An Exact Perturbative Existence and Uniqueness Theorem

Abstract

We investigate singularly perturbed nonlinear complex differential systems of the form $\hslash {\partial }_{x}f=F(x,\hslash ,f)$ where $\hslash$ is a small complex perturbation parameter. Under a geometric assumption on the eigenvalues of the Jacobian matrix of $F$, we prove an Existence and Uniqueness Theorem for exact perturbative solutions; i.e., holomorphic solutions with prescribed perturbative expansions in $\hslash$. In fact, these solutions are the Borel resummation of the formal perturbative solutions.

Significant changes: the paper has been corrected and made much shorter, simpler, and more streamlined. arXiv admin note: text overlap with arXiv:2112.08792