Abstract

Following on from previous work, we analyse solutions to Burgers’ equation in the complex plane, concentrating on the dynamics of the complex singularities and their relationship to the solution on the real line. We use a variety of tools, including matched asymptotic expansions, exponential asymptotics, exact solutions, steepest descents and rational approximations. The small-time limit highlights how infinitely many singularities are born at $t = 0$ and how they orientate themselves to lie increasingly close to anti-Stokes lines in the far-field of the inner problem. This inner problem also reveals whether or not the closest singularity to the real axis moves toward the axis or away. Further analysis characterises the motion of the singularities for intermediate and late times. While Burgers’ equation has an exact solution, we deliberately apply a mix of techniques in our analysis in an attempt to develop methodology that can be applied to other nonlinear partial differential equations that do not.