Time-Delayed Instabilities in Complex Burgers Equations

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu [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 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.