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

Lemma 3.32. The system yxx j = F j (x, y, y x ), j = 1, . . .,m, is equivalentto the flat system Y j XX= 0, j = 1, . . .,m, if and only if there exist localK-analytic functions X(x, y) and Y j (x, y), j = 1, . . .,m, such that it maybe written under the specific form(3.33) ⎧⎪⎨⎪⎩0 = y j xx + □j xx + m∑+l 1 =1m∑l 1 =1 l 2 =1+ y j x[]y l 1x · 2 □ j − xy l 1 δj l 1□ 0 xx +m∑m∑l 1 =1 l 2 =1[]y l 1x y l 2x · □ j y l 1y − l 2 δj l 1□ 0 xy − l 2 δj l 2□ 0 xy+ l 1m∑[ ]y l 1x y l 2x · −□ 0 y l 1y. l 2The complete proof of this lemma involves only linear algebra considerations,although with rather massive terms. This makes it rather lengthy.Consequently, we postpone it to the final Section 5 below.3.34. First auxiliary system. Clearly, if we set⎧G j := −□ j xx,⎪⎨ H j l 1:= −2 □ j(3.35)+ xy l 1 δj l 1□ 0 xx ,L j l 1 ,l 2:= −□ j y⎪⎩l 1y + l 2 δj l 1□ 0 xy l 2+ δ j l 2□ 0 xy 1, lM l1 ,l 2:= □ 0 y l 1y , l 2we immediately see that the first condition of Theorem 1.7 holds true. Moreover,we claim that there are m+1 more square functions than functions G j ,H j l 1, L j l 1 ,l 2and M l1 ,l 2. Indeed, taking account of the symmetries, we enumerate:(3.36) ⎧#{□ j xx} = m, #{□ 0 xx} = 1,⎪⎨#{□ j } = xy l 1 m2 , #{□ 0 xy l 1} = m,⎪⎩ #{□ j y l 1y } = m2 (m + 1), #{□ 0 m(m + 1)l 2 y2l 1y l 2} = ,2whereas(3.37) ⎧⎨ #{G j } = m, #{H j l 1} = m 2 ,⎩ #{L j l 1 ,l 2} = m2 (m + 1)m(m + 1), #{M l1 ,l22} = .2Similarly as in Section 2, for j, l 1 , l 2 = 0, 1, . . ., m, let us introduce functionsΠ j l 1 ,l 2of (x, y 1 , . . .,y m ), symmetric with respect to the lower indices,33