On singularly perturbed small step size difference-differential nonlinear PDEs

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 in near the origin in . 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 (see [4]). 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 Immink [14]), we construct a set of neighbouring sectorial holomorphic solutions and functions whose difference have exponentially and super-exponentially small bounds in the perturbation parameter.

Keywords::

• asymptotic expansion
• Borel–Laplace transform
• difference equation
• integro-differential equation
• nonlinear partial differential equation
• singular perturbation

Keywords::

• 35C10
• 35C20

Acknowledgements

The author is partially supported by the french ANR-10-JCJC 0105 project and the PHC Polonium 2013 project No. 28217SG.