S. Malek
On singularly perturbed small step size difference-differential nonlinear PDEs
(581K, pdf)

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 Gevrey 
order 1 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^{+} 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 version of the so-called Ramis-Sibuya theorem which involves both Gevrey 1 and 1^{+} 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 supra-exponentially small bounds 
in the perturbation parameter.