Modulational instability in equations of KdV type

It is a matter of experience that nonlinear waves in a dispersive medium, propagating primarily in one direction, may appear periodic in small space and time scales, but their characteristics—the amplitude, the phase, the wave number, etc.—slowly vary in large space and time scales. In the 1960s, Whitham developed an asymptotic (WKB) method to study the effects of small “modulations” on nonlinear dispersive waves. Since then, there has been a great deal of work aiming at rigorously justifying the predictions from Whitham’s formal theory. We discuss some recent advances in the mathematical understanding of the dynamics, in particular, the instability, of slowly modulated waves for equations of KdV type.