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On optimal truncation of divergent series solutions of nonlinear differential systems. (English) Zbl 0945.34071

The authors study the optimal approximation problem for asymptotic solutions to the system of nonlinear differential equations \[ y'= f(x, y), \qquad y\in \mathbb C^n, \] near the irregular singularity \(x=\infty\) of rank \(1.\) When the eigenvalues of \((\partial f_i/\partial y_j(\infty,0))\) are different from zero and nonresonant, this system admits a family of formal solutions of the form \[ \widetilde{y}=\widetilde{y}_0+\sum_{k \geq 0; |k|>0}C^{k_1}_1\cdots C^{k_n}_n e^{-( k \cdot \lambda)x}x^{k\cdot m} \widetilde{y}_{k} \] with \[ \widetilde{y}_{k}=x^{- k\cdot ( \beta + m)}\sum_{l=0}^{\infty} a _{k;l}x^{-l}. \] It is shown that, under some suppositions, in every direction, the difference between a true solution \( y\) and the optimal truncation of \(\widetilde{y}_0\) is of the order of magnitude of the least term if and only if \( y\) is the balanced Borel sum of the formal solution with a certain condition on \(C_j.\) Then a similar estimate is given concerning the optimal truncation of the higher series \(\widetilde{y}_{k}\) as well. Furthermore, the Berry-smoothing near a Stokes line is discussed, and the transition is represented by an error function.

MSC:

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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