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 C. 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 [6]. 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 in [18]), we construct a set of neighboring sectorial holomorphic solutions and functions whose difference have exponentially and super-exponentially small bounds in the perturbation parameter.

Some notes on the parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

“…This work is a continuation of a study harvested in the paper [17] dealing with small step size difference-differential Cauchy problems of the form (4) ∂ s ∂ S z X i (s, z, ) = Q(s, z, , {T k, } k∈J , ∂ s , ∂ z )X i (s, z, ) + P (z, , X i (s, z, )) for given initial Cauchy conditions (∂ j z X i )(s, 0, ) = x j,i (s, ), for 0 ≤ i ≤ ν − 1, 0 ≤ j ≤ S − 1, where ν, S ≥ 2 are integers, Q is some differential operator which is polynomial in time s, holomorphic near the origin in z, , that includes shift operators acting on time, T k, X i (s, z, ) = X i (s + k , z, ) for k ∈ J that represents a finite subset of N and P is some polynomial. Indeed, by performing the change of variable t = 1/s, the equation (1) maps into a singularly perturbed linear PDE combined with small shifts T k, , k ∈ I.…”

“…Proof Since the notations used here are rather different from the ones within the result enounced in [17] and in order to explain the part of the proposition concerning 1 and 1 + summability which is not mentioned in our previous work [17], we have decided to present a sketch of proof of the statement.…”

“…We consider the set of sectors E = {E k HJn } k∈ −n,n ∪ {E S dp } 0≤p≤ι−1 constructed in Section 3.3 that fufills the constraints 3),4) and 5). The set E forms a so-called good covering in C * as given in Definition 3 of [17].…”

“…We rephrase the version of the Ramis-Sibuya which entails both 1−Gevrey and 1 + −Gevrey asymptotics displayed in [17] for the specific covering E with additional informations concerning 1 and 1 + summability.…”

On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms

“…where α, k > 0 are real numbers, Q(X), R(X), Q 1 (X), Q 2 (X) stand for polynomials with complex coefficients and P (t, , {U κ } κ∈I , V 1 , V 2 ) represents a polynomial in t, V 1 , V 2 , linear in U κ , with holomorphic coefficients w.r.t near the origin in C, where the symbol m κ,t, denotes a Moebius operator acting on the time variable through m κ,t, u(t, z, ) = u( t 1 + κ t , z, ) arXiv:1807.07453v1 [math.CV] 19 Jul 2018…”