Dynamics of localized structures

We describe qualitative behaviour of solutions of the Gross–Pitaevskii equation in 2D in terms of motion of vortices and radiation. To this end we introduce the notion of the intervortex energy. We develop a rather general adiabatic theory of motion of well separated vortices and present the method of effective action which gives a fairly straightforward justification of this theory. Finally we mention briefly two special situations where we are able to obtain a rather detailed picture of the vortex dynamics. Our approach is rather general and is applicable to a wide class of evolution nonlinear equation which exhibit localized, stable static solutions. It yields description of general time-dependent solutions in terms of dynamics of those static solutions “glued” together.