Self-trapping and breathers in disordered nonlinear oscillatory lattices

We present a comparative study of two classes of solutions to Fröhlich–Spencer–Wayne chain model with random spatial inhomogeneity (disorder): self-trapped wave packets on one hand, and discrete breathers (localized in space, time-periodic solutions) on the other. Wave packets are obtained by numerical integration of dynamical equations with single-site initial conditions. When given sufficient energy, the packet remains localized in space throughout the observation time. Breather solutions are constructed by continuation of a periodic orbit with coupling parameter increased from zero in successive small steps. Found solutions are examined for linear stability. We demonstrate that the great majority of disorder realizations exhibit linearly stable breathers on an interval of coupling parameter values from zero up to a finite realization-dependent threshold. The disappearance of a discrete breather is associated with the bifurcation in which a complex-conjugate pair of Floquet multipliers becomes equal to +1. When a discrete breather exists, self-trapping of wave packets depends upon the proximity of the corresponding trajectory in the phase space to the breather orbit. These observations allow us to associate the well-known selftrapping effect with the existence of stable breather orbits and to explain this effect by the influence of breather orbits upon the phase space structure in their neighbourhood. The presented results are of interest for developing the theoretical description of physical systems characterized by the simultaneous presence of nonlinearity, spatial discreteness and disorder (Bose-Einstein condensates, lattices of coupled optical waveguides, microand nanomechanical systems etc.).