Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems (1999)
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BibTeX
@MISC{Rust99existenceand,
author = {C.J. Rust and G. J. Reid and A. D. Wittkopf},
title = {Existence and Uniqueness Theorems for Formal Power Series Solutions of Analytic Differential Systems},
year = {1999}
}
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Abstract
We present Existence and Uniqueness Theorems for formal power series solutions of analytic systems of pde in a certain form. This form can be obtained by a finite number of differentiations and eliminations of the original system, and allows its formal power series solutions to be computed in an algorithmic fashion. The resulting reduced involutive form (rif 0 form) produced by our rif 0 algorithm is a generalization of the classical form of Riquier and Janet, and that of Cauchy-- Kovalevskaya. We weaken the assumption of linearity in the highest derivatives in those approaches to allow for systems which are nonlinear in their highest derivatives. A new formal development of Riquier's theory is given, with proofs, modeled after those in Grobner Basis Theory. For the nonlinear theory, the concept of relative Riquier Bases is introduced. This allows for the easy extension of ideas from the linear to the nonlinear theory. The essential idea is that an arbitrary nonlinear system can ...
