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

1Now, taking account of the factor , we can re-express all the terms[∆] 2of (3.55) as sums of square functions:⎧ ( ) ( )□ j − □ j =y l 1y l 2 y l 1y l 3(3.56)⎪⎨⎪⎩=−−+y l 3m∑k=0; k≠jy l 2□ k y l 3y k □ j y l 1y l 2 −m∑□ j y l 1y l 2 □k y l 3y− kk=0m∑k=0; k≠j□ k y l 2y k □ j y l 1y l 3 +m∑□ j y l 1y l 3 □k y l 2y. kk=0m∑k=0; k≠jm∑k=0; k≠j□ k y l 1y l 2□ j y l 3y k −□ k y l 1y l 3□ j y l 2y k +Finally, we observe that in the two pairs of sums having k ≠ j appearingin the lines 2 and 4 just above, we can include the term k = j in each pair,because these two terms are immediately killed inside the correspondingpair. In conclusion, after a final obvious killing of four (among six) completesums in this modification of (3.56), we obtain the desired formula (3.41),with two sums. This completes the proof of Lemma 3.40 and also at thesame occasion, the proof of Lemma 2.31.3.57. Compatibility conditions for the first auxiliary system. Accordingto the (approximate) identities (3.3), taking account of the explicit definitions(3.30) of the square functions, we have{ □0 ∼ xx = Xxx , □ 0 ∼xy l 1 = Xxy l 1 , □ 0 ∼y l 1y l 2 = Xy l 1y l 2 ,(3.58)□ j xx ∼ = Y jxx, □ j xy l 1∼ = Yjxy l 1 , □j y l 1y l 2∼ = Yjy l 1y l 2 .Consequently, the first auxiliary system (3.38) looks approximatively like acomplete second order system of partial differential equations in the (m+1)independent variables (x, y) and in the (m+1) dependent variables (X, Y ).By means of elementary algebraic operations, one may transform this systemin a true second order complete system, solved with respect to the toporder derivatives, namely of the form{Xxx = Λ 0 0,0 , X xy l 1 = Λ 0 0,l 1, X y l 1y l 2 = Λ 0 l 1 ,l 2,(3.59)Y jxx = Λ j 0,0,Y jxy l 1 = Λj 0,l 1, Y y l 1y l 2 = Λ j l 1 ,l 2,where the Λ k 1j 1 ,j 2are local K-analytic functions of (x, y l 1, X, Y j , X x , X y l 1 , Y jx , Y jy l 1 ).For such a system, the compatibility conditions [which are necessary andsufficient for the existence of a solution (X, Y )] follow by obvious cross39