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

Restarting from §4.1, let ϕ a Lie symmetry of (E ), namely ϕ ∆E stabilizesF ∆E . The diffeomorphism A defined by (2.9) transforms F v to F ∆E . Conjugating,we get the self-transformation A −1 ◦ϕ ∆E ◦A of the (x, a, b)-space thatmust stabilize also the foliation F v . Equivalently, it must have expression:(6.2) [A −1 ◦ ϕ ∆E ◦ A ] (x, a, b) = ( θ(x, a, b), f(a, b), g(a, b) ) ∈ K n × K p × K m ,where, importantly, the last two components are independent of the coordinatex, because the leaves of F v are just {a = cst., b = cst}.Lemma 6.3. To every Lie symmetry ϕ of (E ), there corresponds a transformationof the parameters(6.4) (a, b) ↦−→ ( f(a, b), g(a, b) ) =: h(a, b)meaning that ϕ transforms the local solution y a,b (x) := Π(x, a, b) to thelocal solution y h(a,b) (x) = Π(x, h(a, b)).Unfortunately, the expression of A −1 ◦ϕ ∆E ◦A does not clearly show thatf and g are independent of x. Indeed, reminding the expressions of A andof Φ, we have:(6.5) (ϕ ∆E ◦A(x, a, b) =ϕ(x, Π(x, a, b)), Φ j(q)β(q)255(xi 1, Π j 1(x, a, b), Π j(q 1)x β(q 1 ) (x, a, b) )) .To compose with A −1 whose expression is given by (2.21), it is useful tosplit ϕ = (φ, ψ) ∈ K n × K m , so above we write(6.6) ϕ(x, Π(x, a, b)) = ( φ(x, Π(x, a, b)), ψ(x, Π(x, a, b)) ) ,and finally, droping the arguments:(6.7)[A −1 ◦ ϕ ∆E ◦ A ] (x, a, b) =(φ i , A q( φ i 1, ψ j 1, Φ j(q 1)β(q 1 )), Bj ( φ i 1, ψ j 1, Φ j(q 1)β(q 1 )) ) .In case (E ) = (E 1 ), is an exercise to verify by computations that the A q (·)and B j (·) are independent of x. In general however, the explicit expressionof Φ j i 1 ,...,i λis unknown. Unfortunately also, nothing shows how( )f(a, b), g(a, b) is uniquely associated to ϕ(x, y). Further explanations areneeded.6.8. Determination of parameter transformations. At first, we state ageometric reformulation of the preceding lemma.Lemma 6.9. Every Lie symmetry (x, y) ↦→ ϕ(x, y) of (E ) induces a localK-analytic diffeomorphism(6.10) (x, y, a, b) ↦−→ ( ϕ(x, y), h(a, b) )