Hamiltonian mechanics of "magnetic'' solitons in two-component Bose-Einstein condensates

We consider motion of a "magnetic'' soliton in two-component condensates along a non-uniform and time-dependent backgrounds in framework of the Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related with 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 the Newton-like equation. The theory is illustrated by several examples of concrete soliton's dynamics.