**Abstract:** |
Modulational instability (MI), namely the instability of a constant background to long-wavelength perturbations, is a ubiquitous nonlinear phenomenon discovered in the mid 1960's. Until recently, however, a characterization of the nonlinear stage of MI, namely, the behavior of solutions once the perturbations have become comparable with the background, was missing. This talk will present recent work on this subject. I will first show how MI is manifested in the inverse scattering transform for the focusing nonlinear Schrodinger (NLS) equation. Then I will characterize the nonlinear stage of MI by computing the long-time asymptotics of the NLS equation for localized perturbations of a constant background. For long times, the xt-plane divides into three regions: a left far field and a right far field, in which the solution is approximately constant, and a central region in which the solution is described by a slowly modulated traveling wave. Finally, I will show that this kind of behavior is not limited to the NLS equation, but it is shared among many different models (including PDEs, nonlocal systems and differential-difference equations). |