Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed non-linear $q$-difference-differential Cauchy problem
S. Malek Université de Lille, Laboratoire Paul Painlevé, France
Аннотация:
We investigate a singularly perturbed
$q$-difference differential Cauchy problem with polynomial coefficients in complex time
$t$ and space
$z$ and with quadratic non-linearity. We construct local holomorphic solutions on sectors in the complex plane with respect to the perturbation parameter
$\varepsilon$ with values in some Banach space of formal power series in
$z$ with analytic coefficients on shrinking domains in
$t$. Two aspects of these solutions are addressed. One feature concerns asymptotic expansions in
$\varepsilon$ for which a Gevrey type structure is unveiled. The other fact deals with confluence properties as
$q>1$ tends to
$1$. In particular,
the built up Banach valued solutions are shown to merge in norm to a fully bounded holomorphic map in all the variables
$t$,
$z$ and
$\varepsilon$ that solves a non-linear partial differential Cauchy problem.
Ключевые слова:
asymptotic expansion, confluence, formal power series, partial differential equation,
$q$-difference equation.
MSC: 35R10,
35C10,
35C15,
35C20 Поступило в редакцию: 18.06.2024
Исправленный вариант: 14.03.2025
Язык публикации: английский
DOI:
10.4213/im9650
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