Soliton chaos in the nonlinear Schrödinger equation with spatially periodic perturbations
Abstract
We perturb the one-dimensional nonlinear Schr\"odinger equation with a time-independent spatially periodic potential with a period large compared to the spatial width of solitons present in the system. A collective-coordinate approximation maps the system to a nonintegrable many-particle dynamics with an effective Hamiltonian, which we derive under the assumption that we can neglect three-soliton collisions as well as two-soliton collisions with vanishing relative velocity. We give an estimate for the power radiated by a single soliton in the presence of the perturbation and show that radiative effects can be neglected for perturbations with sufficiently small amplitude and large spatial period. We show that the nonintegrability of this perturbed nonlinear Schr\"odinger equation manifests itself already in the two-soliton sector. We use the effective many-particle Hamiltonian to investigate soliton depinning. The effective Hamiltonian results are compared with numerical simulations of the full perturbed equation.