A preliminary numerical study of solitary wave propagation in a disordered medium

In this paper we present a preliminary numerical study of the propagation of a soliton in a weakly disordered potential consisting of a number of randomly placed spatially localized scatterers. By conducting the experiment in the locally stationary frame of the soliton we are able to follow the soliton through a large number of scattering events, thus obtaining very good information on the long-distance propagation properties of a soliton in such a medium. The observed behavior is in agreement with the theoretical predictions of Kivshar et al. [Phys. Rev. Lett. 61 (15) (1990) 1693] and Bronski [J. Nonlinear Sci. (1996)]. For solitons whose incident mass is relatively small compared with their velocity (the nearly linear regime) we observe numerically an exponential decay in the mass of the soliton, while the mass of the soliton approaches a constant after a large number of scattering events. When the ratio of the incident soliton mass to the incident soliton velocity is sufficiently large, however, a very different behavior is observed. The mass of the solitary wave is observed to asymptotically approach a constant, while the velocity decays slowly to zero, as predicted by the theory. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena of total reflection and trapping. Copyright (C) 1998 Elsevier Science B.V.