Time-delayed instabilities in complex Burgers equations

For Burgers equations with real data and complex forcing terms, Lerner, Morimoto and Xu proved that only analytical data generate local C^2 solutions. These 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 viscous complex Burgers equations with small viscosity O(ε), initial data with large frequencies O(1/ε) generate solutions that are bounded in time O(1), before exhibiting an exponential growth in time. This phenomenon is not specific to Burgers: considering more generally first-order operators that experience a transition from hyperbolicity to non-hyperbolicity, for which the results of Lerner, Nguyen and Texier give strong, instantaneous instabilities, we show that the introduction of a small O(ε) viscous term can imply uniform bounds in time O( ε^1/2)