Parametric asymptotic expansions and confluence for Banach valued solutions to some singularly perturbed non-linear $q$-difference-differential Cauchy problem
Abstract:
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.
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