NONLINEAR-WAVE PROPAGATION IN A DISORDERED MEDIUM

In this paper we consider the problem of solitary wave propagation in a weakly disordered potential. Through a series of careful numerical e xperiments we have observed behavior which is in agreement with the th eoretical predictions of Kivshar et al., Bronski, and Garnier. In part icular we observe numerically the existence of two regimes of propagat ion. In the first regime the mass of the solitary wave decays exponent ially, while the velocity of the solitary wave approaches a constant. This exponential decay is what one would expect from known results in the theory of localization for the linear Schrodinger equation. In the second regime, where nonlinear effects dominate, we observe the anoma lous behavior which was originally predicted by Kivshar et al. In this regime the mass of the solitary wave approaches a constant, while the velocity of the solitary wave displays an anomalously slow decay. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena of total reflection and trapping.