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An Exact Perturbative Existence and Uniqueness Theorem

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arxiv 2201.04526 v3 pith:2YXTKCD4 submitted 2022-01-12 math.CA math-phmath.APmath.MP

classification math.CAmath-phmath.APmath.MP
keywords hbarperturbativesolutionscomplexexactexistencetheoremuniqueness
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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.

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    nlin.PS 2026-08 conditional novelty 6.0 of 10

    For Möbius-type slow-fast systems such as the overdamped Josephson junction, canards exist in parameter windows of width exp(-S_inst/2ω), where S_inst is the β-cycle instanton action.

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