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arxiv: 1907.04130 · v1 · pith:M3VPVJSWnew · submitted 2019-07-09 · 🧮 math.CV · math.AP

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

Alberto Lastra , St\'ephane Malek This is my paper

Pith reviewed 2026-05-24 23:55 UTC · model grok-4.3

classification 🧮 math.CV math.AP
keywords q-difference-differential equationssingular perturbationasymptotic expansionsLambert W functioncomplex time variablesanalytic solutionsouter and inner solutions
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The pith

Outer and inner analytic solutions are constructed for singularly perturbed q-difference-differential equations in two complex time variables, with asymptotic expansions of different natures involving the -1-branch of the Lambert W function

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs analytic solutions and their formal asymptotic expansions for a family of singularly perturbed q-difference-differential equations in the complex domain. These come as outer and inner solutions whose expansions in the perturbation parameter differ in essential ways. The work is presented as a q-analog of prior results on singularly perturbed partial differential equations with two complex times. The -1-branch of the Lambert W function enters crucially in the description of the inner solutions.

Core claim

For the given family of singularly perturbed q-difference-differential equations, outer analytic solutions admit asymptotic expansions in powers of the perturbation parameter while inner analytic solutions exhibit expansions of a different nature that crucially involve the -1-branch of the Lambert W function, yielding a q-analog to the boundary-layer expansions obtained for the corresponding partial differential equations.

What carries the argument

The construction of outer and inner analytic solutions whose asymptotic expansions are analyzed via the -1-branch of the Lambert W function to capture the irregular singular perturbation in the q-difference setting.

If this is right

  • Outer solutions provide regular asymptotic expansions in sectors away from boundary layers.
  • Inner solutions capture the singular behavior through the Lambert W asymptotics.
  • The q-analog preserves the distinction between outer and inner expansions seen in the PDE case.
  • The solutions supply approximations valid in the complex domain for small values of the perturbation parameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The outer-inner split may extend directly to other families of q-difference equations with irregular singularities.
  • The appearance of the Lambert W function points toward possible links with exponential asymptotics in q-calculus.
  • The method could be tested on concrete coefficient choices to produce explicit numerical checks of the expansions.

Load-bearing premise

The specific form of the family of q-difference-differential equations admits the construction of outer and inner analytic solutions with the stated asymptotic properties.

What would settle it

An explicit member of the equation family for which either no outer or inner analytic solution exists or the inner solution's asymptotic expansion does not involve the -1-branch of the Lambert W function.

Figures

Figures reproduced from arXiv: 1907.04130 by Alberto Lastra, St\'ephane Malek.

Figure 1
Figure 1. Figure 1: Initial (left) and deformed path (right) for the difference of two consecutive solutions [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs analytic solutions together with their formal asymptotic expansions for a stated family of singularly perturbed q-difference-differential equations in two complex time variables. These are presented as a q-analogue of earlier work on singularly perturbed PDEs; outer and inner analytic solutions are built whose expansions in the perturbation parameter are of essentially different character, with the -1 branch of the Lambert W function appearing in the inner expansion via a matching procedure in the complex domain.

Significance. If the constructions hold under the stated hypotheses on q, the coefficients, and the sectorial domains, the work supplies an explicit example of how Lambert-W asymptotics arise in the matching of outer and inner solutions for multi-variable q-difference equations of irregular type. This supplies a concrete template that may be tested on other families of q-difference-differential equations and contributes a q-analogue perspective to the existing literature on boundary-layer expansions in several complex variables.

minor comments (3)
  1. [§1] §1 (Introduction): the precise form of the family of equations (including the precise dependence on the two time variables and the coefficients multiplying the q-shifts and derivatives) should be displayed explicitly before the statement of the main existence theorem, so that the reader can verify at once which terms produce the irregular singularity.
  2. The sectorial domains and the admissible ranges for the q-parameter are invoked repeatedly; a single consolidated statement of all standing assumptions (with explicit constants) would improve readability and make the error estimates easier to track.
  3. The choice of the -1 branch of Lambert W is asserted to be forced by the matching; a short paragraph showing why the principal branch is incompatible with the inner equation would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript, including the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs outer and inner analytic solutions together with their formal asymptotic expansions directly from an explicitly stated family of singularly perturbed q-difference-differential equations in two complex time variables, under explicit assumptions on the q-parameter, coefficients, and sectorial domains. The -1 branch of the Lambert W function appears through the matching procedure performed on the inner solution; this step follows from the equation itself rather than from any fitted input or prior result. The self-citation to the authors' submitted 2019 work on the non-q case is used only for motivation and context, not as a load-bearing premise that defines or forces the q-analog constructions. No self-definitional reductions, renamed empirical patterns, or smuggled ansatzes occur.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the existence of a suitable family of q-difference-differential equations that permit outer/inner analytic solutions with the claimed expansions. No free parameters are introduced in the abstract; the equation family itself is the modeling choice. No new entities are postulated beyond the standard Lambert W function.

axioms (1)
  • domain assumption The family of singularly perturbed q-difference-differential equations admits analytic outer and inner solutions whose asymptotic expansions differ in nature with respect to the perturbation parameter.
    This is the setting stated in the abstract as the object of the constructions.

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