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

This lemma is left to the reader; anyway, we shall complete the proof ofa generalization of Lemma 2.31 to the case of m 1 dependent variables(y 1 , . . .,y m ) in Section 2 below (Lemma 3.40).Coming back to the first auxiliary system (2.20), we therefore have obtaineda necessary and sufficient condition for the existence of (X, Y ): thefunctions Π k 1j 1 ,jshould satisfy the following system of first order partial differentialequations, just obtained from (2.30) by replacing the square func-2tions by the Pi functions:(2.33) ⎧( )Π0 − ( )0,0 Π 0 y 0,1 = − x Π1 0,0 · Π0 1,1 + Π1 0,1 · Π0 0,1( ),⎪⎨ Π0 − ( )0,1 Π 0 y 1,1 = − x Π0 0,1 · Π 0 0,1 − Π 1 0,1 · Π 0 1,1 + Π 0 1,1 · Π 0 0,0 + Π 1 1,1 · Π 0 0,1,( )Π1 − ( )0,0 Π 1 y 0,1 = − x Π0 0,0 · Π1 0,1 − Π1 0,0 · Π1 1,1 + Π0 0,1 · Π1 1,1 + Π1 0,1 · Π1 0,1 ,⎪⎩( )Π1 − ( )0,1 Π 1 y 1,1 = − x Π0 0,1 · Π1 0,1 + Π0 1,1 · Π1 0,0 .172.34. Second auxiliary system. It is now time to come back to the functionsG, H, L and M and to get rid of the auxiliary “Pi” functions. Unfortunately,we cannot invert directly the linear system (2.18), hence we must choose twospecific square functions as principal unknowns, and the best, from a combinatorialpoint of view, is to choose □ 0 xx and □ 1 yy. Remind that by (2.20),we have □ 0 xx = Π 0 0,0 and □ 1 yy = Π 1 1,1. For clarity, it will be useful to adoptthe notational equivalences(2.35) Θ 0 ≡ Π 0 0,0 and Θ 1 ≡ Π 1 1,1.We may therefore quasi-inverse the linear system (2.18), obtaining that thefour functions Π 1 0,0 , Π1 0,1 , Π0 0,1 and Π0 1,1 may be expressed in terms of thefunctions G, H, L and M and in terms of the remaining two principal unknowns(2.35), which yields:⎧Π 1 0,0 = □1 xx = −G(2.36)⎪⎨ Π 1 0,1 = □1 xy = −1 2 H + 1 2 Θ0 ,Π 0 0,1⎪⎩= □0 xy = 1 2 L + 1 2 Θ1 ,Π 0 1,1 = □0 yy = M.Replacing now each of these four expressions in the compatibility conditionsof the first auxiliary system (2.33), solving the four equations with respectto Θ 1 y , Θ0 x , Θ1 x and Θ0 y , we get after hygienic simplifications what we shallcall the second auxiliary system, which is a complete system of first order