13-52 S. Malek
On singularly perturbed small step size difference-differential nonlinear PDEs (593K, pdf) Jun 6, 13
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Abstract. We study a family of singularly perturbed small step size difference-differential nonlinear equations in the complex domain. We construct formal solutions to these equations with respect to the perturbation parameter which are asymptotic expansions with 1-Gevrey order of actual holomorphic solutions on some sectors near the origin in the complex domain. However, these formal solutions can be written as sums of formal series with a corresponding decomposition for the actual solutions which may possess a different Gevrey order called 1^{+}-Gevrey in the literature. This phenomenon of two levels asymptotics has been already observed in the framework of difference equations by B. Braaksma, B. Faber and G. Immink. The proof rests on a new version of the so-called Ramis-Sibuya theorem which involves both 1-Gevrey and 1^{+}-Gevrey orders. Namely, using classical and truncated Borel-Laplace transforms (introduced by G. Immink), we construct a set of neighboring sectorial holomorphic solutions and functions whose difference have exponentially and super-exponentially small bounds in the perturbation parameter.

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