Perturbation theory for the nonlinear Schrödinger equation in the solitonless sector

The nonlinear Schrödinger equation with a perturbation of polynomial type is considered. A perturbation theory in the solitonless situation is developed on the basis of the perturbed equations of motion for canonical variables constructed from scattering data. It is shown that closed equations of the perturbation theory containing only canonical variables can be obtained in the case of a “quasi-asymptotic” initial condition. The cases in which these equations can be solved iteratively are established. Also considered is the ease of an initial condition with spectrum cut off in the “infrared” region. In this case, averaging over the rapid unperturbed motions makes it possible to reduce the equations of the perturbation theory to a closed form as well. A solution to these equations is obtained in implicit form.