1 Y x 2 y Y y∣ ∣∣∣∣∣⎫⎬⎭ +3672.24. Appropriate formalism. To describe the combinatorics underlyingformulas (2.10), (2.22) and (2.23), as in [Me2004], let us introduce the followingnotation for the Jacobian determinant:(2.25) ∆(x 1 |x 2 |y) :=∣∣X 1 xX 1 1 xX 1 2 y ∣∣∣∣∣X 2 xX 2 1 xX 2 2 y .Y x 1 Y x 2 Y yHere, in the notation ∆(x 1 |x 2 |y), the three spaces between the two verticallines | refer to the three columns of the Jacobian determinant, and the termsx 1 , x 2 , y in (x 1 |x 2 |y) designate the partial derivatives appearing in eachcolumn. Accordingly, in the following two examples of modified Jacobiandeterminants:(2.26)⎧∆(x 1 x 2 |x 2 |y) :=⎪⎨∣∆(x 1 |x 2 |x 1 y) :=⎪⎩∣X 1 x 1 x 2 X 1 x 2 Xy1X 2 x 1 x 2 X 2 x 2 Xy2Y x 1 x 2 Y x 2 Y yX 1 x 1 X 1 x 2 X 1 x 1 yX 2 x 1 X 2 x 2 X 2 x 1 yY x 1 Y x 2 Y x 1 y∣,∣andwe simply mean which column of first order derivatives is replaced by acolumn of second order derivatives in the original Jacobian determinant.As there are 6 possible second order derivatives (·) x 1 x 1, (·) x 1 x 2, (·) x 1 x y,(·) x 2 x 2, (·) x 2 y and (·) yy together with 3 columns, we obtain 3 × 6 = 18