# On a $q-$analog of a singularly perturbed problem of irregular type with two complex time variables

## TL;DR

This paper develops analytic solutions and formal asymptotic expansions for a family of $q$-difference-differential equations with two complex time variables, extending previous work on singularly perturbed PDEs and highlighting the role of Lambert W function branches.

## Contribution

It introduces a $q$-analog of singularly perturbed PDEs with two complex time variables, constructing solutions with different asymptotic behaviors and emphasizing the significance of Lambert W function branches.

## Key findings

- Construction of outer and inner analytic solutions
- Solutions exhibit different asymptotic expansions
- Lambert W function's $-1$-branch is crucial

## Abstract

Analytic solutions and their formal asymptotic expansions for a family of the singularly perturbed $q-$difference-differential equations in the complex domain are constructed. They stand for a $q-$analog of the singularly perturbed partial differential equations considered in our recent work [A. Lastra, S. Malek, Boundary layer expansions for initial value problems with two complex time variables, submitted 2019]. In the present work, we construct outer and inner analytic solutions of the main equation, each of them showing asymptotic expansions of essentially different nature with respect to the perturbation parameter. The appearance of the $-1$-branch of Lambert $W$ function will be crucial in this respect.