  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.
