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

166The set of infinitesimal symmetries of (E ) forms a Lie algebra, since wehave the relation [X, X ′ ] (κ) = [X (κ) , X ′ (κ) ] (cf. [Ol1986]). We will denoteby Sym(E ) this Lie algebra. The aim of the forecoming Section is to obtainprecise bounds on the dimension of the Lie algebra Sym(E ) of infinitesimalsymmetries of (E ). For simplicity we start with the case n = m = 1.4. OPTIMAL UPPER BOUND ON DIM K Sym(E ) WHEN n = m = 1.4.1. Defining equations for Sym(E ). Applying the Lie criterion, the tangencycondition of X (κ) to ∆ E is equivalent to the identity:(7.28)R κ −[Q ∂F∂x + R ∂F∂u∂F ∂F+ R1 + R2∂U1 ∂U + · · · + 2 Rκ−1]∂F≡ 0,∂U κ−1on the subvariety ∆ E , that is to a formal identity in K{x, u, U 1 , . . ., U κ−1 },in which we replace the variable U κ by F(x, u, U 1 , . . .,U κ−1 ) in the twomonomials U κ and U 1 U κ of R κ , cf. Lemma 8.1. Expanding F and its partialderivatives in power series of the variables (U 1 , . . .,U κ−1 ) with analyticcoefficients in (x, u), we may rewrite (7.28) as follows:⎧ ∑ [⎪⎨ Φµ1 ,...,µ κ−1(x, u, (Q x k u l) k+l≤κ, (R x k u l) k+l≤κ) ] ×(7.28) µ 1 ,...,µ κ−1 ≥0⎪⎩×(U 1 ) µ 1. . .(U κ−1 ) µ κ−1≡ 0,where the expressions(7.28) Φ µ1 ,...,µ κ−1(x, u, (Q x k u l) k+l≤κ, (R x k u l) k+l≤κ)are linear with respect to the partial derivatives ((Q x k u l) k+l≤κ, (R x k u l) k+l≤κ),with analytic coefficients in (x, u). By construction these coefficients essentiallydepend on the expansion of F . The tangency condition (7.28)is equivalent to the following infinite linear system of partial differentialequations, called defining equations of Sym(E ):(7.28) Φ µ1 ,...,µ κ−1(x, u, (Q x k u l(x, u)) k+l≤κ, (R x k u l(x, u)) k+l≤κ) = 0,satisfied by (Q(x, u), R(x, u)). The Lie method consists in studying thesolutions of this linear system of partial differential equations.4.2. Homogeneous system. As mentioned in the introduction, we focusour attention on the case κ ≥ 3. Denote by (E 0 ) the homogeneous equationu x κ = 0 of order κ. The general solution u = ∑ κ−1l=0 λ l x l consists ofpolynomials of degree ≤ κ − 1 and the defining equation (7.28) reduces toR κ = 0. Using the expression (7.28), expanding (7.28), (7.28) and consideringonly the coefficients of the five monomials ct., U κ−2 , U κ−1 , U 1 U κ−1 and