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

242When M is the submanifold of solutions of a system (E ), it is automaticallysolvable with respect to the variables, the pairs (j(q), β(q)) being thesame as in the arguments of the right hand sides Fα j in (E ). Proceeding asin §1.16, we may associate to M a system of the form (E ). Since we needintroduce some new notation, let us repeat the argument.Considering y = y(x) = Π(x, a, b) as a function of x with extra parameters(a, b) and applying ∂ |α|/ ∂x α , we get y j x α(x) = Πj xα(x, a, b). Writingonly the relevant (m + p) equations:{y j (x) = Π j (x, a, b),(2.14)y j(q) = Π j(q) (x, a, b),x β(q) x β(q)the assumption of solvability with respect to parameters enables to get(2.15)⎧⎨⎩a q = A q( x i , y j , y j(q 1)β(q 1 )),b j = B j( x i , y j 1, y j(q)β(q)).For every (j, α) ≠ (j, 0) and ≠ (j(q), β(q)), we then replace (a, b) in y j x = αΠ j x α:(y j x(2.16)α(x) = Πj x x, A ( x i , y j 1(x), y j(q)α β(q) (x)) , B ( x i , y j 1(x), y j(q)β(q) (x)))(=: Fαj x i , y j 1(x), y j(q) (x) ) .x β(q)Proposition 2.17. There is a one-to-one correspondence(2.18) (E M ) = (E ) ←→ M = M (E ) ,between completely integrable systems of partial differential equations ofthe general form (E ) and submanifolds (of solutions) M of the form (2.11)which are solvable with respect to the parameters. Of course( )(2.19) EM(E) = (E ) and M(EM ) = M.2.20. Transfer of total differentiations. We notice that the auxiliary functionsA q and B j enable to express the inverse of A:(2.21)A −1 :(xi 1, y j 1, y j(q )1)β(q 1 ) ↦−→(x i , A q( x i 1, y j 1, y j(q )1)(β(q 1 ) , Bjx i 1, y j 1, y j(q 1)β(q 1 )) ) .More importantly, the total differentiation operator considerably simplifieswhen viewed on M . This observation is useful for translating differentialinvariants of (E ) as differential invariants of M .Lemma 2.22. Through A, for i = 1, . . .,n, the pull-back of the total differentiationoperator D i is simply ∂ , or equivalently:∂x( i ∂)(2.23) A ∗ = D∂x i i .