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Exponential asymptotics, transseries, and generalized Borel summation for analytic, nonlinear, rank-one systems of ordinary differential equations. (English) Zbl 0841.34005

Consider an \(n\)-dimensional, rank-one, level-one vector differential equation \[ y'= f_0(x)- \Lambda y- \textstyle{{1\over x}} By+ g(x, y) \] in the neighborhood of an irregular singularity \(x= \infty\), where the vector \(f_0(x)= O(x^{- 2})\) for large \(x\), \(B\) and \(\Lambda\) are \(n\times n\) matrices with constant coefficients. The author proves that the formal exponential series solutions (transseries) are Borel summable and functions obtained by resummation of the transseries are precisely the solutions of the differential equation that decay in a specified sector in the complex plane. Further the dependence is shown of the correspondence between the solutions of the differential equation and transseries as the ray in the complex plane changes (local Stokes phenomenon). Simple analytic identities lead to resurgence relations and to an averaging formula having the property of preserving exponential growth at infinity.
Reviewer: F.Rühs (Freiberg)

MSC:

34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
34M37 Resurgence phenomena (MSC2000)
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34C29 Averaging method for ordinary differential equations
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