An Exact Perturbative Existence and Uniqueness Theorem

2022/01/12 by Nikolaev, Nikita · 1 citation
#34A12 #34A34 #34E15 #34E20 #34M04 #34M30 #34M6 (primary) #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2201.04526

Abstract

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

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