Hamiltonian mechanics of “magnetic” solitons in two-component Bose–Einstein condensates

We consider the motion of a “magnetic” soliton in two-component condensates along a nonuniform and time-dependent background in the framework of Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related to its inverse half-width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to a Newton-like equation. The theory is illustrated by several examples of concrete soliton's dynamics.