We study the modulational stability of the nonlinear Schr\"odinger equation (NLS) using a time-dependent variational approach. Within this framework, we derive ordinary differential equations (ODEs) for the time evolution of the amplitude and phase of modulational perturbations. Analyzing the ensuing ODEs, we re-derive the classical modulational instability criterion. The case (relevant to applications in optics and Bose-Einstein condensation) where the coefficients of the equation are time-dependent, is also examined.
@article{arxiv.cond-mat/0404601,
title = {Variational Approach to the Modulational Instability},
author = {Z. Rapti and P. G. Kevrekidis and A. Smerzi and A. R. Bishop},
journal= {arXiv preprint arXiv:cond-mat/0404601},
year = {2009}
}