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

complementary views on the same object. In fact, Lie symmetries, equivalenceproblems, Cartan connections, normal forms and classification listsmay be endeavoured on both sides, yielding essentially equivalent results,though the translation is seldom straightforward. In Section 3, 4 and 5, wereview some features from the side (E ), before studying some aspects fromthe side of M . A more systematic and complete approach shall appear as amonography.245§3. CLASSIFICATION PROBLEMS3.1. Transformations of PDE systems. Through a local K-analytic changeof variables close to the identity (x, y) ↦→ ϕ(x, y) =: (x ′ , y ′ ), the system (E )transforms to a similar system, with primes:((E ′ ) y ′ jx ′α(x′ ) = F ′ jα x ′ , y ′ (x ′ ), ( y ′ j(q)(x ′ ) ) ).x ′β(q) 1qpExample 3.2. Coming back temporarily to the notations of §1.12(II), withn = m = κ = 1, assume that y xx = f(x, y, y x ) transforms to( Y XX = F(X, Y, Y) X ) through a local diffeomorphism (x, y) ↦→ (X, Y ) =X(x, y), Y (x, y) . How F is related to f ? By symmetry, it suffices tocompute f in terms of F , X, Y . The prolongation to J1,1 2 of the diffeomorphismhas components ([BK1989, Me2004]):(3.3) Y X = Y x + y x Y yX x + y x X y,and(3.4)1Y XX = [ ] 3(y xx ·∣ X ∣ x X y ∣∣∣ +Xx + y x X yY x Y y∣ X ∣x X xx ∣∣∣+Y x Y xx{+y x · 2∣ X ∣ x X xy ∣∣∣ −Y x Y xy∣ X ∣}xx X y ∣∣∣+Y xx Y y{∣ ∣ ∣∣∣ X+y x y x · x X yy ∣∣∣ − 2Y x Y yy∣ X ∣}xy X y ∣∣∣+Y xy Y y{+y x y x y x · −∣ X ∣})yy X y ∣∣∣.Y yy Y y