Complex singularities for Burgers' equation with piecewise-continuous initial conditions
Abstract
There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at t=0+ and propagate for t>0, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at t=0+, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-W function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and N-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.
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BibTeX
@misc{gentner2026complex,
title = {{Complex singularities for Burgers' equation with piecewise-continuous initial conditions}},
author = {Jacob C. Gentner and Michael C. Dallaston and Scott W. McCue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04691},
eprint = {2609.04691},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04691},
}
RIS
TY - UNPB
TI - Complex singularities for Burgers' equation with piecewise-continuous initial conditions
AU - Jacob C. Gentner
AU - Michael C. Dallaston
AU - Scott W. McCue
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04691
UR - https://arxiv.org/abs/2609.04691
AB - There is a body of research devoted to understanding how complex singularities of solutions of nonlinear partial differential equations (pdes) spontaneously emerge at t=0^+ and propagate for t>0, and how their behaviour affects the solution on the real axis. Despite the importance of the small-time limit in these studies, there is still a lack of understanding of how complex singularities are born at t=0^+, including for initial conditions that are not analytic functions of the spatial variable. In this paper, we use Burgers' equation as a prototype nonlinear pde and study the complex-plane singularities for initial conditions that are piecewise smooth. Using matched asymptotic expansions, we show how infinitely many singularities emerge from points of discontinuity in a pattern that can be described using branches of the Lambert-W function. For various initial conditions, we observe how these singularities rearrange themselves to align with the appropriate exactly-described long-time behaviour, including sigmoid-shaped travelling waves, constant-area (triangular wave) similarity solutions and N-wave solutions. In terms of Burgers' equation, our small-time asymptotic analysis of the singularity propagation for piecewise-continuous initial conditions illustrates the types of generic behaviours that arise for inner regions when diffusion dominates advection. More generally, this work is a step towards understanding complex-plane behaviour of solutions of nonlinear partial differential equations with non-analytic initial conditions.
ER -