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

198For the time being, with the aim of extending such a kind of characterizationto a broader scope, following §2 of [19], let us now recall how one mayin a natural way construct a sumanifold of solutions M E associated to thedifferential equation (E ) which, when (E ) comes from a Levi nondegeneratelocal real analytic hypersurface M ⊂ C 2 , regives without any modificationits complex defining equation w = Θ ( z, z, w).To begin with, in the first-order jet space (x, y, y x ) that we simply drawas a common three-dimensional space:y xyD0abxyDM (E)0a, bxDDDexp(xD)(0,a, b)we duplicate the two dependent coordinates (y, y x ) by introducing a newsubspace of coordinates (a, b) ∈ K × K, and we draw a vertical plane containingthe two new axes that are just parallel copies (for the moment, justlook at the left-hand side). Then the leaves of the local foliation associatedto the integral curves of the vector field D are uniquely determined by theirintersection with this plane, because thanks to the presence of ∂ in D, all∂xthese curves are approximately directed by the x-axis in a neighborhood ofthe origin: no tangent vector can be vertical. But we claim that all such intersectionpoints of coordinates (0, b, a) ∈ K×K×K correspond bijectivelyto the two initial conditions y(0) ≡ b and y x (0) = a for solving uniquely thedifferential equation. In fact, the flow of D at time x starting from all suchpoints (0, b, a) of the duplicated vertical plane:exp(x D)(0, b, a) =: ( x, Q(x, a, b), S(x, a, b) )(see again the diagram) expresses itself in terms of two certain local K-analytic functions Q and S that satisfy, by the very definition of the flow ofour vector field ∂ x + y x ∂ y + F ∂ yx , the following two differential equations:ddx Q(x, a, b) = S(x, a, b) and: ddx S(x, a, b) = F( x, Q(x, a, b), S(x, a, b) )together with the (obious) initial condition for x = 0:(0, b, a) = exp(0 D)(0, b, a) = ( 0, Q(0, a, b), S(0, a, b) ) .