Time-delayed instabilities in complex Burgers equations

TL;DR

This paper investigates time-delayed instabilities in complex Burgers equations, showing that numerical diffusion causes a delay in exponential growth, with solutions initially growing linearly before becoming unstable.

Contribution

It demonstrates that numerical diffusion explains the observed delay in instability growth in complex Burgers equations with small viscosity and complex forcing.

Findings

Solutions grow linearly before exponential growth

Numerical diffusion causes the time delay in instability

Oscillating data lead to delayed exponential growth

Abstract

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu [{\it Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems}, Amer.~J.~Math.~2010] proved that only analytical data generate local $C^2$ solutions. The corresponding instabilities are however not observed numerically; rather, numerical simulations show an exponential growth only after a delay in time. We argue that numerical diffusion is responsible for this time delay, as we prove that for Burgers equations in the torus with small viscosity and a complex forcing, oscillating data generate solutions which grow linearly in time before growing exponentially. Numerical simulations illustrate the results.