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

152for (j, α) ≠ (j(1), β(1)), . . ., (j(p), β(p)) and j = 1, . . .,m, |α| ≤ κ.Clearly, the natural coordinates on the submanifold ∆ E of Jn,m κ are then + m + p coordinates()(7.28)x, u, (U j(q)β(q) ) 1≤q≤p .Let h = h(x, u) be a local K-analytic diffeomorphism of K n+m close to theidentity mapping and let π κ : J κ n,m → Kn+m be the canonical projection.According to [Ol1986] (Chapter 2) there exists a unique lift h (κ) of h toJ κ n,m such that π κ ◦h (κ) = h◦π κ . The components of h (κ) may be computedby means of universal combinatorial formulas and they are rational functionsof the jet variables (7.28), their coefficients being partial derivatives of thecomponents of h, see for instance §3.3.5 of [BK1989]. By definition, h is alocal symmetry of (E ) if h transforms the graph of every local solution of(E ) into the graph of another local solution of (E ). This definition seems tobe rather uneasy to handle, because of the abstract quantification of “everylocal solution”, but we have the following concrete characterization for h tobe a local symmetry of (E ), cf. Chapter 2 in [Ol1986].Lemma 8.1. The following conditions are equivalent:(1) The local transformation h is a local symmetry of (E ).(2) Its κ-th prolongation h (κ) is a local self-transformation of the skeleton∆ E of (E ).These considerations have an infinitesimal version. Indeed, let X =∑ nl=1 Ql (x, u) ∂/∂x l + ∑ mj=1 Rj (x, u) ∂/∂u j be a local vector field withK-analytic coefficients which is defined in a neighbourhood of the origin inK n+m . Let s ∈ K and consider the flow of L as the one-parameter familyh s (x, u) := exp(s X)(x, u) of local transformations. We recall that X isan infinitesimal symmetry of (E ) if for every small s ∈ K, the mappingh s (x, u) := exp(s X)(x, u) is a local symmetry of (E ). By differentiatingwith respect to s the κ-th prolongation (h s ) (κ) of h s at s = 0, we obtaina unique vector field X (κ) on the κ-th jet space, called the κ-th prolongationof X and which satisfies (π k ) ∗ (X (κ) ) = X. In Subsections 3.1 and 3.2below, we shall analyze the combinatorial formulas for the coefficients ofX (κ) , since they will be needed to prove Theorem 6.4.Let X E be the projection to the restricted jet space K m+n+p , equippedwith the coordinates (7.28), of the restriction of X (κ) to ∆ E , namely(7.28) X E := (π κ,p ) ∗ (X (κ) | ∆E ).The following Lemma, called the Lie criterion, is the concrete characterizationfor X to be an infinitesimal symmetry of (E ) and is a direct corollaryof Lemma 8.1, cf. Chapter 2 in [Ol1986]. This criterion will be central inthe next Sections 3, 4 and 5.