Burgers' equation in the complex plane

TL;DR

This paper investigates the complex singularities of Burgers' equation solutions, analyzing their dynamics and relationship to real-line behavior using asymptotics, exact solutions, and numerical methods, aiming to develop broadly applicable analytical techniques.

Contribution

It introduces a combined analytical and numerical approach to study complex singularities in Burgers' equation, providing insights into their motion and influence on real-line solutions.

Findings

Singularities originate at t=0 and align near anti-Stokes lines.

The closest singularity's movement correlates with solution steepness.

Methodology can be adapted to other nonlinear PDEs.

Abstract

Burgers' equation is a well-studied model in applied mathematics with connections to the Navier-Stokes equations in one spatial direction and traffic flow, for example. 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. For an initial condition with a simple pole in each of the upper- and lower-half planes, we apply formal asymptotics in the small- and large-time limits in order to characterise the initial and later motion of the singularities. 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. For intermediate times, we use the exact solution, apply method of steepest descents, and implement the AAA approximation to track the complex singularities. Connections are made between the motion of the closest singularity to the real axis and the steepness of the solution on the real line. 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.