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

264§7. EQUIVALENCE PROBLEMS AND NORMAL FORMS7.1. Equivalences of submanifolds of solutions. As in §3.1, let (E ) and(E ′ ) be two PDE systems and assume that ϕ transforms (E ) to (E ′ ). DefiningA ′ similarly as A, it follows that(7.2) A ′ −1 ◦ Φ E ,E ′ ◦ A(x, a, b) ≡ ( θ(x, a, b), f(a, b), g(a, b) ) =: (x ′ , a ′ , b ′ )transforms F v to F ′ v, hence induces a map (a, b) ↦→ (a ′ , b ′ ). The arguments ofSection 6 apply here with minor modifications to provide two fundamentallemmas.Lemma 7.3. Every equivalence (x, y) ↦→ (x ′ , y ′ ) between to PDE systems(E ) and (E ′ ) comes with an associated transformation (a, b) ↦→ (a ′ , b ′ ) ofthe parameter spaces such that(7.4) (x, y, a, b) ↦−→ (x ′ , y ′ , a ′ , b ′ )is an equivalence between the associated submanifolds of solutions M (E ) →M ′(E ′ ) .Conversely, let M and M ′ be two submanifolds of K n x × K m y × K p a × K m band of K n x ′ × Km y ′ × Kp a ′ × K m b ′ represented by y = Π(x, a, b) and by y′ =Π ′ (x ′ , a ′ , b ′ ), in the same dimensions. Assume both are solvable with respectto the parameters.Lemma 7.5. Every equivalence(7.6) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )between M and M ′ belonging to G v,p induces by projection the equivalence(x, y) ↦→ ϕ(x, y) between the associated PDE systems ( E M)and(E′M ′).7.7. Classification problems. Consequently, classifying PDE systems underpoint transformations (Section 3) is equivalent to the following.Equivalence problem 7.8. Find an algorithm to decide whether two givensubmanifolds (of solutions) M and M ′ are equivalent through an elementof G v,p .Classification problem 7.9. Classify submanifolds (of solutions) M ,namely provide a complete list of all possible such equations, including theirautomorphism group Aut v,p (M ) ⊂ G v,p .7.10. Partial normal forms. Both problems above are of high complexity.At least as a preliminary step, it is useful to try to simplify somehowthe defining equations of M , by appropriate changes of coordinates belongingto G v,p . To begin with, the next lemma holds for M defined byy = Π(x, a, b) with the only assumption that b ↦→ Π(0, 0, b) has rank m atb = 0.