Hamiltonian theory of motion of dark solitons in the theory of nonlinear Schrödinger equation

We develop a method for deriving Hamilton's equations describing the dynamics of solitons when they move along an inhomogeneous and time-varying large-scale background for nonlinear wave equations that are completely integrable in the Ablowitz–Kaup–Newell–Segur (AKNS) scheme. The method is based on the development of old Stokes' ideas that allow analytically continuing the relations for linear waves into the soliton region, and is practically implemented in the example of the defocusing nonlinear Schrödinger equation. A condition is formulated under which the external potential is only to be taken into account when describing the evolution of the background, and that this case, the Newton equation is obtained for the soliton dynamics in an external potential.