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

358is of the specific cubic polynomial form:(1.10)n∑y x j 1x j 2 = G j1 ,j 2+k 1 =1for j 1 , j 2 = 1, . . .,n.y x k 1(H k 1j 1 ,j 2+ 1 2 y x j 1 L k 1j 2+ 1 2 y x j 2 L k 1j 1+ y x j 1 y x j 2 M k 1It may seem quite paradoxical and counter-intuitive (or even false?) thatevery system (1.10), for arbitrary choices of functions G j1 ,j 2, H k 1j 1 ,j 2, L k 1j 1and M k 1, is automatically equivalent to Y X j 1X j 2 = 0. However, a stronghidden assumption holds: that of complete integrability. Shortly, this crucialcondition amounts to say that(1.11) D j3(Fj 1 ,j 2)= Dj2(Fj 1 ,j 3),for all j 1 , j 2 , j 3 = 1, . . ., n, where, for j = 1, . . ., n, the D j are the totaldifferentiation operators defined by(1.12) D j := ∂∂x j + y x j∂n∑∂y +l=1F j,l∂.∂y x lThese conditions are non-void precisely when n 2. More concretely,developing out (1.11) when the F j 1,j 2are of the specific cubic polynomialform (1.10), after some nontrivial manual computation, we obtain the complicatedcubic differential polynomial in the variables y x k. Equating to zeroall the coefficients of this cubic polynomial, we obtain four familes (I’), (II’),(III’) and (IV’) of first order partial differential equations satisfied by G j1 ,j 2,H k 1j 1 ,j 2, L k 1j 1and M k 1:{(I’)0 = G j1 ,j 2 ,x j 3 − G j1 ,j 3 ,x j 2 +⎧n∑k 1 =1H k 1j 1 ,j 2G k1 ,j 3−n∑k 1 =10 = δ k 1j 3G j1 ,j 2 ,y − δ k 1j 2G j1 ,j 3 ,y + H k 1j 1 ,j 2 ,x j 3 − Hk 1j 1 ,j 3 ,x j 2 +H k 1j 1 ,j 3G k1 ,j 2.),(II’)⎪⎨⎪⎩+ 1 2 G j 1 ,j 3L k 1j 2− 1 2 G j 1 ,j 2L k 1+ 1 2 δk 1j 1+ 1 2 δk 1j 2+n∑k 2 =1n∑k 2 =1n∑k 2 =1j 3+G k2 ,j 3L k 2j 2− 1 2 δk 1j 1G k2 ,j 3L k 2j 1− 1 2 δk 1j 3H k 1k 2 ,j 3H k 2j 1 ,j 2−n∑k 2 =1n∑k 2 =1n∑k 2 =1H k 1k 2 ,j 2H k 2j 1 ,j 3.G k2 ,j 2L k 2j 3+G k2 ,j 2L k 2j 1+