ON THE STABILITY OF TIME-HARMONIC LOCALIZED STATES IN A DISORDERED NONLINEAR MEDIUM

We study the problem of localization in a disordered one-dimensional n onlinear medium modeled by the nonlinear Schrodinger equation. Devilla rd and Souillard have shown that almost every time-harmonic solution o f this random PDE exhibits localization. We consider the temporal stab ility of such time-harmonic solutions and derive bounds on the locatio n of any unstable eigenvalues. By direct numerical determination of th e eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex pla ne. The distributions are distinctly different for focusing and defocu sing nonlinearities. We argue further that these instabilities are con nected with resonances in a Schrodinger problem, and interpret the ear lier numerical simulations of Caputo, Newell, and Shelley, and of Shel ley in terms of these instabilities. Finally, in the defocusing case w e are able to construct a family of asymptotic solutions which include s the stable limiting time-harmonic state observed in the simulations of Shelley.