Nonlinear self-modulation: An exactly solvable model

The cubic Schr\"odinger equation (CSE) (${\mathrm{iu}}_{t}$+${u}_{\mathrm{xx}}$\ifmmode\pm\else\textpm\fi{}2\ensuremath{\Vert}u${\ensuremath{\Vert}}^{2}$u=0) is a generic model equation used in the study of modulational problems in one spatial dimension. The CSE is exactly solvable using inverse-scattering techniques. Periodic solutions of the focusing CSE (``+'' sign in the above equation) are also well known to be subject to modulational instabilities. This unique mixture of solvability and instability allows the development of a complete and explicit analytical theory for the long-time behavior of the instabilities. Among the results to be discussed are (i) a method for calculating the growth rates of instabilities around (spatially nonuniform) initial states, (ii) a discussion of recurrence phenomena for systems with finite spatial period, and (iii) a method for calculating the recurrence time.