An Exact Perturbative Existence and Uniqueness Theorem

TL;DR

This paper proves an existence and uniqueness theorem for exact solutions to certain singularly perturbed nonlinear complex differential systems, showing these solutions are Borel resummations of formal perturbative solutions.

Contribution

It establishes a rigorous link between formal perturbative solutions and actual holomorphic solutions via Borel resummation under geometric eigenvalue conditions.

Findings

Proves existence and uniqueness of exact perturbative solutions.

Shows solutions are Borel resummations of formal series.

Provides conditions on eigenvalues for solution validity.

Abstract

We investigate singularly perturbed nonlinear complex differential systems of the form $\hbar \partial_x f = F (x, \hbar, f)$ where $\hbar$ 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 $\hbar$. In fact, these solutions are the Borel resummation of the formal perturbative solutions.