Hamilton theory of NLS equation soliton motion

We suggest the method of derivation of Hamilton equations which describe the motion of solitons along non-uniform and time dependent large-scale background in case of wave dynamics described by the completely integrable equations in the Ablowitz-Kaup-Newell-Segur scheme. The method is based on development of old Stokes' argumentation which allows one to continue analytically some relationships derived for linear waves to the soliton region. It is presented here for a particular case of the defocusing nonlinear Schr\"{o}dinger equation. We formulate the condition when the external potential should only be taken into account for the background evolution, and in this case we obtain the Newton equation for the soliton dynamics.