Locating complex singularities of Burgers' equation using exponential asymptotics and transseries

TL;DR

This paper develops exponential asymptotics and transseries techniques to accurately locate complex singularities of Burgers' equation, revealing how subdominant exponentials influence pole and zero formation in the complex plane.

Contribution

It introduces a systematic transseries approach to analyze small-time behavior and locate singularities in Burgers' equation, extending asymptotic methods to nonlinear differential equations.

Findings

Identifies the emergence of poles and zeros from initial singularities.

Shows subdominant exponentials become significant along anti-Stokes curves.

Provides a transseries framework for asymptotic pole and zero approximation.

Abstract

Burgers' equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers' equation shows an infinite stream of simple poles born at t = 0^+, emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t > 0. We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.