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

Let λ ∈ N be an arbitrary integer. For i ′ = 1, . . ., n, let D λ i ′ denotes the i′ -thλ-th order total differentiation operators, defined precisely by:(1.22) ⎧⎪⎨⎪⎩D λ i ′ := ∂∂x i′ ++ · · · +m∑j ′ =1m∑j ′ =1y j′i ′∂∂y j′ +n∑m∑j ′ =1n∑i ′ 1 =1 y j′i ′ ,i ′ 1i ′ 1 ,i′ 2 ,...,i′ λ−1 =1 y j′i ′ ,i ′ 1 ,i′ 2 ,...,i′ λ−1∂∂y j′i ′ 1+m∑j ′ =1∂∂y j′i ′ 1 ,i′ 2 ,...,i′ λ−1n∑i ′ 1 ,i′ 2 =1 y j′i ′ ,i ′ 1 ,i′ 2.309∂∂y j′i ′ 1 ,i′ 2Then, for i = 1, . . .,n, the expressions of Y j are given by theX i 1 ···X i λ−1Xi following compact formulas (again [BK1989]):(1.23) ⎛⎛⎞⎜⎝Y j ⎞DX i 1 ···X i λ−1X 11 1⎟X1 · · · D 1 ⎞ ⎛1 Xn−1D1 λ. ⎠ = ⎝. · · · .⎠ ⎜ Y j X i 1 ···X i λ−1⎝ .Y j D 1X i 1 ···X i λ−1XnX 1 · · · DnX 1 n D λ n n Y j X i 1 ···X i λ−1Again, these inductive formulas are incomplete and unsatisfactory.Problem 1.24. Find totally explicit complete formulas for the κ-th prolongationϕ (κ) .⎟⎠ .Except in the cases κ = 1, 2, we have not been able to solve this problem.The case κ = 1 is elementary. Complete formulas in the particularcases κ = 2, n = 1, m 1 and n 1, m = 1 are implicitely providedin [Me2004] and in Section ?(?), where one observes the appearance of somemodifications of the Jacobian determinant of the diffeomorphism ϕ, insertedin a clearly understandable combinatorics. In fact, there is a nice dictionarybetween the formulas for ϕ (2) and the formulas for the second prolongationL (2) of a vector field L which were written in equation (43) of [GM2003a](see also equations (2.6), (3.20), (4.6) and (5.3) in the next paragraphs). Inthe passage from ϕ (2) to L (2) , a sort of formal first order linearization maybe observed and the reverse passage may be easily guessed. However, forκ 3, the formulas for ϕ (κ) explode faster than the formulas for the κ-thprolongation L (κ) of a vector field L . Also, the dictionary between ϕ (κ)and L (κ) disappears. In fact, to elaborate an appropriate dictionary, webelieve that one should introduce before a sort of formal (κ − 1)-th orderlinearizations of ϕ (κ) , finer than the first order linearization L (κ) . To be optimistic,we believe that the final answer to Problem 1.24 is, nevertheless,accessible after hard work.The present article is devoted to present totally explicit complete formulasfor the κ-th prolongation L (κ) of a vector field L to J κ n,m , for n 1arbitrary, for m 1 arbitrary and for κ 1 arbitrary.+