PERTURBATION EXPANSION FOR 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.