Forcing assumptions ensuring holomorphy and convergence of the δ-series for Burgers’ equation

Identify sufficient conditions on the external forcing term f in the Cole–Hopf-transformed linear homotopy Burgers’ equation that guarantee the solution depends holomorphically on the deformation parameter at δ = 0, and determine additional restrictions on f under which the Taylor series in δ converges up to δ = 1.

Background

The authors rigorously establish analyticity in δ and infinite radius of convergence for the unforced case using the Cole–Hopf transform. However, when an external forcing f is present, the analytical dependence on δ and the convergence radius are not established.

Clarifying the minimal assumptions on f needed to ensure holomorphy at δ = 0 and convergence at δ = 1 would extend the series approach to forced Burgers’ dynamics and strengthen theoretical guarantees for the method.

References

Rigorous analysis in the case of f \neq 0 appears far more delicate. For instance, it is not even immediately clear what assumptions must be made on f in order to ensure the solution of Eq.~eq:BurgerLinearHomotopy_ColeHopf is holomorphic at \delta = 0, let alone what additional restrictions must be placed on f for the radius of convergence of the Taylor expansion of \delta \mapsto u(t,x;\delta) about \delta = 0 to include \delta = 1.

— Robust series linearization of nonlinear advection-diffusion equations  (2512.12019 - Kieffer et al., 12 Dec 2025) in Section 2.1 (The linear deformation, series expansion, and proof of analyticity)