Complete Systems of Partial Differential Equations

Abstract

Any infinitesimal transformation $X = \sum _{ i = 1}^n \, \xi _i ( x) \, \frac{ \partial }{ \partial x_i}$ can be considered as the first order analytic partial differential equation $X \omega = 0$ with the unknown $\omega$. After a relocalization, a renumbering and a rescaling, one may suppose $\xi _n (x) \equiv 1$. Then the general solution $\omega$ happens to be any (local, analytic) function $\varOmega \big ( \omega _1, \dots , \omega _{ n-1} \big )$ of the $(n-1)$ functionally independent solutions defined by the formula:

$$ \omega _k(x) := \exp \big (-x_nX\big )(x_k) \ \ \ \ \ \ \ \ \ \ \ \ \ {\scriptstyle {(k\,=\,1\,\cdots \,n-1)}}. $$

What about first order systems $X_1 \omega = \cdots = X_q \omega = 0$ of such differential equations? Any solution $\omega$ also trivially satisfies $X_i \big ( X_k ( \omega ) \big ) - X_k \big ( X_i ( \omega ) \big ) = 0$. But it appears that the subtraction in the Jacobi commutator $X_i \big ( X_k ( \cdot ) \big ) - X_k \big ( X_i ( \cdot ) \big )$ 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 $X_1 \omega = \cdots = X_q \omega = 0$, becomes complete in the sense of Clebsch, namely satisfies, locally in a neighborhood of a generic point $x^0$: (i) for all indices $i, k = 1, \dots , q$, there are appropriate functions $\chi _{ ik \mu } ( x)$ such that $X_i\big ( X_k ( f) \big ) - X_k \big ( X_i ( f ) \big ) = \chi _{ ik1} ( x) \, X_1 ( f) + \cdots + \chi _{ ik q} ( x) \, X_q ( f)$; (ii) the rank of the vector space generated by the $q$ vectors $X_1\big \vert _x, \dots , X_q \big \vert _x$ is constant equal to $q$ for all $x$ near the central point $x^0$. Under these assumptions, it is shown in this chapter that there are $n - q$ functionally independent solutions $x_1^{ ( q)}, \dots , x_{ n- q}^{ (q)}$ of the system that are analytic near $x_0$ such that any other solution is a suitable function of these $n-q$ fundamental solutions.