Complete Systems of Partial Differential Equations

Any infinitesimal transformation $$X = \sum _{ i = 1}^n \, \xi _i ( x) \, \frac{ \partial }{ \partial x_i}$$ X = ∑ i = 1 n ξ i ( x ) ∂ ∂ x i can be considered as the first order analytic partial differential equation $$X \omega = 0$$ X ω = 0 with the unknown $$\omega $$ ω . After a relocalization, a renumbering and a rescaling, one may suppose $$\xi _n (x) \equiv 1$$ ξ n ( x ) ≡ 1 . Then the general solution $$\omega $$ ω happens to be any (local, analytic) function $$\varOmega \big ( \omega _1, \dots , \omega _{ n-1} \big )$$ Ω ( ω 1 , ⋯ , ω n - 1 ) of the $$(n-1)$$ ( n - 1 ) functionally independent solutions defined by the formula: $$ \omega _k(x) := \exp \big (-x_nX\big )(x_k) \ \ \ \ \ \ \ \ \ \ \ \ \ {\scriptstyle {(k\,=\,1\,\cdots \,n-1)}}. $$ ω k ( x ) : = exp ( - x n X ) ( x k ) ( k = 1 ⋯ n - 1 ) . What about first order systems $$X_1 \omega = \cdots = X_q \omega = 0$$ X 1 ω = ⋯ = X q ω = 0 of such differential equations? Any solution $$\omega $$ ω also trivially satisfies $$X_i \big ( X_k ( \omega ) \big ) - X_k \big ( X_i ( \omega ) \big ) = 0$$ X i ( X k ( ω ) ) - X k ( X i ( ω ) ) = 0 . But it appears that the subtraction in the Jacobi commutator $$X_i \big ( X_k ( \cdot ) \big ) - X_k \big ( X_i ( \cdot ) \big )$$ X i ( X k ( · ) ) - X k ( X i ( · ) ) kills all the second-order differentiation terms, so that one may freely add such supplementary first-order differential equations to the original system, continuing again and again, until the system, still denoted by $$X_1 \omega = \cdots = X_q \omega = 0$$ X 1 ω = ⋯ = X q ω = 0 , becomes complete in the sense of Clebsch, namely satisfies, locally in a neighborhood of a generic point $$x^0$$ x 0 : (i) for all indices $$i, k = 1, \dots , q$$ i , k = 1 , ⋯ , q , there are appropriate functions $$\chi _{ ik \mu } ( x)$$ χ i k μ ( x ) such that $$X_i\big ( X_k ( f) \big ) - X_k \big ( X_i ( f ) \big ) = \chi _{ ik1} ( x) \, X_1 ( f) + \cdots + \chi _{ ik q} ( x) \, X_q ( f)$$ X i ( X k ( f ) ) - X k ( X i ( f ) ) = χ i k 1 ( x ) X 1 ( f ) + ⋯ + χ i k q ( x ) X q ( f ) ; (ii) the rank of the vector space generated by the $$q$$ q vectors $$X_1\big \vert _x, \dots , X_q \big \vert _x$$ X 1 | x , ⋯ , X q | x is constant equal to $$q$$ q for all $$x$$ x near the central point $$x^0$$ x 0 . Under these assumptions, it is shown in this chapter that there are $$n - q$$ n - q functionally independent solutions $$x_1^{ ( q)}, \dots , x_{ n- q}^{ (q)}$$ x 1 ( q ) , ⋯ , x n - q ( q ) of the system that are analytic near $$x_0$$ x 0 such that any other solution is a suitable function of these $$n-q$$ n - q fundamental solutions.