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

x κ−1 χ κ−1 + O(|x| κ ) + O(|χ| 2 ). It may be checked that l 0 = κ − 1, l ∗ 0 = 1and µ 0 = 3, hence κ 0 = 3κ. Then the dimension estimate in (7.28) is:dim K Sym(E ) ≤ 2 C2+3κ + κ C3κ 4κ . This bound is much larger than the optimalbound dim K Sym(E ) ≤ κ + 4 due to S. Lie (cf. [5]; see also the casen = m = 1 of Theorem 6.4).Untill now we focused on providing the set of Lie symmetries of a generalsystem of partial differential equations with a local Lie group structure. Asa byproduct we obtained the (non optimal) dimensional upper bound (7.28)of Theorem 6.4. In the next Sections 3, 4 and 5, using the classical Liealgorithm based on the Lie criterion (see Lemma 8.1), we provide an optimalbound for some specific systems of partial differential equations, answeringan open problem raised in [Ol1995] page 206.1593. LIE THEORY FOR PARTIAL DIFFERENTIAL EQUATIONS3.1. Prolongation of vector fields to the jet spaces. Consider the followingK-analytic system (E ) of non linear partial differential equations:)(7.28) u j x k1···x kκ(x) = F j k 1 ,...,k κ(x, u(x), u i 1xl1(x), . . .,u i 1xl1···x lκ−1(x) ,where 1 ≤ k 1 ≤ · · · ≤ k κ ≤ n, 1 ≤ j ≤ m, and F j k 1 ,...,k κare analyticfunctions of n + m Cn+κ−1 variables, defined in a neighbourhood of theorigin. We assume that (E ) is completely integrable. The Lie theory consistsin studying the infinitesimal symmetries X = ∑ n∑ l=1 Ql (x, u) ∂/∂x l +mj=1 Rj (x, u) ∂/∂u j of (E ). Consider the skeleton of (E ), namely thecomplex subvariety ∆ E of codimension m Cκ+n−1 κ in the jet space Jn,m,κdefined by)(7.28) U j k 1 ,...,k κ= F j k 1 ,...,k κ(x, u, U i 1l 1, . . ., U i 1l 1 ,...,l κ−1,where j, i 1 = 1, . . ., m and k 1 , . . .,k κ , l 1 , . . .,l κ−1 = 1, . . ., n. For k =1, . . ., n let D k be the k-th operator of total differentiation, characterized bythe property that for every integer λ ≥ 2 and for every analytic functionP = P(x, u, U i 1l 1, . . ., U i 1l 1 ,...,l λ−1) defined in the jet space Jn,m λ−1 , the operatorD k is the unique formal infinite differential operator satisfying the relation(7.28)⎧⎪⎨⎪⎩([D k P]x, u(x), u i 1xl1(x), . . . , u i 1∂[P∂x k)xl1···x lλ−1(x) ≡(x, u(x), u i 1xl1(x), . . .,u i 1xl1···x lλ−1(x))].