Length-scale competition for the one-dimensional nonlinear Schrödinger equation with spatially periodic potentials

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Abstract
The effect of spatially periodic on-site potentials on solitons of the (1+1)-dimensional nonlinear Schrödinger equation depends not only on the amplitude of the perturbation but also on two length ratios—between the wavelength of the potential and either the spatial width of the soliton or its phase-modulation wavelength. For solitons which are slowly moving or at rest, only the first ratio is important: Solitons which are narrow compared to the wavelength of the potential move like particles in an effective potential; wide solitons move like ‘‘renormalized’’ particles; and for competing length scales, solitons are easily destabilized by the perturbation and break up into smaller localized excitations and radiation. For initially rapidly moving solitions in short-wavelength potentials, a different resonance effect is described which can inhibit soliton propagation. Away from this resonance the solitons can move in a ‘‘dressed’’ form radiating very slowly. Furthermore, it is shown numerically that two such ‘‘dressed’’ solitons reemerge after collision, essentially unchanged. We also comment on the structural stability of completely integrable dynamics under spatially periodic perturbations. Finally, we remark on the relevance of our results for the dynamical behavior of large polarons and bipolarons in disordered ionic lattices.