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

210with G := Φ and with T := Θ zz , we exactly get the expression [ ] AJ 4 (Θ) ofthe Introduction, and then its further derivative ∂ yx ∂ yx Gyxyx = Gyxyxy xy xis exactly:(− Θ w0 ≡Θ z Θ zw − Θ w Θ zz=:AJ 6 (Θ)[Θ z Θ zw − Θ w Θ zz ] 7 .∂∂z +Θ zΘ z Θ zw − Θ w Θ zz) 2∂ [AJ 4 (Θ) ]∂wAs we have said, the vanishing of the second invariant of w zz (z) =Φ ( z, w(z), w z (z) ) amounts to the complex conjugation of the above equation,which is then obviously redundant. Thus, the proof of the Main Theoremis now complete, but we will nevertheless discuss in a specific finalsection what AJ 6 (Θ) would look like in purely expanded form.§5. SOME COMPLETE EXPANSIONS:EXAMPLES OF EXPRESSION SWELLINGSComing back to the non-CR context with the submanifold of solutionsM (E) = { y = Q(x, a, b) } , let us therefore figure out how to expand theexpression differentiated twice:G yxy xy xy x =„− Q b∆+ Ta∆ 3 „+ T b∆ 3 „∂∂a + Qa∆« 2 j∂ Qb Q bT∂b ∆ 2 aa − 2 Qa Q b Qa QaT∆ 2 ab + T∆ 2 bb +Q bbQQ a Q bQa˛˛˛˛ Q xb Q xbb˛˛˛˛ − 2 Q a Q b Q ab Qb˛˛˛˛ Q xb Q xab˛˛˛˛ + Q b Q b Q aab˛˛˛˛ Q xaa˛˛˛˛Q bbQ xbQ aQ a Q− Q a Q a˛˛˛˛ Q xa Q xbb˛˛˛˛ + 2 Q a Q ab Q a Q aab˛˛˛˛ Q xa Q xab˛˛˛˛ − Q b Q b˛˛˛˛ Q xaa˛˛˛˛which would make the Main Theorem a bit more precise and explicit.First of all, we notice that, in the formulas for G yxy x, for G yyx , for G yy ,all the appearing 2 × 2 determinants happen to be modifications of the basicJacobian-like ∆-determinant:∆ ( a|b ) := ∆ =∣ Q ∣a Q b ∣∣∣,Q xa Q xband we will denote them accordingly by employing the following (formallyand intuitively clear) notations:∆ ( b|bb ) :=∣ Q ∣b Q bb ∣∣∣∆ ( b|ab ) :=Q xb Q xbb∣ Q ∣b Q ab ∣∣∣∆ ( b|aa ) :=Q xb Q xab∣ Q ∣b Q aa ∣∣∣Q xb Q xaa∆ ( a|bb ) :=∣ Q ∣a Q bb ∣∣∣∆ ( a|ab ) :=Q xa Q xbb∣ Q ∣a Q ab ∣∣∣∆ ( a|aa ) :=Q xa Q xab∣ Q ∣a Q aa ∣∣∣,Q xa Q xaathe bottom line always coinciding with the differentiation with respect tox of the top line. These abbreviations will be very appropriate for the nextQ xa«+«ff,