A Perturbation Expansion for the Zakharov–Shabat Inverse Scattering Transform

For any nonlinear evolution equation which is reasonably close to a nonlinear evolution equation that can be exactly solved by the Zakharov–Shabat inverse scattering transform, the total time evolution of the scattering data can be given and also be expanded in a perturbation expansion. Thus, at least for moderate times, one can determine the effects of small dissipations, relaxations, etc., on the evolution of these exactly solvable nonlinear evolution equations, and in particular, the effects on the soliton states. A simple example is given to illustrate the method, where a soliton slowly decays and slowly radiates its energy into the continuum.