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A fundamental property of Monge characteristics in involutive systems of nonlinear partial differential equations and its application. (English) Zbl 0563.58038

Let \(R_{\ell}\) be a nonlinear involutive system of partial differential equations of order \(\ell\) with several unknown functions of n independent variables. Let \(\Sigma\) be that exterior differential system on the space \(J_{\ell}\) of \(\ell\)-jets of unknown functions which is canonically constructed in such a way that the solutions of \(R_{\ell}\) correspond to its n-dimensional integral manifolds (solution manifolds of \(R_{\ell})\). First, some new aspect of the notion of Monge characteristics for \(R_{\ell}\) is clarified, introducing the notion of Monge characteristic vectors. It is shown that, for a system \(R_{\ell}\) to which the theory may be successfully applied, one can construct several Pfaffian systems on \(J_{\ell}\) each of which possesses these properties: (i) It is composed of all the Pfaffian equations in \(\Sigma\) and several Pfaffian equations linearly independent from them, (ii) one has a method of constructing a family of its integral curves generating any given solution manifold. In dealing with Monge characteristics, a module M called the characteristic module of \(R_{\ell}\) is introduced. It is a submodule of a certain Noetherian module L over a polynomial ring R in n variables. It is an irredundant primary decomposition \(M=\cap^{\nu}_{j=1}Q_ j\) in L that play a fundamental role. Let \(Q_ j\) be \({\mathfrak P}_ j\)-primary. The ideals \({\mathfrak P}_ j\) in R define the characteristic variety of \(R_{\ell}\). If the Cartan characters \(s_ i\) of the system \(R_{\ell}\) satisfy \(s_ p>0\), \(s_{p+1}=...=s_ n=0\), then the maximum of \(proj \dim {\mathfrak P}_ j\) is equal to p-1. The notion of Monge characteristics is then used to deduce an existence theorem of \(C^{\infty}\) local solutions for an involutive system \(R_{\ell}\) satisfying these conditions: \(s_ 1>0\), \(s_ 2=...=s_ n=0\), the zeros of the homogeneous ideals \({\mathfrak P}_ j\) are all real, and \({\mathfrak P}_ jL\subset Q_ j\) \((j=1,...,\nu)\). This existence theorem may be applied, in particular, to a strictly hyperbolic involutive system, that is, a system whose characteristic projective variety is composed of \(s_ 1\) real distinct points.

MSC:

58J99 Partial differential equations on manifolds; differential operators
58A17 Pfaffian systems
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References:

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