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

Similarly, the second equation takes the form:(3.19) ⎧{ }0 = yxx 2 + ∆(x|y1 |xx)∆(x|y 1 |y 2 ) + y1 x · 2 ∆(x|y1 |xy 1 )+∆(x|y 1 |y 2 ){}+ yx 2 · 2 ∆(x|y1 |xy 2 )∆(x|y 1 |y 2 ) − ∆(xx|y1 |y 2 )+ y∆(x|y 1 |y 2 x 1 yx 1 ·){}⎪⎨ + yx 1 y2 x · 2 ∆(x|y1 |y 1 y 2 )− 2 ∆(xy1 |y 1 |y 2 )+∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 ){ }∆(x|y+ yx 2 yx 2 2 |y 2 y 2 )·− 2 ∆(xy2 |y 1 |y 2 )+∆(x|y 1 |y 2 ) ∆(x|y 1 |y 2 ){}+ yx 1 yx 1 yx 2 · − ∆(y1 y 1 |y 1 |y 2 )+ y∆(x|y 1 |y 2 x 1 yx 2 yx 2 ·){}⎪⎩ + yx 2 y2 x y2 x · − ∆(y2 y 2 |y 1 |y 2 ).∆(x|y 1 |y 2 )29{ } ∆(x|y 1 |y 1 y 1 )+∆(x|y 1 |y 2 ){−2 ∆(y1 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )Since the formulas are still of a consequent size, analogously to what wasachieved in Section 2, we shall introduce a new family of square functions asfollows. We first index the coordinates (x, y 1 , . . .,y m ) as (y 0 , y 1 , . . ., y m ),namely we introduce the notational equivalence(3.20) y 0 ≡ x ,which will be very convenient in the sequel, especially in order to write generalformulas. With this convention at hand, our eighteen square functions□ k 1y l 1y , defined for 0 j l 2 1, j 2 , k 1 2 are defined by(3.21)⎧⎪⎨⎪⎩□ 0 xx := ∆(xx|y1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 1 y 1 := ∆(y1 y 1 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xx := ∆(x|xx|y2 )∆(x|y 1 |y 2 )□ 1 y 1 y 1 := ∆(x|y1 y 1 |y 2 )∆(x|y 1 |y 2 )□ 2 xx := ∆(x|y1 |xx)∆(x|y 1 |y 2 )□ 2 y 1 y 1 := ∆(x|y1 |y 1 y 1 )∆(x|y 1 |y 2 )□ 0 xy 1 := ∆(xy1 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 1 y 2 := ∆(y1 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xy 1 := ∆(x|xy1 |y 2 )∆(x|y 1 |y 2 )□ 1 y 1 y 2 := ∆(x|y1 y 2 |y 2 )∆(x|y 1 |y 2 )□ 2 xy 1 := ∆(x|y1 |xy 1 )∆(x|y 1 |y 2 )□ 2 y 1 y 2 := ∆(x|y1 |y 1 y 2 )∆(x|y 1 |y 2 )}+□ 0 xy 2 := ∆(xy2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 0 y 2 y 2 := ∆(y2 y 2 |y 1 |y 2 )∆(x|y 1 |y 2 )□ 1 xy 2 := ∆(x|xy2 |y 2 )∆(x|y 1 |y 2 )□ 1 y 2 y 2 := ∆(x|y2 y 2 |y 2 )∆(x|y 1 |y 2 )□ 2 xy 2 := ∆(x|y1 |xy 2 )∆(x|y 1 |y 2 )□ 2 y 2 y 2 := ∆(x|y1 |y 2 y 2 )∆(x|y 1 |y 2 )Obviously, the square functions are symmetric with respect to the lowerindices: □ k 1y l 1y = l 2 □k 1. Here, the upper index designates the columny l 2y l 1