Stéphane Malek

On -Gevrey Asymptotics for Singularly Perturbed -Difference-Differential Problems with an Irregular Singularity

We study a -analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open -spirals to the origin. By means of a -Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the -Gevrey asymptotic expansion (of certain type) of the actual solutions.

On singularly perturbed small step size difference-differential nonlinear PDEs

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.

On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

We study a family of singularly perturbed -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

On the Reducibility of the Schlesinger Equations

In this paper we give a constructive way to obtain a reducible isomonodromic Schlesinger deformation from a given isomonodromic Schlesinger deformation.

On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schafke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

On multiscale Gevrey and Gevrey asymptotics for some linear difference differential initial value Cauchy problems

Abstract We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution. The proof leans on a two level novel version of Ramis–Sibuya theorem under Gevrey and q-Gevrey orders.

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

This paper is a slightly modified, abridged version of the work (Lastra and Malek, Adv Differ Equ 21:767–800, 2016). It deals with some questions made to the authors during the conference Analytic, Algebraic and Geometric Aspects of Differential Equations, held in Bedlewo (Poland) during the second week of September, 2015.

On Singular Solutions to PDEs with Turning Point Involving a Quadratic Nonlinearity

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in . We construct a family of sectorial meromorphic solutions obtained as a small perturbation in of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

On Parametric Gevrey Asymptotics for Some Cauchy Problems in Quasiperiodic Function Spaces

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with 2π-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

On Complex Singularity Analysis for Some Linear Partial Differential Equations in

We investigate the existence of local holomorphic solutions