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

2.23. Compatibility conditions for the first auxiliary system. Indeed, tobegin with, let us remind that the ∆(·|·) are determinant, hence we have theskew-symmetry relation ∆(x a y b |x c y d ) = −∆(x c y d |x a y b ) and the followingtwo formulas for partial differentiation{ [∆(x a y b |x c y d ) ] = x ∆(xa+1 y b |x c y d ) + ∆(x a y b |x c+1 y d ),(2.24)[∆(x a y b |x c y d ) ] y = ∆(xa y b+1 |x c y d ) + ∆(x a y b |x c y d+1 ).With these formal rules at hand, as an exercise, let us compute for instancethe following cross differentiation (remember that the lower index in thesquare functions is not a partial derivative):(2.25) ⎧⎪⎨⎪⎩(□0xx)==)( ) ∆(xx|y)− ∂∆(x|y) ∂x( ) ∆(xy|y)=∆(x|y)−( □ 0 y xy = ∂ x∂y1{∆(xxy|y) · ∆(x|y)[∆(x|y)] 2 a+ ∆(xx|yy) · ∆(x|y)−−∆(xy|y) · ∆(xx|y) b− ∆(x|yy) · ∆(xx|y)−−∆(xxy|y) · ∆(x|y) a− ∆(xy|xy) · ∆(x|y) c+}+∆(xy|y) · ∆(xx|y) b+ ∆(xy|y) · ∆(x|xy) =1{∆(xx|yy) · ∆(x|y) − ∆(x|yy) · ∆(xx|y)+[∆(x|y)]2+∆(xy|y) · ∆(x|xy)}.Crucially, we observe that the third order derivatives kill each other anddisappear, see the underlined terms with a appended. Also, two productsof two determinants ∆(·|·) involving a second order derivative upon onecolumn of each determinant kill each other: they are underlined with bappended. Finally, by antisymmetry of determinants, the term ∆(xy|xy) ·∆(x|y) vanishes gratuitously: it is underlined with c appended.However, there still remains one term involving second order derivativesupon the two columns of a determinant: it is ∆(xx|yy).We must transform this unpleasant term ∆(xx|yy) · ∆(x|y) and expressit as a product of two determinants, each involving a second order derivativeonly in one column. To this aim, we have:Lemma 2.26. The following three relations between the differential determinants∆(·|·) hold true:(2.27) ⎧⎪⎨∆(xx|xy) · ∆(x|y) = ∆(xx|y) · ∆(x|xy) − ∆(xy|y) · ∆(x|xx),∆(xx|yy) · ∆(x|y) = ∆(xx|y) · ∆(x|yy) − ∆(yy|y) · ∆(x|xx),⎪⎩∆(xy|yy) · ∆(x|y) = ∆(xy|y) · ∆(x|yy) − ∆(yy|y) · ∆(x|xy).15