Abstract
The cubic Schrödinger equation (CSE) (+±2‖uu=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.
- Received 10 August 1987
DOI:https://doi.org/10.1103/PhysRevA.37.815
©1988 American Physical Society