Direct perturbation theory for the dark soliton solution to the nonlinear Schrodinger equation with normal dispersion

After finding the basic solutions of the linearized nonlinear Schrodinger equation by the method of separation of variables, the perturbation theory for the dark soliton solution is constructed by linear Green's function theory. In application to the self-induced Raman scattering, the adiabatic corrections to the soliton's parameters are obtained and the remaining correction term is given as a pure integral with respect to the continuous spectral parameter.