Burgers’ equation in the complex plane

Burgers' equation is a well-studied model in applied mathematics with connections to the Navier-Stokes equations one spatial direction and traffic flow, for example. Following on from previous work, we analyse solutions complex plane, concentrating dynamics of singularities their relationship solution real line. For an initial condition simple pole each upper- lower-half planes, apply formal asymptotics small- large-time limits order characterise later motion singularities. The small-time limit highlights how infinitely many are born at $t=0$ they orientate themselves lie increasingly close anti-Stokes lines far-field inner problem. This problem also reveals whether or not closest singularity axis moves toward away. intermediate times, use exact solution, method steepest descents, implement AAA approximation track Connections made between steepness While has deliberately mix techniques our analysis attempt develop methodology that can be other nonlinear partial differential do not.