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

for some nonnegative integers A, B, α, β ∈ N. Sometimes A is zero, butB is zero only for the (constant, with respect to pure jet variables) termY x κ. Importantly, X is differentiated once more with respect to x and Y isdifferentiated once more with respect to y. Again, this may be confirmed byreading all the terms in the formulas for Y 1 , Y 2 , Y 3 , Y 4 and Y 5 .In addition, we claim that there is a link between the couple (α, β) andthe collection {µ 1 , λ 1 , . . .,µ d , λ d }. To discover it, let us write some of themonomials appearing in the expressions of Y 4 (first column) and of Y 5(second column), for instance:(2.16)⎧[6 Y x 2 y 2 − 4 X x 3 y](y 1 ) 2 , [5 Y xy 4 − 10 X x 2 y 3] (y 1) 4 ,⎪⎨[12 Y xy 2 − 18 X x 2 y]y 1 y 2 , [30 Y xy 3 − 60 X x 2 y 2] (y 1) 2 y 2 ,[−10 X y 3] (y 1 ) 3 y 2 , [−15 X y 4] (y 1 ) 4 y 2 ,[4 Y y 2 − 16 X xy ] y 1 y 3 , [10 Y y 2 − 50 X xy ] y 2 y 3 ,⎪⎩[−10 X y 2] (y 1 ) 2 y 3 , [−60 X y 2] y 1 y 2 y 3 .After some reflection, we discover the hidden intuitive rule: the partialderivatives of Y and of X associated with the monomial (y λ1 ) µ1 · · ·(y λd ) µ dare, respectively:{Yx κ−µ 1 λ 1 −···−µ d λ d y µ 1 +···+µ d ,(2.17)X x κ−µ 1 λ 1 −···−µ d λ d +1 y µ 1 +···+µ d −1.This may be checked on each of the 10 examples (2.16) above.Now that we have explored and discovered the combinatorics of the purejet monomials, of the partial derivatives and of the complete sum giving Y κ ,we may express that it is of the following general form:(2.18)⎧Y κ = Y x κ +⎪⎨⎪⎩∑κ+1d=1[A(µ 1 ,λ 1 ),...,(µ d ,λ d )κ∑1λ 1